Econometrica, Vol. 68, No. 3 ŽMay, 2000., 525᎐574
OPTIMAL NONPARAMETRIC ESTIMATION OF
FIRST-PRICE AUCTIONS
BY EMMANUEL GUERRE, ISABELLE PERRIGNE,
AND
QUANG VUONG1
This paper proposes a general approach and a computationally convenient estimation
procedure for the structural analysis of auction data. Considering first-price sealed-bid
auction models within the independent private value paradigm, we show that the underlying distribution of bidders’ private values is identified from observed bids and the number
of actual bidders without any parametric assumptions. Using the theory of minimax, we
establish the best rate of uniform convergence at which the latent density of private values
can be estimated nonparametrically from available data. We then propose a two-step
kernel-based estimator that converges at the optimal rate.
KEYWORDS: First-price auctions, independent private value, nonparametric identification, two-stage nonparametric estimation, kernel estimation, minimax theory.
1.
INTRODUCTION
IN RECENT YEARS THE EMPIRICAL ANALYSIS of auction data has attracted a lot of
attention Žsee Porter Ž1995. and Laffont Ž1997. for recent surveys.. First,
auctions are frequently used to exchange commodities and to allocate public
projects through procurements. Thus many auction data are now available.
Second, the theory of auctions has considerably expanded since Vickrey’s Ž1961.
seminal paper due to the development of the Bayesian Nash equilibrium
concept by Harsanyi Ž1967.. Third, after many years of intense theoretical
developments, modern industrial organization is facing the challenge of its
empirical usefulness. As argued by Sutton Ž1993., auction models, which emphasize asymmetric information and strategic behavior, are the most favorable case
for facing this challenge.
Starting with Paarsch Ž1992., a few empirical papers have adopted a fully
structural econometric approach, which has exclusively relied upon a parametric
specification of the bidders’ private values distribution. As is well known, even
for first-price sealed-bid auctions with independent private values, the structural
analysis has suffered from the numerical complexity associated with the compu1
We are grateful to J. J. Laffont for introducing us to auction models. We thank D. Brown, W.
Hardle, T. Li, H. Paarsch, M. Simioni, J. Wu, three anonymous referees, the co-editor whose
comments greatly improved the paper, and H. Raynal for computational assistance. Various versions
were presented at the Xth Journees
Appliquee,
´ de Microeconomie
´
´ Sfax, June 1993, the Far Eastern
Meeting of the Econometric Society, Taipei, June 1993, the World Congress of the Econometric
Society, Tokyo, August 1995, the European Meeting of the Econometric Society, Toulouse, August
1997, the University of California ŽSanta Barbara, Riverside, Berkeley, Los Angeles., Stanford, USC,
Yale, Cornell, GREMAQ-INRA, Weierstrass Institute and Malinvaud econometric seminars. We
thank IDEI for its hospitality during the completion of this paper. Financial support from the
National Science Foundation under Grants SBR-9409569 and SBR-9631212 is gratefully acknowledged.
525
526
E. GUERRE, I. PERRIGNE, AND Q. VUONG
tation of the Bayesian Nash equilibrium strategy. As a result, only very simple
parametric specifications of the latent distribution of private values have been
entertained. More recently, Laffont and Vuong Ž1993. and Laffont, Ossard, and
Vuong Ž1995. have proposed some computationally convenient estimation methods based on simulations that allow for general parametric specifications.
The main purpose of this paper is to address three fundamental issues at the
heart of the structural analysis: Ži. Does a theoretical auction model place any
restrictions on observable data to be testable? Žii. Does a structural approach
require a priori parametric information about the structural elements to identify
the model under consideration? Žiii. Can an estimation procedure be proposed
that does not rely upon parametric assumptions and that is computationally
feasible? We answer these issues for the symmetric first-price auction model
within the independent private value ŽIPV. paradigm.
In particular, we propose an indirect two-step procedure for estimating the
distribution of bidders’ private values from observed bids, which requires neither
parametric assumptions nor the computations of the Bayesian Nash equilibrium
strategy. The crucial idea of our method is that each private value can be
expressed as a function of the corresponding bid, the distribution of observed
bids, and the corresponding density function. The first step then consists in
constructing a sample of pseudo private values based on kernel estimates of the
distribution and density functions of observed bids. In a second step, this pseudo
sample is used to estimate nonparametrically the density of bidders’ private
values. We establish the uniform consistency of our estimator and show its
optimality in the sense that it attains the best uniform convergence rate for
estimating the latent density of private values from observed bids.
Our results are important for several reasons. First, on economic grounds,
policy conclusions based on our fully nonparametric procedure are necessarily
robust to misspecifications of the underlying distribution. Moreover, because
economic theory does impose some restrictions on the distribution of observed
bids, one can test in principle the validity of the theoretical auction model
without making strong parametric assumptions on its structural elements. Second, from a statistical point of view our estimator applies nonparametric
techniques in each step of a structural estimation method.2 Our first main
statistical contribution is to derive the best rate of uniform convergence of
nonparametric estimates of the density of Žunobserved. private values from
observed bids. To this end, we apply the minimax theory as developed by
Ibragimov and Has’minskii Ž1981.. To our knowledge, such a theory has been
seldom applied in econometric work. We show that our two-step nonparametric
estimator attains this optimal rate using suitably chosen bandwidths. Third, on
computational grounds, our estimation procedure avoids the numerical diffi2
Though in recent years nonparametric techniques have been used in two-step estimation
procedures, typically the second step concentrates on the estimation of a finite dimensional
parameter. Only a few authors have used nonparametric techniques in both steps of an estimation
procedure. For an example in a regression context, see Ahn Ž1995. who does not characterize the
lower bound for his problem and whose two-step nonparametric estimator is suboptimal in the
minimax sense.
527
FIRST PRICE AUCTIONS
culties that have plagued the structural analysis of auction data. Indeed, because
it is indirect and fully parametric, our procedure requires neither the numerical
determination of the Bayesian Nash equilibrium bid function nor iterative
optimization algorithms. Such computational advantages are especially crucial
when the Bayesian Nash equilibrium strategy is the solution of a differential
equation that cannot be solved explicitly. As a result, the structural analysis of
many auction models that are known to be untractable become readily accessible through our indirect method.
The paper is organized as follows. In Section 2, we present the first-price
sealed-bid auction model with independent private values. To present the basic
ideas, we consider first a nonbinding reservation price, and illustrate our
procedure with some Monte Carlo experiments. In Section 3, we establish its
asymptotic properties allowing for a varying number of bidders and heterogeneity across auctions. Specifically, we derive the optimal uniform convergence rate
for estimating the density of private values from observed bids. We prove the
uniform convergence of the pseudo sample as well as the optimality of our
two-step nonparametric estimator. In Section 4, we consider the case where the
reservation price is binding. In Section 5, we stress the generality of our
procedure and indicate some future lines of research. Three Appendices contain
the proofs of our results.
2.
MODEL, IDENTIFICATION, ESTIMATION
2.1. The First-Price Auction Model
A single and indivisible object is auctioned. All bids are collected simultaneously. The object is sold to the highest bidder who pays his bid to the seller,
provided the bid is at least as high as a reservation price p 0 . In such an
institutional framework each bidder does not know others’ bids when forming
his bid. Within the IPV paradigm, each potential buyer i s 1, . . . , I is assumed to
have a private value ¨ i for the auctioned object. Each bidder does not know
other bidders’ private values but knows that all private values including his own
have been drawn independently from a common distribution F Ž⭈., which is
absolutely continuous with density f Ž⭈. and support w ¨ , ¨ x ; ⺢q. The distribution
of private values F Ž⭈., the number of potential bidders I, and the reservation
price p 0 are common knowledge with p 0 g w ¨ , ¨ x. In particular, all bidders are
identical ex ante and the game is said to be symmetric. Each bidder is assumed
to be risk neutral.
The Žunique. symmetric differentiable Bayesian Nash equilibrium of the
corresponding game of incomplete information was characterized by Riley and
Samuelson Ž1981. among others. Specifically, assuming I G 2, the equilibrium
bid bi of the ith bidder is
Ž1.
bi s s Ž ¨ i , F , I, p 0 . ' ¨ i y
1
Ž F Ž ¨ i ..
¨i
Iy1
Hp Ž F Ž u ..
0
Iy 1
du
528
E. GUERRE, I. PERRIGNE, AND Q. VUONG
if ¨ i G p 0 . If ¨ i - p 0 , then bi can be any value strictly less than the reservation
price p 0 . This strategy is obtained by solving the first-order differential equation
in sŽ⭈.:
f Ž¨i .
1
Ž2.
1 s Ž ¨ i y s Ž ¨ i ..Ž I y 1 .
,
X
F Ž¨i . s Ž¨i .
with boundary condition sŽ p 0 . s p 0 . The equilibrium strategy Ž1. is strictly
increasing in ¨ i on w p 0 , ¨ x, and expresses the equilibrium bid as a function of the
bidder’s private value, the distribution of private values, the number of bidders,
and the reservation price.
In general, bids are observed while private values are unobserved. The
preceding theoretical model leads to a closely related structural econometric
model. Specifically, because bi is a function of ¨ i , which is random and
distributed as F Ž⭈., then bi is also random with a distribution GŽ⭈. Žsay. that is
uniquely determined by Ž1.. This simple observation is the basis of the structural
analysis of auction data. It has led to the development of maximum likelihood
estimation methods Žsee Donald and Paarsch Ž1993, 1996.. and simulation based
estimation methods Žsee Laffont and Vuong Ž1993., Laffont, Ossard, and Vuong
Ž1995., and Li and Vuong Ž1997... In particular, simulation based methods are
convenient and computationally advantageous when moments of bids can be
simulated without determining numerically the equilibrium strategy sŽ⭈, F, I, p 0 .
or its inverse sy1 Ž⭈, F, I, p 0 ..
All preceding methods can be viewed as direct methods as they all start from
a parametric specification for F Ž⭈. so as to derive expressions for the distribution
and moments of observed bids. In contrast, the estimation method proposed in
this paper is indirect and starts from the estimation of the distribution of
observed bids so as to construct an estimate of the distribution of bidders’
private values. The crucial idea upon which our identification result and estimation procedure rest is to use the differential equation Ž2. to express each private
value as a known function of the corresponding bid, its distribution and density,
the number of bidders, and the reservation price. As a result, an important
advantage of our procedure is that it requires neither solving the differential
equation Ž2. nor computing the equilibrium strategy Ž1. within each iteration of
an optimization procedure. To clarify both the conceptual and the technical
issues, we begin with a nonbinding reservation price, i.e., p 0 s ¨ so that sŽ ¨ . s ¨
until Section 4.
2.2. Nonparametric Identification
A fundamental issue in structural estimation is whether the structural elements of the economic model are identified from available observations. Because the reservation price is nonbinding, the number I of potential bidders is
equal to the number of actual bidders. Hence I and bi , i s 1, . . . , I, are
observed. Thus the only unknown structural element of the model is the latent
distribution F Ž⭈., and the identification problem reduces to whether this distribution is uniquely determined from observed bids.
FIRST PRICE AUCTIONS
529
Though the equilibrium relation that links the observed bid bi to the underlying private value ¨ i is strictly monotonic, the identification problem is nontrivial.
This is because the distribution GŽ⭈. of bi depends on the underlying distribution F Ž⭈. in two ways: directly through ¨ i , which is distributed as F Ž⭈., and
indirectly through the equilibrium strategy sŽ⭈., which depends on F Ž⭈. Žsee Ž1...
The next result solves the identification problem by stating that the distribution
F Ž⭈. is unique whenever it exists. In addition, it gives a necessary and sufficient
condition on the distribution GŽ⭈. for the existence of a distribution F Ž⭈. of
bidders’ private values that can ‘‘rationalize’’ GŽ⭈..
Our result relies upon the fact that the first derivative sX Ž⭈. and the distribution F Ž⭈. with its density f Ž⭈. can be eliminated simultaneously from the
differential equation Ž2. by introducing the distribution GŽ⭈. of bi and its density
g Ž⭈.. Specifically, for every bg w b, b x s w ¨ , sŽ ¨ .x we have GŽ b . s PrŽ ˜
bF b . s PrŽ ¨˜
F sy1 Ž b .. s F Ž sy1 Ž b .. s F Ž ¨ ., where the last equality uses bs sŽ ¨ .. It follows
that the distribution GŽ⭈. is absolutely continuous with support w ¨ , sŽ ¨ .x and
density g Ž b . s f Ž ¨ .rsX Ž ¨ ., where ¨ s sy1 Ž b .. Taking the ratio gives g Ž b .rGŽ b .
s Ž1rsX Ž ¨ .. f Ž ¨ .rF Ž ¨ .. Thus the differential equation Ž2. becomes
1 G Ž bi .
Ž3.
¨ i s ␰ Ž bi , G, I . ' bi q
.
I y 1 g Ž bi .
Equation Ž3. now expresses the individual private value ¨ i as a function of the
individual’s equilibrium bid bi , its distribution GŽ⭈., its density g Ž⭈., and the
number of bidders I. Specifically, Ž3. states that if bi is the equilibrium bid, as it
is assumed in the structural approach, then the bidder’s private value ¨ i
corresponding to bi must satisfy Ž3..3
We define the set P of probability distributions P Ž⭈. on ⺢q as
P s P Ž ⭈ . is absolutely continuous with an interval support in ⺢q 4 .4
As usual, we restrict ourselves to strictly increasing and differentiable Bayesian
Nash equilibrium strategies. 5 Let GŽ⭈. denote the joint distribution of Ž b1 , . . . , bI ..
THEOREM 1: Let I G 2. Let GŽ⭈. belong to the set P I with support w b, b x I. There
exists a distribution of bidders’ pri¨ ate ¨ alues F Ž⭈. g P such that GŽ⭈. is the
distribution of the equilibrium bids in a first-price sealed-bid auction with independent pri¨ ate ¨ alues and a nonbinding reser¨ ation price if and only if:
I
C1: GŽ b1 , . . . , bI . s Ł is1
GŽ bi ..
Though equations similar to Ž3. have appeared in the decision theoretic literature starting with
Friedman Ž1956., a fundamental difference is that in decision theory each bidder plays as if hershe
were alone, while Ž3. is the result of the differential equation Ž2., where bidders are in a Nash game,
and each bidder’s expectations about how others bid agree with their actual bidding behavior.
Consequently, in decision theory, F Ž⭈. and GŽ⭈. need not be related by F Ž⭈. s Gw sŽ⭈.x, and bidders
need not be at the Bayesian Nash equilibrium. See Laffont Ž1997.. See also Section 4, where our
idea of expression ¨ i in terms of observables leads to Ž25..
4
The support is defined as the closure of x : pŽ x . ) 04 where pŽ⭈. is a density of P Ž⭈.. As a result,
any distribution in P is strictly increasing on its support.
5
Throughout we assume that the second-order conditions hold. Thus the Bayesian Nash equilibrium strategy is fully characterized by the first-order condition Ž2..
3
530
E. GUERRE, I. PERRIGNE, AND Q. VUONG
C2: The function ␰ Ž⭈, G, I . defined in Ž3. is strictly increasing on w b, b x and its
in¨ erse is differentiable on w ¨ , ¨ x ' w ␰ Ž b, G, I ., ␰ Ž b, G, I .x.
Moreo¨ er, when F Ž⭈. exists, it is unique with support w ¨ , ¨ x and satisfies F Ž ¨ . s
GŽ ␰y1 Ž ¨ , G, I .. for all w ¨ , ¨ x. In addition, ␰ Ž⭈, G, I . is the quasi in¨ erse of the
equilibrium strategy in the sense that ␰ Ž b, G, I . s sy1 Ž b, F, I . for all b g w b, b x.
Theorem 1 is important for many reasons. First, it shows that the theoretical
auction model does impose some restrictions on the distribution of observed
bids. These restrictions can constitute the basis of a formal test of the theory
Žsee Section 5 for a discussion.. Specifically, Condition C1 says that bids are
independent and identically distributed as GŽ⭈.. Condition C2 says that, given I,
the distribution GŽ⭈. of observed bids can be rationalized by a distribution of
private values F Ž⭈. only if ␰ Ž⭈, G, I . is strictly increasing. For instance, any
log-concave distribution GŽ⭈., i.e., such that g Ž b .rGŽ b . is strictly decreasing,
satisfies Condition C2 and thus can be rationalized.6 On the other hand,
densities that exhibit deep U-shaped parts can violate Condition C2. An example is the distribution GŽ b . s w brŽ5 y 4 b .x1r5 defined on w0, 1x with I s 2. Other
examples include some highly peaked multimodal densities.
Second, assuming that buyers behave as predicted by the model of Section 2.1,
Theorem 1 establishes that the distribution F Ž⭈. of bidders’ private values is
identified from the distribution of observed bids. In particular, it shows that
identification of the structural model does not require a priori parametric
specifications. Moreover, because it is nonparametric in nature, our identification result applies to parametric identification as well Žsee Donald and Paarsch
Ž1996. for a recent contribution on parametric identification.. As Roehrig Ž1988.
has argued, parametric identification may be achieved through misspecified
parametric specifications, and hence can be misleading.7
Third, it is useful to note that the function ␰ Ž⭈, G, I . is completely determined
from the knowledge of GŽ⭈. and I. Because ␰ Ž⭈, G, I . is the quasi inverse of
sŽ⭈, F, I ., one has neither to solve the differential equation Ž2. nor to apply
numerical integration in Ž1. so as to determine the buyer’s equilibrium strategy
sŽ⭈, F, I .. This remark is important because it underlies the principle and the
computational advantages of our indirect estimation method presented next.
2.3. Nonparametric Estimation
The basic idea of our estimation procedure is straightforward. If one knew the
distribution GŽ⭈. and density g Ž⭈., then one could use Ž3. to recover every
bidder’s ¨ i so as to estimate f Ž⭈.. Unfortunately, GŽ⭈. and g Ž⭈. are unknown, but
As a matter of fact, by differentiating Ž3. with respect to bi , it can be shown that Condition C2 is
equivalent to g Ž⭈.rG I Ž⭈. strictly decreasing.
7
Identification of F Ž⭈. can be proved from Roehrig’s Ž1988. Condition 3.2 applied to b y ˜
sŽ ¨ , I . s
0, where ˜
sŽ ¨ , I . ' sŽ ¨ , F, I ., when ¨ and I are independent conditionally upon the exogenous
variables x in F Ž⭈< ⭈ .. The latter independence condition, however, is not used in our proof.
Moreover, our proof is constructive as it gives F Ž⭈.. In addition, our identification result is global
instead of local.
6
FIRST PRICE AUCTIONS
531
they can be estimated from observed bids. This suggests the following two-step
estimator. In the first step we construct a sample of pseudo private values from
Ž3. using nonparametric estimates of the distribution and density functions of
observed bids. In a second step, this pseudo sample is used to estimate
nonparametrically the density of bidders’ private values.
To clarify ideas, we consider L homogeneous auctions with the same number
of bidders I. These assumptions will be relaxed in the next sections. Let l index
the lth auction. Our procedure is as follows. In the first step, we use the
observations Bp l , ps 1, . . . , I, l s 1, . . . , L4 to estimate nonparametrically GŽ⭈.
and g Ž⭈. by the empirical distribution and the kernel density estimator, respectively, i.e., by
Ž4.
Ž5.
˜Ž b . s
G
˜g Ž b . s
L
1
IL
I
Ý Ý
| Ž Bp l F b . ,
ls1 ps1
L
1
I
Ý Ý
ILh g
Kg
ls1 ps1
ž
b y Bp l
hg
/
,
where h g is a bandwidth and K g Ž⭈. is a kernel with a compact support. The
kernel density estimator ˜
g Ž⭈. is, however, biased at the boundaries of the
support.8 Indeed, let ␳ g - ⬁ be the length of the support of K g Ž⭈.. For bs
by ␭␳ g h gr2 where ␭ g w0, 1., the expectation of Ž5. gives Ew ˜
g Ž by ␭␳ g h gr
Ž
byb.r
h
y
␳
r2
2.x s Hy␭ ␳ g r2 g g K g Ž u. g Ž by ␭␳ g h gr2 y h g u. du using the change of variable B s b y ␭␳ g h gr2 y h g u. Hence Ew ˜
g Ž b y ␭␳ g h gr2.x y g Ž b y ␭␳ g h gr
q⬁
Ž
.
Ž .
2.Hq⬁
K
u
du
goes
to
zero
as
L
ª
⬁.
Because
Hy
y␭ ␳ g r2
g
␭ ␳ g r2 K g u du/ 1, the
density estimator is asymptotically biased for bg Ž by ␳ g h gr2, b x, and similarly
in w b, bq ␳ g h gr2.. Thus, using Ž3. to estimate private values corresponding to
observed bids close to the boundaries is likely to be problematic.
Let Bm i n and Bm a x be the minimum and maximum of the IL observed bids.
Because b F Bm i n F Bm a x F b, ˜
g Ž⭈. is asymptotically unbiased on w Bm i n q
␳ g h gr2, Bm a x y ␳ g h gr2x. This leads to defining the pseudo private value Vˆp l
corresponding to Bp l as
¡B
Ž6.
Vˆp l s~
pl q
¢q⬁
1
˜Ž Bp l . rg˜Ž Bp l .
G
Iy1
if Bm i n q ␳ g h gr2 F Bp l F Bm a x y ␳ g h gr2,
otherwise
for p s 1, . . . , I and l s 1, . . . , L. The pseudo sample of private values Vˆp l ,
p s 1, . . . , I, l s 1, . . . , L4 is used to estimate the density of private values by
Ž7.
fˆŽ ¨ . s
1
ILh f
L
I
Ý Ý
ls1 ps1
Kf
ž /
¨ y Vˆp l
hf
,
8
The support of g Ž⭈. is always finite. Indeed 0 F b F b. Moreover, from Laffont, Ossard, and
Vuong Ž1995., b s H¨¨ y Ž I y 1. f Ž y . F Iy 2 Ž y . dy F Ž I y 1.H¨¨ yf Ž y . dy - ⬁ because Ew ¨ i x - ⬁.
532
E. GUERRE, I. PERRIGNE, AND Q. VUONG
where h f is a bandwidth and K f Ž⭈. is a kernel. Because our kernels have
compact supports, Ž6. in effect trims observed bids that do not belong to
w Bm i n q ␳ g h gr2, Bm a x y ␳ g h gr2x.
The asymptotic properties as L ª ⬁ and I fixed of such a two-step nonparametric estimator are obtained in Section 3. When private values are observed
and when f Ž⭈. has R bounded continuous derivatives, the optimal uniform
convergence rate for estimating f Ž⭈. is Ž Lrlog L. R rŽ2 Rq1. Žsee Stone Ž1982... In
our case, private values are unobserved while bids are observed. As a result, this
rate cannot be attained. In Theorem 2, we show that the optimal rate is
Ž Lrlog L. R rŽ2 Rq3., which is smaller than the optimal rate when private values
are directly observed. By choosing appropriately the vanishing rates of the
bandwidths, namely h g s c g Žlog LrL.1rŽ2 Rq3. and h f s c f Žlog LrL.1rŽ2 Rq3., we
show in Theorem 3 that this optimal rate can be attained by our two-step
estimator.
2.4. Monte Carlo Experiments
To illustrate our two-step nonparametric procedure, we conduct a limited
Monte Carlo study. We consider L s 200 auctions, each having I s 5 bidders,
which gives 1000 observed bids. These numbers correspond to realistic sizes of
auction data. Our Monte Carlo experiment consists of 1000 replications. The
true distribution F of private values is log-normal with parameters zero and
one, truncated at 0.055 and 2.5 to satisfy Assumption A2 of Section 3.1, which
corresponds to leaving out 20% approximately of the original log-normal distribution. For each replication, we first generate randomly IL private values from
this truncated distribution. We then compute numerically the corresponding
bids Bp l using Ž1. with p 0 s ¨ .
Next, we apply our estimation procedure for each replication. First, we
estimate the distribution function and density of observed bids using Ž4. and Ž5..
In a second step we compute the pseudo private values Vˆp l using Ž6.. From this
pseudo data we estimate the private values density function using Ž7.. Specifically, we consider that the latent density f Ž⭈. is once-continuously differentiable
so that R s 1. To satisfy Assumption A3 on the kernels in Section 3, we choose
the triweight kernel Ž35r32.Ž1 y u 2 . 3|Ž< u < F 1. for K g Ž⭈. and K f Ž⭈. so that
␳ g s ␳ f s 2. As indicated above, the orders of the bandwidths are Ly1 r5. They
are chosen as h g s 1.06 ␴ˆb Ž IL.y1 r5 and h f s 1.06 ␴ˆ¨ Ž ILT .y1 r5, where ␴ˆb and ␴ˆ¨
are the estimated standard deviations of observed bids and trimmed pseudo
private values, respectively, and LT is the number of auctions remaining after
the trimming Ž6.. The factor 1.06 follows from the so-called rule of thumb Žsee
Hardle Ž1991... The use of I arises because we have I bidders per auction.
The program is written in FORTRAN. For each replication, the execution
time lasted less than one minute, which reduces by a factor of 1000 the
execution time of a parametric method Žnonlinear least squares. because the
latter requires the numerical integration of Ž1.. Each replication gives us two
estimated functions: Ži. the estimated inverse ␰ˆŽ⭈. of the equilibrium strategy
FIRST PRICE AUCTIONS
533
FIGURE 1.ᎏTrue and estimated equilibrium strategies.
function Žsee Ž3.. and Žii. the estimated density function fˆŽ⭈., each evaluated at
500 equally spaced points on w b, b x s w sŽ0.055., sŽ2.5.x and w0.055, 2.5x, respectively. Our Monte Carlo results are summarized in Figures 1 and 2.
Figure 1 displays the true equilibrium strategy bs sŽ ¨ . in plain line. We
display for each value of bg w sŽ0.055., sŽ2.5.x the mean, the 5% percentile, and
the 95% percentile of the 1000 estimates ␰ˆŽ b .. This gives the Žpointwise. 90%
confidence interval for ␰ Ž b . s sy1 Ž b .. Figure 2 displays the true density of the
truncated log-normal distribution in plain line, and for each value of ¨ g
w0.055, 2.5x, the mean, the 5% percentile, and the 95% percentile of the 1000
estimates fˆŽ ¨ .. This gives the Žpointwise. 90% confidence interval for f Ž ¨ .. The
striking features are that, on the interval bordered by the horizontalrvertical
lines, Ži. the true curve Žthe equilibrium strategy or the density. falls within the
confidence band and Žii. the mean of the estimates perfectly matches the true
curve.9
9
In Figure 1, the horizontal lines are defined by Bm i n q h g and Bm a x y h g corresponding to the
trimming Ž6.. In Figure 2, the lower Župper. vertical line is defined on average by this trimming to
which one h f is added Žsubtracted . to eliminate remaining boundary effects. For instance, the lower
vertical line corresponds to the average of ␰ˆŽ Bm i n q h g . q h f .
534
E. GUERRE, I. PERRIGNE, AND Q. VUONG
FIGURE 2.ᎏTrue and estimated densities of private values.
3.
ASYMPTOTIC PROPERTIES
In practice, the auctioned objects can be heterogeneous and the number of
potential bidders can vary across auctions. These considerations modify our
estimator and raise new technical difficulties. In particular, estimation of the
boundaries and trimming near the boundaries are not as simple as in Ž6.. To
clarify these issues, we still assume in this section that the reservation price is
nonbinding.10
3.1. Regularity Assumptions and Key Properties
Let X l denote the vector of relevant characteristics for the lth auctioned
object, and Il be the number of bidders in the lth auction.11 The distribution of
bidders’ private values Vp l for the lth auction is the conditional distribution
F Ž⭈< X l , Il . of private values given Ž X l , Il .. Similarly, the distribution of observed
10
This arises when the reservation price does not constitute an effective screening device such as
in Outer Continental Shelf gas and leases auctions Žsee McAfee and Vincent Ž1992...
11
Following the common knowledge of F Ž⭈. in the theoretical model, the vector X l is common
knowledge to all parties. This is frequently justified as the auctioned objects are fully described in a
freely available booklet. We then assume that none of X l is omitted to control heterogeneity across
auctioned objects. Thus unobserved heterogeneity comes only from differences in bidders’ private
values, which are the unobserved random terms in the structural econometric model.
535
FIRST PRICE AUCTIONS
bids in the lth auction is GŽ⭈< X l , Il .. Thus Ž1. and Ž3. become
1
Ž8.
Bp l ' s Ž Vp l , X l , Il . s Vp l y
Ž9.
Vp l ' ␰ Ž Bp l , X l , Il . s Bp l q
F Ž Vp l < X l , Il .
1
H
Vpl
I ly1
¨l
G Ž B p l < X l , Il .
Il y 1 g Ž B p l < X l , Il .
F Ž ¨ < X l , Il .
I ly1
d¨ ,
,
where ¨ l ' ¨ Ž X l , Il . is the lower bound of the support of F Ž⭈< X l , Il . and g Ž⭈< ⭈ , ⭈ .
is the density of GŽ⭈< ⭈ , ⭈ ..
The next assumptions concern the underlying generating process as well as
the smoothness of the latent joint distribution of Ž Vp l , X l , Il . for any ps 1, . . . , Il .
ASSUMPTION A1:
Ži. The Ž dq 1.-dimensional ¨ ectors Ž X l , Il ., l s 1, 2, . . . , are independently and
identically distributed as FmŽ⭈, ⭈ . with density f mŽ⭈, ⭈ ..
Žii. For each l the ¨ ariables Vp l , ps 1, . . . , Il , are independently and identically
distributed conditionally upon Ž X l , Il . as F Ž⭈< ⭈ , ⭈ . with density f Ž⭈< ⭈ , ⭈ ..
In particular, privates values are independent across auctions. As is well
known, dependent private values across auctions introduce dynamic considerations that invalidate the Bayesian Nash equilibrium solution Ž1., and hence are
outside the scope of this paper. Note that we do not assume that X l and Il are
independent from each other. Thus we allow the number of bidders to depend
upon the characteristics of the auctioned object.
Let I be the set of possible values for Il . Throughout we denote by SŽ). the
support of ), and by Si Ž). the support when the number of bidders is equal to i.
ASSUMPTION A2: I is a bounded subset of 2, 3, . . . 4 , and:
Ži. for each i g I , Si Ž F . s Ž ¨ , x . : xg w x, x x, ¨ g w ¨ Ž x ., ¨ Ž x .x4 , with x - x;
Žii. for Ž ¨ , x, i . g SŽ F ., f Ž ¨ < x, i . G c f ) 0, and for Ž x, i . g S Ž Fm ., f mŽ x, i . G c f )
0;
Žiii. for each i g I , F Ž⭈< ⭈ , i . and f mŽ⭈, i . admit up to R q 1 continuous bounded
partial deri¨ ati¨ es on Si Ž F . and S i Ž Fm ., with R G 1.
Without loss of generality, we can assume that x and x are known.12 On the
other hand, the boundary functions ¨ Ž⭈. and ¨ Ž⭈. are typically unknown. Next,
because Ž3. is used to recover private values, it is convenient that g Ž⭈< ⭈ , ⭈ . be
bounded away from zero. From Proposition 1 below, this is achieved if the
12
Because the boundaries x and x can be estimated at a faster rate than our estimation
procedure, our statistical results are not affected. To simplify, we assume that X is a vector of
continuous variables. If some X ’s are discrete, our results still hold with d replaced by the number
of continuous variables, and the nonparametric estimators Ž13., Ž14., and Ž20. ᎐ Ž22. modified
appropriately following Bierens Ž1987..
536
E. GUERRE, I. PERRIGNE, AND Q. VUONG
density f Ž⭈< ⭈ , ⭈ . is bounded away from zero, which is the purpose of A2-Žii..13
Lastly, standard assumptions in the nonparametric literature deal with the
smoothness of f Ž⭈< ⭈ , ⭈ .. In particular, A2-Žiii. implies that f Ž⭈< ⭈ , i . admits up to R
bounded continuous partial derivatives on Si Ž F .. Consequently, the best uniform convergence rate for estimating f Ž⭈< ⭈ , ⭈ . is Ž Lrlog L. R rŽ2 Rqdq1. if private
values were observed Žsee Stone Ž1982...
Since private values are unobserved, estimation of the density f Ž⭈< ⭈ , ⭈ . must be
based on observed bids. This is called an in¨ erse or indirect estimation problem;
see, e.g., Groeneboom Ž1996.. To determine the best uniform convergence rate
for estimating the latent density f Ž⭈< ⭈ , ⭈ . from observed bids, an important step is
to study the implied smoothness of the bid density g Ž⭈< ⭈ , ⭈ .. This is the purpose
of the next proposition.
PROPOSITION 1: Gi¨ en A2, the conditional distribution GŽ⭈< ⭈ , ⭈ . satisfies:
Ži. its support SŽ G . is such that Si Ž G . s Ž b, x . : xg w x, x x, bg w bŽ x, i ., bŽ x, i .x4 ,
with inf x g w x, x xŽ bŽ x, i . y bŽ x, i .. ) 0. Moreo¨ er, Ž bŽ⭈, i ., bŽ⭈, i .. admit up to R q 1
continuous bounded deri¨ ati¨ es on w x, x x for each i g I , and bŽ⭈, i . s ¨ Ž⭈.;
Žii. for Ž b, x, i . g S Ž G ., g Ž b < x, i . G c g ) 0;
Žiii. for each i g I , GŽ⭈< ⭈ , i . admits up to R q 1 continuous bounded partial
deri¨ ati¨ es on Si Ž G .;
Živ. for each i g I , if Ci Ž B . is a closed subset of the interior Sio Ž G . of Si Ž G ., then
g Ž⭈< ⭈ , i . admits up to R q 1 continuous bounded partial deri¨ ati¨ es on Ci Ž B ..
The striking feature of Proposition 1 is item Živ.. Specifically, because it has
R q 1 continuous bounded derivatives instead of R, the bid density is smoother
than the private value density. The intuition behind this result comes from the
equality
G Ž b < x, i .
g Ž b < x, i . s
,
Ž i y 1 .Ž ␰ Ž b, x, i . y b .
which follows from Ž9.. Since ␰ Ž⭈, ⭈ , i . s sy1 Ž⭈, ⭈ , i . and sŽ⭈, ⭈ , i . has the same
smoothness as F Ž⭈< ⭈ , i . as suggested by Ž8., then ␰ Ž⭈, ⭈ , i . has R q 1 continuous
bounded derivatives. Now, since GŽ⭈< ⭈ , i . has also R q 1 continuous bounded
derivatives from Žiii., then Živ. follows. As an important consequence, g Ž⭈< ⭈ , ⭈ .
can be estimated uniformly at a faster rate, namely Ž Lrlog L.Ž Rq1.rŽ2 Rqdq3.,
than f Ž⭈< ⭈ , ⭈ . can be.
3.2. Optimal Uniform Con¨ ergence Rate
In this section, we study the optimal rate at which the latent density of private
values can be estimated uniformly from observed bids. Indeed, uniform conver13
If A2-Žii. is not tenable, our results still apply provided there exists a known transformation with
R q 1 continuous bounded derivatives such that the transformed density satisfies A1᎐A2. In this
case, one should use the transformed X l ’s and Vˆp l ’s in Ž13. ᎐ Ž14. and Ž20. ᎐ Ž21. defining our
estimator.
537
FIRST PRICE AUCTIONS
gence results are crucial for recovering the shape of a density. As is well-known,
however, nonparametric estimators typically converge at different rates depending on the choice of their smoothing parameters. An important issue in nonparametric statistics is to determine the best rate at which the functional of
interest can be estimated uniformly. Though the optimal rate of uniform
convergence is known for density estimation Žsee Stone Ž1982.., this rate does
not apply in our case because private values are unobserved. To our knowledge,
such a difficult problem has been seldom addressed in structural estimation.
Hereafter, we focus upon the estimation of the conditional density f Ž ¨ < x ..14
We adopt a minimax approach, as developed by Khas’minskii Ž1976.. We
consider joint densities f Ž ¨ , x, i . satisfying A2. Let f 0 Ž ¨ , x, i . be one such
density. In order to determine the optimal rate of uniform convergence rLU for
estimating the corresponding conditional density f 0 Ž ¨ < x ., we study the quantity
Ž 10.
inf
ž
sup Pr g rL
fˆŽ⭈<⭈. fgU⑀ Ž f 0 .
sup
Ž ¨ , x .g C Ž V .
/
< fˆŽ ¨ < x . y f Ž ¨ < x . < ) ␬ ,
where ␬ is a positive constant, C Ž V . is an arbitrary inner compact subset with
nonempty interior of the support SŽ f 0 Ž ¨ , x .. s Di g I Si Ž F0 . of f 0 Ž ¨ , x ., and
Pr g Ž⭈. denotes the probability distribution of Ž b, x, i . when the underlying
density of Ž ¨ , x, i . is f Ž⭈, ⭈ , ⭈ ..
Because we consider the uniform convergence of estimators, the relevant
discrepancy measure is the sup norm over C Ž V . of the difference between an
arbitrary estimator fˆŽ⭈< ⭈ . and the conditional density f Ž⭈< ⭈ . of interest. The latter
is restricted to belong to the set of densities U⑀ Ž f 0 ., which is a neighborhood of
f 0 defined as
U⑀ Ž f 0 .
½
' f;
sup
Ž ¨ , x , i .gS Ž F 0 .
f Ž ¨ , x, i . y f 0 Ž ¨ , x, i . - ⑀ ,
f Ž ⭈, ⭈ , ⭈ .
R -M
5,
where M) 0 and 5 ⭈ 5 R - M requires densities to have all their derivatives up to
the Rth order bounded by M uniformly on SŽ F0 .. As in the standard theory,
considering the supremum over such a neighborhood avoids superefficient
estimators.
The relevance of Ž10. in determining the optimal uniform convergence rate rLU
can be explained as follows. First, consider an arbitrary estimator fˆŽ⭈< ⭈ .. Intuitively, if rL diverges to infinity sufficiently fast, the probability in Ž10. will tend
to one. On the other hand, if rL does not diverge sufficiently fast, this
probability will tend to zero. Now, to determine the optimal uniform conver14
Though our results also apply to the estimation of the conditional density f Ž ¨ < x, i ., our interest
in f Ž ¨ < x . is justified by the economic model, which assumes that the private values and number of
bidders are independent conditionally on x so that f Ž ¨ < x, i . s f Ž ¨ < x . for every i. If the latter does
not hold, a more complex bidding model with a game of entry should be considered.
538
E. GUERRE, I. PERRIGNE, AND Q. VUONG
gence rate for our problem, we must consider all possible estimators of f Ž⭈< ⭈ ..
Suppose that rL diverges to infinity extremely fast; then the probability in Ž10.
will converge to one for every possible estimator and hence Ž10. will be bounded
away from zero. The optimal rate rLU is the infimum of such rL s.15
The next theorem gives an upper bound for the optimal uniform convergence
rate for estimating f Ž⭈< ⭈ . from observed bids. Its proof adapts the argument used
in Khas’minskii Ž1976. to our problem, and crucially relies on Theorem 1 and
Proposition 1.
THEOREM 2: Assume that A1᎐A2 hold and 5 f 0 Ž⭈, ⭈ , ⭈ .5 R - M. Let C Ž V . be an
inner compact subset of SŽ f 0 Ž ¨ , x .. with nonempty interior. There exists a constant
␬ ) 0 such that
lim
lim
inf
sup Pr g
⑀ª0 Lªq⬁ fˆŽ⭈<⭈. fgU Ž f .
⑀ 0
žž /
L
Rr Ž2 Rqdq3 .
log L
=
sup
Ž ¨ , x .g C Ž V .
/
< fˆŽ ¨ < x . y f Ž ¨ < x . < ) ␬ ) 0.
We consider a lower limit Žor limit inf. as L ª ⬁ because the simple limit of
Ž10. may not exist. Also, because Ž10. is nondecreasing in ⑀ , by taking the limit
as ⑀ ª 0, we establish in fact the desired result for all ⑀ . Specifically, Theorem 2
implies that there exists a strictly positive constant ␬ such that, for any ⑀ ) 0,
any L G L0 Ž ⑀ .:
inf
sup Pr g
fˆŽ⭈<⭈. fgU⑀ Ž f 0 .
žž /
L
log L
Rr Ž2 Rqdq3 .
sup
Ž ¨ , x .g C Ž V .
< fˆŽ ¨ < x . y f Ž ¨ < x . < ) ␬
/
G ␦ ) 0.
Thus the optimal uniform convergence rate rLU and hence the rate of uniform
convergence of any estimator of f Ž⭈< ⭈ . cannot be larger than Ž Lrlog L. R rŽ2 Rqdq3.
over C Ž V ..
Note that this rate is slower than Ž Lrlog L. R rŽ2 Rqdq1., which is the optimal
rate if private values were observed. This slower rate of convergence is specific
to our auction problem, where the variables associated with the density of
interest f Ž⭈< ⭈ . are not observed. Intuitively, this result can be understood as
follows. Since g 0 Ž⭈< ⭈ , ⭈ . has R q 1 derivatives Žsee Proposition 1., g 0 Ž⭈< ⭈ , ⭈ . and
hence ␰ 0 Ž⭈, ⭈ , ⭈ . can be estimated at the rate Ž Lrlog L.Ž Rq1.rŽ2 Rqdq3. from
15
Hence, in the statistical literature, determining optimal rates is referred as finding lower
bounds.
539
FIRST PRICE AUCTIONS
Stone Ž1982.. But
Ž 11.
f 0 Ž ¨ < x, i . s
Ž ¨ , x, i . < x, i .
g 0 Ž ␰y1
0
Ž ¨ , x, i . , x, i .
␰ 0X Ž ␰y1
0
.
Hence the derivative ␰ 0X must be estimated.16 As our results indicate, this is the
hardest statistical estimation problem when estimating f 0 Ž⭈< ⭈ ., because it requires to estimate g X0 in view of Ž9.. Since the best rate for estimating g X0 and
hence ␰ 0X is Ž Lrlog L. R rŽ2 Rqdq3., this actually gives the best rate at which
f 0 Ž⭈< ⭈ . can be estimated from observed bids.
As noted above, Theorem 2 only provides an upper bound to the optimal uniform convergence rate rLU . As usual, such a bound would not be much
useful if it cannot be attained, i.e., if there does not exist an estimator of
f Ž⭈< ⭈ . that converges at the rate Ž Lr log L. R rŽ2 Rqdq3.. In the next subsections,
we establish that our two-step nonparametric estimator converges at the rate
Ž Lrlog L. R rŽ2 Rqdq3. given appropriate choice of the smoothing parameters Žsee
Theorem 3.. As a consequence, the optimal uniform convergence rate rLU for
estimating the latent density f Ž⭈< ⭈ . from observed bids is Ž Lrlog L. R rŽ2 Rqdq3.. It
also follows that our two-step nonparametric estimator converges at the best
possible rate, i.e., is optimal.
3.3. Definition of the Estimator
The purpose of this section is to generalize the two-step procedure presented
in Section 2.3 to heterogeneous auctions. At the same time, we make precise the
assumptions on the kernels and bandwidths, which define our estimator.17
Note that Ž9. can be rewritten as
Ž 12.
Vp l ' ␰ Ž Bp l , X l , Il . s Bp l q
1
G Ž B p l , X l , Il .
Il y 1 g Ž B p l , X l , Il .
,
where GŽ b, x, i . s GŽ b < x, i . f mŽ x, i . s HbŽb x . g Ž u, x, i . du. Hence, using the observations Ž Bp l , X l , Il .; ps 1, . . . , Il , l s 1, . . . , L4 , our first step consists in estimating
˜ ˜ where
the ratio ␺ Ž⭈, ⭈ , ⭈ . s GŽ⭈, ⭈ , ⭈ .rg Ž⭈, ⭈ , ⭈ . by ␺˜s Grg,
Ž 13.
˜Ž b, x, i . s
G
1
L
Ý
d
Lh G
ls1
1
Il
Il
Ý
ps1
| Ž Bp l F b . K G
ž
/
x y X l i y Il
,
,
hG
hG I
16
Throughout, we use prime to denote a derivative with respect to the first argument of a
function.
17
An alternative estimator is to use directly Ž11.. Specifically, estimation of g Ž⭈< ⭈ , ⭈ ., and hence of
␰ Ž⭈, ⭈ , ⭈ ., followed by determination of ␰ X Ž⭈, ⭈ , ⭈ . and ␰y1 Ž⭈, ⭈ , ⭈ . would give an estimate of f Ž⭈< ⭈ .. The
major problem of this procedure is that ␰ˆŽ⭈, ⭈ , ⭈ . may not be invertible. In addition to converging at
the optimal rate, our two-step estimator avoids this drawback.
540
Ž 14.
E. GUERRE, I. PERRIGNE, AND Q. VUONG
˜g Ž b, x, i . s
1
L
Ý
Lh dq1
g
ls1
1
Il
Il
Ý
Kg
ps1
ž
/
by Bp l xy X l i y Il
,
,
,
hg
hg
hgI
which clearly extend Ž4. ᎐ Ž5.. Here h G , h G I , h g , and h g I are some bandwidths,
and K G and K g are kernels with bounded supports. Following Bierens Ž1987.,
we apply kernel techniques to the discrete variable I, which allows for averaging
over values of i. Note that we choose different bandwidths for continuous and
discrete variables.18
In view of Ž12. it would be natural to estimate each private value Vp l by
Ž 15.
V˜p l s Bp l q
1
Il y 1
␺˜ Ž Bp l , X l , Il . ,
and to use these estimates in a second-step estimation of the conditional density
f Ž ¨ < x .. Unfortunately, it is well-known that ␺˜ is an asymptotically biased
estimator of ␺ at the boundaries of the support of Ž B, X, I .. Because of this
boundary effect, we modify slightly Ž15. by introducing a trimming near the
boundaries.
In this aim we estimate the boundary of the support of the joint distribution
of Ž B, X, I ., which is unknown.19 We focus on the estimation of the support
w bŽ x, i ., bŽ x, i .x of the conditional distribution of B given Ž X, I . since the
support of Ž X, I . can be assumed to be known Žor can be readily estimated.. We
propose some nonparametric boundary estimators that generalize Geffroy’s
Ž1984. estimators to the multidimensional case. Let h⭸ ) 0. We consider the
following partition of ⺢ d with a generic hypercube of side h⭸ :
␲ k 1 , . . . , k d s w k 1 h⭸ , Ž k 1 q 1 . h⭸ . = ⭈⭈⭈ = w k d h⭸ , Ž k d q 1 . h⭸ . ,
where Ž k 1 , . . . , k d . runs over ⺪ d. This induces a partition of w x, x x. Given an
integer i and a value x, the estimate of the upper boundary bŽ x, i . is the
maximum of those bids for which Il s i and the corresponding value of X l falls
in the hypercube ␲ k 1 , . . . , k d containing x. The estimate of the lower boundary is
similarly defined. Formally, our boundary estimates of the support of the
conditional density of B given Ž X, I . s Ž x, i . are
Ž 16.
ˆb Ž x, i . s sup B , ps 1, . . . , I , l s 1, . . . , L; X g ␲
4
pl
l
l
k 1 , . . . , k d , Il s i ,
Ž 17.
ˆb Ž x, i . s ˆb Ž x . s inf Bp l , ps 1, . . . , Il , l s 1, . . . , L; X l g ␲ k 1 , . . . , k d 4 .
Because bŽ x, i . s ¨ Ž x . is independent of i, we need not restrict Ž17. to bids such
Because ␺ resembles the hazard rate grŽ1 y G ., various nonparametric estimators of the latter
can be used. See Hassani, Sarda, and Vieu Ž1986. and Singpurwalla and Wong Ž1983. for recent
surveys. Our estimator ␺˜ has the characteristic that it takes into account a repeated aspect of our
data due to the number of bidders Il G 2. In addition, to minimize boundary effects, we have chosen
the sum averaging instead of the more common integral averaging for estimating G.
19
Note that even if the boundary functions ¨ Ž⭈. and ¨ Ž⭈. were known the upper boundary bŽ x, i .
would be unknown since bŽ x, i . s sŽ ¨ Ž x ., x, i ., which depends on the underlying density f Ž⭈< ⭈ , ⭈ ..
18
541
FIRST PRICE AUCTIONS
that Il s i. Our estimator for Si Ž G . is Sˆi Ž G . ' Ž b, x . : bg w ˆ
bŽ x, i ., ˆ
bŽ x, i .x, xg
w x, x x4 .
We now turn to the trimming. Basically, for each i we will trim out observations Ž Bp l , X l , Il . with Il s i that are close to the boundaries of the estimated
support Sˆi Ž G . by less than a distance, which is a function of the smoothing
parameters. Specifically, let SŽ h G . and SŽ h g . be the supports of K G Ž⭈rh G , 0.
and K g Ž⭈rh g , ⭈ rh g , 0., respectively. For instance, if the supports of K G Ž⭈, 0. and
K g Ž⭈, ⭈ , 0. are hypercubes of sides equal to 1, then SŽ h G . and SŽ h g . are
hypercubes of sides h G and h g in ⺢ d and ⺢ dq 1 , respectively. Because we
consider points Ž b, x . in ⺢ dq1, we consider hereafter that SŽ h G . is 0 = SŽ h G .4
; ⺢ dq 1. Instead of Ž15., we define the pseudo private value as
1
Ž 18.
Vˆp l s Bp l q
␺ˆ Ž Bp l , X l , Il . ,
Il y 1
where
Ž 19.
¡␺˜ Ž b, x, i .
␺ˆ Ž b, x, i . '~
¢q⬁
if Ž b, x . q S Ž 2 h G . ; Sˆi Ž G . and
Ž b, x . q S Ž 2 h g . ; Sˆi Ž G . ,
otherwise.
This extends Ž6. to the heterogeneous case.20
In the second step we use the pseudo sample Ž Vˆp l , X l ., ps 1, . . . , Il , l s
1, . . . , L4 , to estimate nonparametrically the density f Ž ¨ < x . by fˆŽ ¨ < x . s fˆŽ ¨ , x .r
fˆŽ x ., where
Ž 20.
Ž 21.
fˆŽ ¨ , x . s
fˆŽ x . s
1
Lh dq1
f
1
L
L
Ý
ls1
Ý KX
Lh dX ls1
ž
1
Il
Il
Ý
ps1
xy X l
hX
Kf
/
ž
¨ y Vˆp l
hf
,
xy X l
hf
/
,
,
h f and h X are bandwidths, and K f and K X are kernels. Because the latter have
compact supports, the contribution of infinite pseudo private values is nil in our
estimate fˆŽ ¨ , x .. Hence, Ž19. can be interpreted as trimming the observations
Ž Bp l , X l , Il . that are too close to the boundary of the estimated support SˆI Ž G ..
l
We turn to the choice of kernels and bandwidths defining our two-step
estimator.
ASSUMPTION A3:
Ži. The kernels K G Ž⭈, ⭈ ., K g Ž⭈, ⭈ , ⭈ ., K f Ž⭈, ⭈ ., K X Ž⭈. are symmetric with bounded
hypercube supports and twice continuous bounded Ž uniformly in I . deri¨ ati¨ es with
respect to their continuous arguments.
20
In fact, as the proof of Proposition 3-Ži. shows, the trimming in Ž19. can be defined using
Ž1 q ⑀ . h G and Ž1 q ⑀ . h g with ⑀ ) 0 instead of 2 h G and 2 h g , respectively. Moreover, in the simple
case where there are no exogenous variables X, this proof shows that we can have ⑀ s 0 because
b F Bm i n and Bm a x F b. This gives the trimming Ž6..
542
E. GUERRE, I. PERRIGNE, AND Q. VUONG
Žii. HK G Ž x, 0. dx s 1, HK g Ž b, x, 0. db dx s 1, HK f Ž ¨ , x . d¨ dx s 1, and
HK X Ž x . dxs 1.
Žiii. K G Ž⭈, 0., K g Ž⭈, ⭈ , 0., K f Ž⭈, ⭈ ., K X Ž⭈. are of order R q 1, R q 1, R, R q 1. Thus
moments of order strictly smaller than the gi¨ en order ¨ anish.
Assumption A3 is standard. The orders of the kernels have been chosen
according to the smoothness of the estimated functions. In particular, K g is of
order R q 1, since the density g Ž⭈, ⭈ , ⭈ . admits up to R q 1 bounded continuous
derivatives from Proposition 1.
ASSUMPTION A4:
Ži. As L ª ⬁, the ‘‘discrete’’ bandwidths h G I and h g I ¨ anish.
Žii. The ‘‘continuous’’ bandwidths h G , h g , h f , and h X are of the form:
h G s ␭G Ž log LrL .
h f s ␭ f Ž log LrL .
1r Ž2 Rqdq2 .
1r Ž2 Rqdq3 .
,
,
h g s ␭ g Ž log LrL .
h X s ␭ X Ž log LrL .
1r Ž2 Rqdq3 .
1r Ž2 Rqdq2 .
,
,
where the ␭’s are strictly positi¨ e constants.
Žiii. The ‘‘boundary’’ bandwidth is of the form h⭸ s ␭⭸ Žlog LrL.1rŽ dq1. if d) 0.
Part Ži. combined with A3-Ži. implies that averaging over observations
Ž Bp l , X l , Il . such that Il / i will disappear from Ž13. and Ž14., as L ª ⬁.
The log L arises because we deal with uniform consistency. From this point of
view, h G , h g , and h X are optimal bandwidths given Proposition 1 and A2-Žiii.
Žsee, e.g., Hardle Ž1991... Hence our kernel estimators Ž13., Ž14., and Ž21. of
GŽ⭈, ⭈ , ⭈ ., g Ž⭈, ⭈ , ⭈ ., and f Ž⭈. converge uniformly at the best possible rate. If the
private values were observed, the optimal bandwidth for estimating f Ž⭈, ⭈ . would
be of order Žlog LrL.1rŽ2 Rqdq1., which is asymptotically smaller than the rate
for h f given in Žii.. Thus our choice of h f implies oversmoothing and would be
suboptimal in this sense. However, private values are unobserved. As we show
below, given the optimal rates for h G , h g , and h X , our choice for h f is the only
rate that achieves the optimal rate of Theorem 2.
3.4. Uniform Consistency
Our next main result establishes the uniform consistency of our two-step
estimator with its rate of convergence. To do so, we need two results, which are
of independent interests. The first proposition establishes the uniform consistency with rates of convergence of our nonparametric boundary estimators
Ž16. ᎐ Ž17..21
The bandwidth h⭸ in A4-Žiii. does not depend upon the smoothness of the upper and lower
bounds bŽ⭈, i . and bŽ⭈, i .. Therefore, the rate of convergence of our boundary estimators can be
improved. See, for instance, Korostelev and Tsybakov Ž1993..
21
543
FIRST PRICE AUCTIONS
PROPOSITION 2: Let r⭸ s Ž Lrlog L.1rŽ dq1.. Under A1᎐A2 and A4-Ž iii .,
sup
Ž x , i .g w x , x x= I
sup
xg w x , x x
ˆb Ž x, i . y b Ž x, i . s O Ž1rr . a.s. and
⭸
ˆb Ž x . y b Ž x . s O Ž1rr⭸ . a.s.
The second proposition studies the rate at which the pseudo private values Vˆp l
converge uniformly to the true values.22
PROPOSITION 3: Let r g s Ž Lrlog L. R rŽ2 Rqdq3., r Ug s Ž Lrlog L.Ž Rq1.rŽ2 Rqdq3..
Under A1᎐A4:
Ži. sup p, l|Ž Vˆp l / q⬁.< Vˆp l y Vp l < s O Ž1rr g . a.s.;
Žii. for any closed inner subset C Ž V . of SŽ f Ž ¨ , x .., we ha¨ e
sup |C ŽV . Ž Vp l , X l . < Vˆp l y Vp l < s O Ž 1rr Ug .
a.s.
p, l
Proposition 3 shows that the pseudo private values Vˆp l converge uniformly to
the true private values, with the exception of those corresponding to observations Ž Bp l , X l , Il . close to the boundaries Žsee Ž19... However, r g is smaller than
r Ug . Hence the rate of convergence in part Ži. is slower than in Žii.. The reason is
that Ži. considers observations that are arbitrary close to the support SŽ G . as
L ª ⬁, while Žii. deals with observations bounded away from the boundary.
Moreover, from Ž12. and Ž18. the rate of uniform convergence of Vˆp l to Vp l
depends mainly on that of ˜
g Ž⭈, ⭈ , i . to g Ž⭈, ⭈ , i .. And we know from Proposition 1
that g Ž⭈, ⭈ , i . is smoother on a closed inner subset of its support.
We now state the main result of this section. Because kernel estimators suffer
from boundary effects, we restrict uniform convergence to inner closed subsets
of SŽ f Ž ¨ , x ...
THEOREM 3: Suppose that A1᎐A4 hold. We ha¨ e
sup
Ž ¨ , x .g C Ž V .
< fˆŽ ¨ < x . y f Ž ¨ < x . < s O Ž Ž log LrL . Rr 2 Rqdq3
Ž
.
.
a.s.
for any closed inner subset C Ž V . of SŽ f Ž ¨ , x ...
In addition to establishing the uniform consistency of our two-step estimator,
Theorem 3 is important for two reasons. First, it implies that the upper bound
Ž Lrlog L. R rŽ2 Rqdq3. of Theorem 2 is in fact the optimal uniform convergence
rate rLU for estimators of the conditional density f Ž⭈< ⭈ . from observed bids.
22
Proposition 3 is interesting in itself because it can also be used for the estimation of the
conditional mean, variance, or quantiles, etc . . . of private values.
544
E. GUERRE, I. PERRIGNE, AND Q. VUONG
Second, it shows that our two-step estimator attains the optimal rate rLU , and
hence is asymptotically optimal.
We present the proof of Theorem 3 as it helps understand the role of
Proposition 3 and our bandwidth choices.
PROOF OF THEOREM 3: We have fˆŽ ¨ < x . s fˆŽ ¨ , x .rfˆŽ x .. Given the optimal
bandwidth choice for h X in A4, we know that fˆŽ x . converges uniformly to f Ž x .
at the rate Ž Lrlog L.Ž Rq1.rŽ2 Rqdq2. on any inner compact subset of its support
Žsee Hardle Ž1991... Because this rate is faster than that of the theorem and
f Ž x . is bounded away from 0 by A2-Žii., it suffices to show that fˆŽ ¨ , x . converges
at the rate Ž Lrlog L. R rŽ2 Rqdq3..
Our proof relies upon the Žinfeasible . nonparametric estimator of the density
of Ž V, X . using the unobserved true private values Vp l :
Ž 22.
f˜Ž ¨ , x . s
L
1
Ý
Lh dq1
f
ls1
Il
1
Ý
Il
Kf
ps1
ž
¨ y Vp l
hf
,
xy X l
hf
/
.
Lemma B2 in Appendix B implies that the suboptimal bandwidth h f leads
to a uniform convergence of f˜Ž ¨ , x . to f Ž ¨ , x . on C Ž V . at the rate Ž Lr
log L. R rŽ2 Rqdq3.. Since fˆŽ ¨ , x . y f Ž ¨ , x . s w fˆŽ ¨ , x . y f˜Ž ¨ , x .x q w f˜Ž ¨ , x . y
f Ž ¨ , x .x, we are left with the first term.
Let C X Ž V . be an inner closed subset of SŽ f Ž ¨ , x .. containing all hypercubes
of size ␦ Žsmall enough. centered at a point Ž ¨ , x . in C Ž V .. Define C Y Ž V .
similarly with respect to C X Ž V .. Hence C Ž V . ; C X Ž V . ; C Y Ž V . ; S Ž f Ž ¨ , x ...
Now, for Ž ¨ , x . g C Ž V . and L large enough, fˆŽ ¨ , x . uses at most observations
Ž Vˆp l , X l . in C X Ž V . and hence for which Ž Vp l , X l . is in C Y Ž V . by Proposition 3-Ži..
Because f˜Ž ¨ , x . uses at most Ž Vp l , X l . in C Y Ž V . for any Ž ¨ , x . in C Ž V ., we
obtain almost surely for L large enough,
fˆŽ ¨ , x . y f˜Ž ¨ , x .
s
L
1
ls1
Il
1
Lh dq1
f
ž ž
Ý
= Kf
Il
Ý
¨ y Vˆp l
hf
|C Y ŽV . Ž Vp l , X l .
ps1
,
xy X l
hf
/ ž
yKf
¨ y Vp l
hf
,
xy X l
hf
Next, a second-order Taylor expansion gives
Ž 23.
< fˆŽ ¨ , x . y f˜Ž ¨ , x . <
F
1
Lh dq1
f
L
1
ls1
Il
Ý
Il
Ý
ps1
ž
|C Y ŽV . Ž Vp l , X l . Vˆp l y Vp l
/
//
.
545
FIRST PRICE AUCTIONS
ž
q
=
F
/
1 ⭸ K f ¨ y Vp l xy X l
,
hf ⭸ ¨
hf
hf
=
L
1
Ý
2 Lh dq1
f
ls1
1
h2f
⭸¨
¨
Ý
Il
⭸ 2K f
sup
Il
1
2
ž
|C Y ŽV . Ž Vp l , X l . Vˆp l y Vp l
ps1
ž
¨,
xy X l
hf
1
hf
1
Lh dq
f
L
1
ls1
Il
q
Ý
ps1
/
sup p , l|C Y ŽV . Ž Vp l , X l . < Vˆp l y Vp l < 2
1
2 h 3f
Lh df
⭸ 2K f
L
= Ý sup
ls1
FO
ž
⭸ K f ¨ y Vp l xy X l
,
⭸¨
hf
hf
Il
⭸¨
¨
žž /
log L
2
ž
¨,
L
1
ls1
Il
qO
Il
Ý
ps1
log L
ž
/
/
1
1
Lh dq
f
⭸ K f ¨ y Vp l xy X l
,
⭸¨
hf
hf
žž /
Ž2 Ry1 .r Ž2 Rqdq3 .
L
L
= Ý sup
ls1
hf
Rr Ž2 Rqdq3 .
L
=Ý
xy X l
¨
⭸ 2K f
⭸¨
2
ž
¨,
2
/
sup p , l|C Y ŽV . Ž Vp l , X l . < Vˆp l y Vp l <
=Ý
/
xy X l
hf
/
/
/
1
Lh df
,
by Proposition 3-Žii. and the definition of h f in A4-Žii.. The two sums appearing
in Ž23. may be viewed as kernel estimators, and hence converge uniformly on
C Ž V . to
f Ž¨ , x.
H
⭸Kf
⭸¨
Ž u, y . du dy and
f Ž x.
H sup
¨
⭸ 2K f
⭸ 2¨
Ž ¨ , y . dy,
respectively. Thus they stay bounded almost surely. Since R G 1 implies Ž2 R y
1.rŽ2 R q dq 3. G RrŽ2 R q dq 3., it follows that fˆŽ ¨ , x . y f˜Ž ¨ , x . s O Žlog Lr
L. R rŽ2 Rqdq3..
Q. E. D.
546
E. GUERRE, I. PERRIGNE, AND Q. VUONG
The above proof shows that, under our bandwidth choice A4, f˜Ž ¨ , x . y f Ž ¨ , x .
and the first-order term in the Taylor expansion of fˆŽ ¨ , x . y f˜Ž ¨ , x . have the
same order of magnitude, namely O ŽŽlog LrL. R rŽ2 Rqdq3. ., while the secondorder term is smaller. In fact, given our choice for h G and h g , it can be shown
that our choice for h f , which implies oversmoothing, is the only choice that
achieves the optimal uniform convergence rate Ž Lrlog L. R rŽ2 Rqdq3. for estimating the density f Ž ¨ , x . for any combination Ž d, R . of dimension and smoothness. In particular, the standard ‘‘optimal’’ choice Žlog LrL.1rŽ2 Rqdq1. for h f
would lead to a suboptimal uniform convergence rate for estimating f Ž ¨ , x ..
There may, however, exist alternative bandwidth choices for Ž h G , h g , h f . leading
to the optimal uniform convergence rate Ž Lrlog L. R rŽ2 Rqdq3.. Our choice for
h G and h g , which corresponds to the optimal bandwidth rates for estimating
GŽ⭈, ⭈ , i . and g Ž⭈, ⭈ , i . Žsee Hardle Ž1991.. have the advantage of providing an
estimate of the inverse equilibrium strategy ␰ Ž⭈, ⭈ , i . for every i with the best
uniform convergence rate.
Theorem 3 is useful in practice only if the support SŽ f Ž ¨ , x .. is known so that
one can choose appropriately the compact set C Ž V .. This support can be
estimated. Specifically, because ¨ Ž x . s bŽ x ., then the lower bound ¨ Ž⭈. is estimated from observed bids using Ž17. directly. Regarding the upper bound ¨ Ž⭈.,
we can use an estimator of the form Ž17. with inf and Bp l replaced by sup and
Vˆp l , respectively. Thus from Proposition 3-Žii. and a proof analogous to that of
Proposition 2, it can be readily shown that the support SŽ f Ž ¨ , x .. can be
estimated uniformly.
Lastly, asymptotic normality of our estimator is not covered by Theorem 3.
The proof of Theorem 3 indicates that classical asymptotic normality results are
likely to be imprecise because only the leading term in the expansion of
fˆŽ ¨ , x . y f Ž ¨ , x . is used. In our case, the second order term of the Taylor
expansion can be close to the first one, especially if the degree of smoothness R
is small. Such drawbacks can be circumvented by establishing an exponential-type
inequality for all L of the form
Ž 24.
Pr
žž /
L
log L
Rr Ž2 Rqdq3 .
sup
Ž ¨ , x .g C Ž V .
/
< fˆŽ ¨ < x . y f Ž ¨ < x . < ) eUf Ž t . F PfU Ž t . ,
for t ) 0 and some positive functions eUf Ž t . and PfU Ž t ., analogous to the
inequalities obtained in Lemma C4. Though we shall not pursue such an issue,
which is outside the scope of the paper, such an inequality is useful in practice
for two reasons.
First, Ž24. can be used to obtain conservative Žuniform. confidence intervals.
Specifically, choose t such that PfU Ž t . F ␣ . Hence the probability that f Ž ¨ < x . is
in the interval
fˆŽ ¨ < x . y eUf Ž t .Ž log LrL .
Rr Ž2 Rq3 .
, fˆŽ ¨ < x . q eUf Ž t .Ž log LrL .
Rr Ž2 Rq3 .
for all Ž ¨ , x . g C Ž V .
547
FIRST PRICE AUCTIONS
is larger than 1 y ␣ . Second, Ž24. can be used to choose the constants ␭
appearing in the definitions of the various bandwidths. Indeed, Fubini’s Theorem yields
E
sup
Ž ¨ , x .g C Ž V .
s
q⬁
H0
< fˆŽ ¨ < x . y f Ž ¨ < x . <
Pr
ž
sup
Ž ¨ , x .g C Ž V .
/
< fˆŽ ¨ < x . y f Ž ¨ < x . < ) t dt.
Combined with Ž24., this gives an upper bound for EwsupŽ ¨ , x .g C ŽV . < fˆŽ ¨ < x . y
f Ž ¨ < x .<x, which can be used to assess the choice of the ␭’s based on the expected
supnorm loss function.
4. Reser¨ ation Price and Number of Bidders
Up to now, we have assumed that the reservation price is nonbinding. In
practice, the seller may announce a reservation price sufficiently high prior to
the bidding as a screening device for participating in the auction. Though the
Bayesian Nash equilibrium strategy is still given by Ž1., a binding reservation
price raises a new difficulty due to the unobserved number I of potential
bidders. This number is typically different from the observed number I U of
actual bidders who have proposed a bid ŽG p 0 .. Hence there is a new structural
element, namely I, in addition to the latent distribution of bidders’ private
values F Ž⭈.. In this section we show how our results can be extended to this
situation.
4.1. Nonparametric Identification
As in Section 2.2, we first consider one auction only and hence we omit the
subscript l. Alternatively, our reasoning can be viewed as conditional upon
Ž x, p 0 , I .. Following the derivation leading to Ž3., we rewrite the differential
equation Ž2. in terms of observables. Unlike Section 2, however, a binding
reservation price p 0 introduces a truncation because a potential bidder with a
private value lower than p 0 does not bid. Let bUi denote the equilibrium bid of
the ith actual bidder, i s 1, . . . , I U , and GU Ž⭈. be its distribution. Thus GU Ž bU . s
PrŽ sŽ ¨˜. F bU < ¨˜G p 0 . s w F Ž ¨ . y F Ž p 0 .xrw1 y F Ž p 0 .x where ¨ s sy1 Ž bU .. Differentiating with respect to bU gives the conditional density g U Ž bU . s Ž1r
sX Ž ¨ ..Ž f Ž ¨ .rŽ1 y F Ž p 0 .... Hence, from Ž2. elementary algebra gives
Ž 25.
¨ i s ␰ Ž bUi , GU , I, F Ž p 0 ..
' bUi q
1
Iy1
ž
GU Ž bUi .
g
U
Ž bUi .
q
F Ž p0 .
1
U
1 y F Ž p 0 . g Ž bUi .
/
,
548
E. GUERRE, I. PERRIGNE, AND Q. VUONG
for i s 1, . . . , I U . Equation Ž25. is the analog of Ž3., but involves I and F Ž p 0 .,
which are unknown. This complicates the identification and estimation of the
model.
The next result solves the identification problem.
THEOREM 4: Let GU Ž⭈. g P with support w p 0 , b x, and ␲ Ž⭈. be a discrete
distribution. There exist a distribution of buyers’ pri¨ ate ¨ alues F Ž⭈. g P and a
number I G 2 of potential buyers such that Ž i . GU Ž⭈. is the truncated distribution of
the equilibrium bid in a first-price sealed-bid auction with reser¨ ation price p 0 g Ž ¨ , ¨ .
and Ž ii . ␲ Ž⭈. is the distribution of the number of actual bidders I U if and only if the
following conditions hold:
C1: ␲ Ž⭈. is Binomial with parameters Ž I, 1 y F Ž p 0 .., where 0 - F Ž p 0 . - 1.
C2: The obser¨ ed bids are i.i.d. as GU Ž⭈. conditionally upon I U and
lim b x p 0 g U Ž b . s q⬁.
C3: The function ␰ Ž⭈, GU , I, F Ž p 0 .. defined in Ž25. is strictly increasing on
w p 0 , b x and its in¨ erse is differentiable on w ¨ , ¨ x ' w ␰ Ž p 0 , GU , I, F Ž p 0 .., ␰ Ž b, GU ,
I, F Ž p 0 ..x.
Moreo¨ er, if Conditions C1᎐C3 hold, then I and F Ž p 0 . are unique while F Ž⭈. is
uniquely defined on w p 0 , ¨ x as F Ž⭈. s F Ž p 0 . q w1 y F Ž p 0 .xGU Ž ␰y1 Ž⭈, GU , I, F Ž p 0 ....
In addition, ␰ Ž⭈, GU , I, F Ž p 0 .. is the quasi in¨ erse of the equilibrium strategy
sŽ⭈, F, p 0 , I . in the sense that ␰ Ž b, GU , I, F Ž p 0 .. s sy1 Ž b, F, p 0 , I . for all bg
w p 0 , b x.
Theorem 4 parallels Theorem 1. In particular, it shows that the game
theoretical model imposes some restrictions on the joint distribution of the
observables Ž bU1 , . . . , bUI U , I U .. Moreover, the number of potential bidders I is
identified, while the latent private values distribution F Ž⭈. is identified nonparametrically on w p 0 , ¨ x.
4.2. Nonparametric Estimation
As in Section 3 we consider heterogeneous auctioned objects characterized by
X l with the difference that the reservation price is now binding. The next
assumption clarifies the nature of the number of potential bidders and the
reservation price.
ASSUMPTION A5:
Ži. The number of potential bidders I G 2 is constant.
Žii. The reser¨ ation price P0 is a possibly unknown deterministic Ž R q 1. continuously differentiable function hŽ⭈. of the characteristics X.
Žiii. For some ⑀ ) 0, ¨ Ž x . q ⑀ F hŽ x . F ¨ Ž x . y ⑀ for all x g w x, x x.
By A5-Ži., the size of the market is assumed constant across auctions. Though
known to potential bidders in the theoretic model, I is unknown to the analyst,
549
FIRST PRICE AUCTIONS
as is typically the case.23 Assumption A5-Žii. is suited when the seller determines
the reservation price as a function of the object’s characteristics, as it is the case
in practice. In particular, this is satisfied when the seller’s private value V0 is a
function of the object’s characteristics, and the reservation price is determined
optimally Žsee Riley and Samuelson Ž1981...24 Assumption A5-Žiii. requires that
the reservation price be bounded away from ¨ Ž x . and ¨ Ž x .. This ensures that
the reservation price is always binding and that there is always a positive
probability of having at least one actual bidder. For instance, given A2, A5-Žiii.
is satisfied when the reservation price is chosen optimally and ¨ Ž X . F V0 F ¨ Ž X ..
Given A5-Žii., the conditional distribution of an observed bid BUp l given
Ž X l , P0 l . is the conditional distribution given X l only, namely GU Ž⭈< X l . with
support w hŽ X l ., bŽ X l .x. Thus, for ps 1, . . . , IlU and l s 1, . . . , L, Ž25. becomes
Ž 26.
Vp l s ␰ Ž BUp l , X l .
' BUp l q
1
Iy1
ž
GU Ž BUp l < X l .
g
U
Ž
BUp l <
Xl .
q
/
⌽ Ž Xl .
1
,
U
1 y ⌽ Ž X l . g Ž BUp l < X l .
where ⌽ Ž X l . ' F Ž P0 l < X l .. Provided one can estimate I and ⌽ Ž X l ., this equation
can be used to develop a two-step estimation procedure analogous to that of
Section 3. A technical difficulty arises as the density g U Ž⭈< X l . is unbounded at
BU s P0 l Žsee C2 of Theorem 4.. This is because sU Ž P0 l , X l . s 0. Hence Theorems 2 and 3 no longer apply.
In fact, the density g U Ž⭈< X l . behaves as 1r bU y P0 l in the neighborhood of
P0 l since the behavior of sŽ⭈, X l . y P0 l is quadratic in the neighborhood of P0 l .
This leads to considering the transformation B† s Ž BU y P0 .1r2 . Hence B† s
s† Ž V, X ., where s† Ž V, X . s w sŽ V, X . y P0 x1r2 with V G P0 and P0 s hŽ X .. Thus
the distribution of B† is G† Ž b† < X . s GU Ž P0 q b†2 < X . with bounded density
g † Ž b† < X . s 2 b† g U Ž P0 q b†2 < X . on its support w0, b† Ž X .x s w0, Ž bŽ X . y P0 .1r2 x.
Under A5-Žii., Ž26. becomes
'
Ž 27.
Vp l s ␰† Ž B† p l , X l .
' P0 l q B†2p l q
2 B† p l
Iy1
ž
G† Ž B† p l , X l .
g † Ž B† p l , X l .
q
/
⌽ Ž Xl .
f Ž Xl .
,
1 y ⌽ Ž X l . g † Ž B† p l , X l .
using G† Ž⭈< ⭈ .rg † Ž⭈< ⭈ . s G† Ž⭈, ⭈ .rg † Ž⭈, ⭈ . and 1rg † Ž⭈< ⭈ . s f Ž⭈.rg † Ž⭈, ⭈ ..
23
In fact, we only need that I be constant on some known subsets of auctions, in which case our
estimation method applies to each subset. Tests of the constancy of I can be based on estimates of I
Žsee below. for each subset. Because of A5-Ži., hereafter I is dropped as an argument from any
function.
24
If there are many sellers across auctions, A5-Žii. may no longer hold as hŽ⭈. may vary across
sellers. Thus P0 given X can be viewed as stochastic, which leads to a random truncation given X
Žsee Wang, Jewell, and Tsai Ž1986... It can be shown that our two-step estimator is still uniformly
consistent on compact subsets at the rate Ž Lrlog L. R rŽ2 Rqdq4..
550
E. GUERRE, I. PERRIGNE, AND Q. VUONG
PROPOSITION 4: Gi¨ en A2 and A5, the conditional distribution G† Ž⭈< ⭈ . satisfies
properties Ž i . ᎐ Ž i¨ . of Proposition 1, where GŽ⭈< ⭈ , ⭈ ., S i Ž G ., bŽ x, i ., bŽ x, i ., and
Ci Ž B . are replaced respecti¨ ely by G† Ž⭈< ⭈ ., SŽ G† . s Ž b† , x . : xg w x, x x, b† g
w0, b† Ž x .x4 , 0, b† Ž x ., and C Ž B† ., which is a closed inner subset of S Ž G† ..
Proposition 4 is important as it is analogous to Proposition 1, which is the
backbone of the optimal rate and uniform consistency of our estimator ŽTheorems 2 and 3..
Analogously to Section 3, Ž27. suggests a two-step estimation procedure. In
the first step, f Ž⭈. is estimated by Ž21., while G† Ž⭈, ⭈ . and g † Ž⭈, ⭈ . are estimated by
˜† Ž b† , x . s
G
˜g † Ž b† , x . s
L
1
Ý
d
Lh G
ls1
Ý
|
IlU ps1
L
1
IlU
1
Ý
Lh dq1
g
ls1
Ž B† p l F b† . K G
IlU
1
Ý
IlU ps1
Kg
ž
ž
xy X l
hG
/
/
,
b† y B† p l xy X l
,
.
hg
hg
U
U
.
This follows because the transformed actual bids Ž B†1
l , . . . , B†I lU l are indepenU
dent of Il conditionally upon X l in view of Condition C2 of Theorem 4.
In the second step, Ž27. is used to recover the pseudo private values provided
one can estimate I and ⌽ Ž X l .. A natural estimator for I is Iˆs max ls1, . . . , L IlU .
Given Condition C1 of Theorem 4, Iˆs I almost surely. To estimate ⌽ Ž X l ., we
note that Ew IlU < X l x s I w1 y ⌽ Ž X l .x. Solving for ⌽ Ž X l . suggests to estimate it by
ˆ Ž X l ., where
⌽
ˆŽ x. s1y
⌽
1
ˆ
ILh dx
fˆŽ x .
L
Ý IlU K x
ls1
ž
xy X l
hx
/
,
using the usual Nadaraya Ž1964. ᎐Watson Ž1964. nonparametric regression estiˆ Ž⭈. is always between 0 and 1.
mator with fˆŽ x . as in Ž21.. Note that ⌽
Let ␰˜† Ž⭈, ⭈ . be given by Ž27., where I, ⌽ Ž⭈., f Ž⭈., G† Ž⭈, ⭈ ., and g † Ž⭈, ⭈ . are
replaced by their estimators. Following Ž16., let ˆ
b† Ž x . s sup B† p l , ps 1, . . . , IlU ,
l s 1, . . . , L, X l g ␲ k 1 , . . . , k d 4 . The estimated support of G† Ž⭈, ⭈ . is SˆŽ G† . '
Ž b, x . : bg w0, ˆ
b† Ž x .x, x g w x, x x4 . The pseudo private values are given by Vˆp l s
␰ˆ† Ž B† p l , X l ., where
¡␰˜ Ž b , x .
if Ž b† , x . q S Ž 2 h G . ; SˆŽ G† . ,
~
Ž b† , x . q S Ž 2 h g . ; SˆŽ G† . ,
†
␰ˆ† Ž b† , x . '
¢q⬁
†
xq S Ž 2 h f . g w x, x x ,
otherwise,
for p s 1, . . . , IlU , l s 1, . . . , L. The nonparametric estimator of the conditional
ˆ Ž x .x fˆU Ž ¨ < x ., where fˆU Ž ¨ < x . is the estimator of
density f Ž ¨ < x . is fˆŽ ¨ < x . ' w1 y ⌽
the truncated conditional density f U Ž ¨ < x . s f Ž ¨ < x .rw1 y ⌽ Ž x .x and is obtained
FIRST PRICE AUCTIONS
551
as in Section 3 with Il replaced by IlU in Ž20.. Let SU ' Ž ¨ , x . : x g w x, x x,
¨ g w hŽ x ., ¨ Ž x .x4 be the support of f U Ž⭈< ⭈ .. The next result gives the asymptotic
properties of the estimator fˆŽ⭈< ⭈ ..
THEOREM 5: Suppose that A1᎐A5 hold. Then fˆŽ⭈< ⭈ . is uniformly consistent with
optimal rate Ž Lrlog L. R rŽ2 Rqdq3. on any closed inner subset of SU with nonempty
interior.
In view of Theorem 4 and A5-Žii., SU is the largest subset of the support of
f Ž⭈< ⭈ . where the latter is identified.
5.
CONCLUSION
For first-price sealed-bid auctions with binding or nonbinding reservation
prices, we have shown that the underlying distribution of bidders’ private values
within the independent private values paradigm is identified from observables,
namely, observed bids and the number of actual bidders, without any parametric
assumptions. Moreover, using the recently developed minimax theory, we have
established the best rate of uniform convergence at which the latent density of
private values can be estimated from available data. We then have proposed a
computationally convenient two-step nonparametric estimator of this density
that converges at this optimal rate.25
As a matter of fact, our results go well beyond the auction mechanism and
paradigm studied here as they can be generalized to other auction models as
considered by Milgrom and Weber Ž1982.. For instance, as shown by Guerre,
Perrigne, and Vuong Ž1995. for the independent private value paradigm, they
apply to descending or Dutch auctions, where only the winning bid, if any, is
observed. Alternatively, our results can be extended to the affiliated private
value paradigm, which is the most general auction model that can be identified
from observed bids Žsee Laffont and Vuong Ž1996... Li, Perrigne, and Vuong
Ž1996, 1999. show how our two-step nonparametric procedure can be modified
accordingly despite the complexity of the Bayesian Nash equilibrium strategy.
More generally, because the determination of the equilibrium strategy is
avoided, our indirect estimation procedure is especially convenient when the
equilibrium strategy cannot be obtained explicitly. In such situations, direct
Žparametric. estimation methods become cumbersome, if not computationally
25
Alternatively, our general indirect two-step principle can also be used to estimate parametrically
the distribution of observed bids in the first step and the distribution of private values in the second
step. Besides some technical difficulties arising from the relationship between GŽ⭈. and F Ž⭈., such a
parametric two-step procedure would constitute a powerful alternative to existing parametric
methods such as the simulation-based method proposed in Laffont, Ossard, and Vuong Ž1995..
Specifically, this new parametric method will be computationally much easier when the equilibrium
strategy cannot be easily simulated or cannot be solved explicitly as in asymmetric auctions.
552
E. GUERRE, I. PERRIGNE, AND Q. VUONG
infeasible, since the differential equation must be solved numerically for the
equilibrium strategy for each trial value of the parameters. An example of such
a situation arises with a random reservation price. This occurs when the seller
adopts a random rejection rule of the highest bid, or when the reservation price
is announced after bidding. Within the independent private values paradigm,
Elyakime, Laffont, Loisel, and Vuong Ž1994, 1997. extend our results to the
latter situation. Another important case arises when bidders are asymmetric ex
ante due to joint bidding, informational advantages such as in Outer Continental Shelf auctions Žsee Hendricks and Porter Ž1988., Hendricks, Porter, and
Wilson Ž1994.., the large size of a bidder, and more generally collusion among
some bidders Žsee Graham and Marshall Ž1987... In this case, the equilibrium
strategies are solutions of a system of differential equations that cannot be
solved explicitly Žsee Maskin and Riley Ž1983... Using results similar to ours,
Campo, Perrigne, and Vuong Ž1998. and Laffont, Li, and Vuong Ž1999. show
how to circumvent this difficulty. Asymmetry also arises in procurement actions
when firms bid in both price and quality. Applying our approach, Laffont,
Oustry, Simioni, and Vuong Ž1996. identify and estimate the resulting model.
Lastly, an important feature of our identification results is that they provide
necessary and sufficient conditions for the existence of a latent distribution that
can ‘‘rationalize’’ the observed bid distribution. Because they are the restrictions
imposed by the game theoretic model on observables, these conditions can
constitute the basis of a test of the theory. They are of two types. The first type
deals with independence of bids Žsee conditions C1 and C2 of Theorems 1 and 4,
respectively., which relates to the paradigm and can be tested nonparametrically
using, e.g., the Blum, Kiefer, and Rosenblatt Ž1961. test. The second type deals
with the monotonicity of an estimable function Žsee conditions C2 and C3 of
Theorem 1 and 4, respectively., which can be used to test whether bidders adopt
the symmetric Bayesian Nash equilibrium strategy. A major difficulty in developing a nonparametric test, however, is to estimate the function ␰ Ž⭈. under
monotonicity constraint. Though progress has been made in the regression
context Žsee, e.g., Wright and Wegman Ž1980. and Utreras Ž1985.., the different
nature of ␰ Ž⭈. calls for further work.
To conclude, by proposing a general and computationally convenient estimation principle, this paper contributes to the structural analysis of auction data
that was plagued by the numerical complexities associated with the equilibrium
strategies. The estimation of auction models can also be considered as a first
step in the estimation of asymmetric information models used in the theory of
regulation and contracts. Hence our paper contributes to the agenda of the new
empirical industrial organization calling for the data evaluation of game theoretic models Žsee Laffont and Tirole Ž1993. and Salanie
´ Ž1997...
Uni¨ ersite´ de Jussieu, LSTA-ISUP, Boite 158, 4, Place Jussieu, 75252 Paris Cedex
05, France,
FIRST PRICE AUCTIONS
553
Dept. of Economics, Uni¨ ersity of Southern California, Los Angeles, CA 900890253, U.S.A.
and
Dept. of Economics, Uni¨ ersity of Southern California, Los Angeles, CA 900890253, U.S.A.
Manuscript recei¨ ed April, 1995; final re¨ ision recei¨ ed April, 1998.
APPENDIX A
PROOFS
OF
MATHEMATICAL PROPERTIES
This Appendix gives Ži. the proofs of our identification results ŽTheorems 1 and 4. and Žii.
establishes the model’s regularity properties ŽPropositions 1 and 4., when the reservation price is
nonbinding and binding respectively. Lemmas are proved in Appendix C.
A.1. Proofs of Identification Results
PROOF OF THEOREM 1: First, we prove that Conditions C1 and C2 are necessary. Because
bi s sŽ ¨ i , F, I . and the ¨ i ’s are i.i.d., it follows that the bi ’s are i.i.d. so that Condition C1 must hold.
To prove Condition C2, let sŽ⭈, F, I . be the strictly increasing differentiable and symmetric Bayesian
Nash equilibrium strategy corresponding to F Ž⭈. with support w ¨ , ¨ x Žsay.. Let GŽ⭈. be the distribution defined by GŽ b . s F Ž sy1 Ž b, F, I .. for every bg w b, b x ' w sŽ ¨ , F, I ., sŽ ¨ , F, I .x. Note that GŽ⭈.
must be the distribution of observed Žequilibrium. bids. Now, sŽ⭈, F, I . must solve the first-order
differential equation Ž2.. But, because Ž3. follows from Ž2., then sŽ⭈, F, I . must satisfy ␰ Ž sŽ ¨ , F, I .,
G, I . s ¨ for all ¨ g w ¨ , ¨ x. Making the change of variable bs sŽ ¨ , F, I ., we obtain ␰ Ž b, G, I . s
sy1 Ž b, F, I . for all b g w b, b x. Hence Condition C2 must hold because sy1 Ž⭈, F, I . is strictly
increasing on w b, b x and sŽ⭈, F, I . differentiable on w ¨ , ¨ x s w ␰ Ž b, G, I ., ␰ Ž b, G, I .x.
To prove sufficiency, let GŽ⭈. belong to P I with support w b, b x I. By Condition C1, the bi ’s are
i.i.d., each distributed as some GŽ⭈. with support w b, b x. Now note that
Ž A.1.
lim ␰ Ž b, G, I . s b.
bx b
This follows from Ž3. and the fact that Ži. b is finite ŽG 0., Žii. lim b x b log GŽ b . s y⬁, and Žiii.
g Ž b .rGŽ b . s d log GŽ b .rdb, so that lim b x b g Ž b .rGŽ b . s q⬁.
Next define F Ž⭈. ' F Ž ␰y1 Ž⭈, G, I .. on w ¨ , ¨ x, where ¨ ' ␰ Ž b, G, I . s b by ŽA.1. and ¨ ' ␰ Ž b, G, I ..
Thus F Ž⭈. is a valid distribution because ␰ Ž⭈, G, I . is strictly increasing on w b, b x by Condition C2.
Moreover, because GŽ⭈. is strictly increasing on w b, b x Žsee footnote 3., then F Ž⭈. is strictly increasing
on w ¨ , ¨ x. Thus the support of F Ž⭈. is w ¨ , ¨ x, which is an interval of ⺢q. Lastly, because ␰y1 Ž⭈, G, I . is
differentiable and GŽ⭈. is absolutely continuous, then F Ž⭈. is absolutely continuous. Therefore F Ž⭈.
belongs to P as required.
It remains to show that this distribution F Ž⭈. of buyers’ private values can rationalize GŽ⭈. in a
first-price sealed-bid auction with no reservation price, i.e., that GŽ⭈. s F Ž sy1 Ž⭈, F, I .. on w b, b x,
where sŽ⭈, F, I . solves Ž2. with the boundary condition sŽ ¨ , F, I . s ¨ . By construction of F Ž⭈., we
have GŽ⭈. s F Ž ␰ Ž⭈, G, I ... Thus it suffices to show that ␰y1 Ž⭈, G, I . solves Ž1. with boundary
condition ␰y1 Ž ¨ , G, I . s ¨ . It is easy to see X that the boundary condition is satisfied. From the
construction of F Ž⭈. note that f Ž⭈.rF Ž⭈. s ␰y1 Ž⭈, G, I . g Ž ␰y1 Ž⭈, G, I ..rGŽ ␰y1 Ž⭈, G, I .. after differen-
554
E. GUERRE, I. PERRIGNE, AND Q. VUONG
tiating and taking a ratio. Thus ␰y1 Ž⭈, G, I . solves Ž2. if
Ž A.2.
1 s Ž ¨ y ␰y1 Ž ¨ , G, I ..Ž I y 1 .
g Ž ␰y1 Ž ¨ , G, I ..
᭙¨ g w¨ , ¨ x.
G Ž ␰y1 Ž ¨ , G, I ..
But ŽA.2. clearly holds by definition of ␰ Ž⭈, G, I .. This completes the proof of sufficiency.
It remains to establish the last part of Theorem 1. From the proof of necessity, we know that
␰ Ž⭈, G, I . s sy1 Ž⭈, F, I . holds when F Ž⭈. exists. Since F Ž⭈. s GŽ sŽ⭈, F, I .., then F Ž⭈. s GŽ ␰y1 Ž⭈, G, I ...
Because ␰ Ž⭈, G, I . is uniquely determined by GŽ⭈., it follows that F Ž⭈. is unique given GŽ⭈..
Moreover, its support must be w ␰ Ž b, G, I ., ␰ Ž b, G, I .x s w b, ␰ Ž b, G, I .x by ŽA.1..
Q. E. D.
PROOF OF THEOREM 4: Consider necessity. Condition C1 must hold since a potential bidder bids
if and only if his private value is at least p 0 . Thus 0 - F Ž p 0 . - 1 follows from p 0 g Ž ¨ , ¨ ..
i
Regarding Condition C2, let Bj s 0 if Vj - p 0 for j s 1, . . . , I. For any Ž b1 , . . . , bi . g ⺢q
,
U
<
.
Pr Ž B1U F b1 , . . . , BU
i F bi , I s i I, p 0
s
Ý
1Fr 1 / ⭈⭈⭈ /r iFI
s
s
I!
i! Ž I y i . !
I!
i! Ž I y i . !
Pr Ž p 0 F Br 1 F b1 , . . . , p 0 F Br i F bi , Bj s 0, j f r1 , . . . , ri 4< I, p 0 .
Pr Ž p 0 F B1 F b1 , . . . , p 0 F Bi F bi , Biq1 s ⭈⭈⭈ s BI s 0 < I, p 0 .
i
Pr Iy i Ž Biq 1 s 0 < I, p 0 .
Ł Pr Ž p0 F Br F br < I, p0 . ,
rs1
because Bj ’s are iid given Ž I, p 0 .. Since I U is binomial with parameters Ž I, 1 y F Ž p 0 .., then
< U
.
Pr Ž B1U F b1 , . . . , BU
i F bi I s i , I, p 0 s
i
Ł
rs1
i
s
Ł
rs1
Pr Ž p 0 F Br F br < I, p 0 .
1 y F Ž p0 .
y1 Ž
FŽ s
br .. y F Ž p 0 .
1 y F Ž p0 .
,
X
as desired. Moreover, because sŽ p 0 . s p 0 and F Ž p 0 . ) 0 it follows from Ž2. that lim ¨ x p 0 s Ž ¨ .rf Ž ¨ .
X
s 0. Since g U Ž b . s f Ž sy1 Ž b ..rw s Ž sy1 Ž b ..Ž1 y F Ž p 0 .x, then lim b x p 0 g U Ž b . s q⬁. Regarding the
necessity of Condition C3, the proof is similar to that of Condition C2 in Theorem 1, noting that
␰ Ž⭈, GU , I, F Ž p 0 .. s sy1 Ž⭈, F, I, p 0 . on w p 0 , b x s w p 0 , sŽ ¨ , F, I, p 0 .x.
Turning to sufficiency, choose ¨ - p 0 and 0 - F Ž p 0 . - 1. The proof then follows that of Theorem
1 by replacing GŽ⭈. and F Ž⭈. by the truncated distributions GU Ž⭈. and F U Ž⭈. ' w F Ž⭈. y F Ž p 0 .xrw1 y
F Ž p 0 .x defined on w p 0 , b x and w p 0 , ¨ x, respectively. The condition lim b x p 0 ␰ Ž b, GU , I, F Ž p 0 .. s p 0 ,
which is analogous to ŽA.1., follows now from Ž25. and lim b x p 0 g U Ž b . s q⬁. In particular, if
Conditions C1᎐C3 hold, we can establish that there exists a distribution F Ž⭈. on w ¨ , ¨ x with
¨ s ␰ Ž b, GU , I, F Ž p 0 .. that Ži. is absolutely continuous on w p 0 , ¨ x and Žii. can rationalize GU Ž⭈. in a
first-price sealed-bid auction with reservation price p 0 . A distribution F Ž⭈. g P is obtained by
extension in an absolute continuous fashion to the interval w ¨ , p 0 x.
To prove the last part of Theorem 4, note that I and F Ž p 0 . are identified from ␲ Ž⭈.. Moreover, similar to the end of the proof of Theorem 1, we have ␰ Ž⭈, GU , I, F Ž p 0 .. s sy1 Ž⭈, F, I, p 0 . on
w p 0 , b x, and the truncated distribution F U Ž⭈. ' w F Ž⭈. y F Ž p 0 .xrw1 y F Ž p 0 .x is unique and equal to
GU Ž ␰y1 Ž⭈, GU , I, F Ž p 0 ... with support w p 0 , ¨ x s w p 0 , ␰ Ž b, GU , I, F Ž p 0 ..x. Because I and F Ž p 0 . are
identified, it follows that F Ž⭈. is uniquely determined on w p 0 , ¨ x and equal to F Ž p 0 . q w1 y
F Ž p 0 .xGU Ž ␰y1 Ž⭈, GU , I, F Ž p 0 ... on this interval.
Q. E. D.
A.2. Proofs of Regularity Properties
To prove Proposition 1 we need two lemmas. Lemma A1 studies the regularity of the boundaries
¨ Ž⭈. and ¨ Ž⭈. as well as that of the bid function sŽ⭈, ⭈ , i .. Lemma A2 translates these regularity
properties into regularity of bŽ⭈, i ., bŽ⭈, i ., and ␰ Ž⭈, ⭈ , i ..
555
FIRST PRICE AUCTIONS
LEMMA A1: Under A2:
Ži. The boundaries ¨ Ž⭈. and ¨ Ž⭈. admit up to R q 1 continuous bounded deri¨ ati¨ es on w x, x x, and
inf x g w x, x xŽ ¨ Ž x . y ¨ Ž x .. ) 0.
Žii. For each i g I , sŽ⭈, ⭈ , i . admits up to R q 1 continuous bounded partial deri¨ ati¨ es on Si Ž F ..
X
Moreo¨ er, for any Ž ¨ , x . g Si Ž F ., s Ž ¨ , x, i . G c s ) 0.
LEMMA A2: Under A2, for each i g I , we ha¨ e the following:
Ži. The boundaries bŽ x, i . and bŽ x, i . admit up to R q 1 continuous bounded deri¨ ati¨ es on w x, x x,
and inf x g w x, x xŽ bŽ x, i . y bŽ x, i .. ) 0.
Žii. ␰ Ž⭈, ⭈ , i . admits up to R q 1 continuous bounded partial deri¨ ati¨ es on Si Ž G .. Moreo¨ er, for any
Ž b, x . g Si Ž G ., ␰ X Ž b, x, i . G c␰ ) 0.
PROOF
Ž A.3.
OF
PROPOSITION 1: Ži. is Lemma A2-Ži. plus the boundary condition. Next,
g Ž b < x, i . s
f Ž ␰ Ž b, x, i . < x, i .
X
s Ž ␰ Ž b, x, i . , x, i .
.
X
Because f is bounded away from 0 by A2-Žii. and s is bounded by Lemma A1-Žii., then Žii. follows.
To prove Žiii., it suffices to note that GŽ b < x, i . s F Ž ␰ Ž b, x, i .< x, i ., where F Ž⭈< ⭈ , i . and ␰ Ž⭈, ⭈ , i . have
R q 1 continuous bounded derivatives on Si Ž F . and Si Ž G . by A2-Žiii. and Lemma A2-Žii., respectively. Lastly, to prove Živ. we note that Ž9. gives
Ž A.4.
g Ž b < x, i . s
G Ž b < x, i .
Ž i y 1 .Ž ␰ Ž b, x, i . y b .
with ␰ Ž b, x, i . y b) 0 for Ž b, x . g Ci Ž B .. Because GŽ⭈< ⭈ , i . and ␰ Ž⭈, ⭈ , i . admit up to R q 1 bounded
continuous derivatives by Žiii. and Lemma A2-Žii., the desired result follows.
Q. E. D.
The proof of Proposition 4 follows that of Proposition 1. In particular, the following lemmas
similar to Lemmas A1 and A2 are needed. As I is constant by A5-Ži., then I s I 4 below.
LEMMA A3: Under A2 and A5, properties Ž i . and Ž ii . of Lemma A1 hold with ¨ Ž⭈., Si Ž F . and
sŽ ¨ , x, i . replaced by hŽ⭈., SU ' Ž ¨ , x . : x g w x, x x, ¨ g w hŽ x ., ¨ Ž x .x4 and s† Ž ¨ , x ., respecti¨ ely.
LEMMA A4: Under A2 and A5, properties Ž i . and Ž ii . of Lemma A2 hold with bŽ x, i ., bŽ x, i .,
Si Ž G ., and ␰ Ž b, x, i . replaced by hŽ x ., b† Ž x ., SŽ G† ., and ␰† Ž b† , x ., respecti¨ ely.
PROOF OF PROPOSITION 4: The proof is similar to that of Proposition 1, using F U Ž ¨ < x . s w F Ž ¨ < x .
y F Ž hŽ x .< x .xrw1 y F Ž hŽ x .< x .x, G† Ž b† < x ., Lemmas A3 and A4 instead of F Ž ¨ < x, i ., GŽ b < x, i ., Lemmas
A1 and A2, respectively. In particular, instead of ŽA.4. we have from Ž27.
Ž A.5.
g † Ž b† < x . s
2 b†
1
␰† Ž b† , x . y p 0 y b†2 I y 1
ž
G† Ž b† < x . q
F Ž p0 < x .
1 y F Ž p0 < x .
with p 0 s hŽ x ..
/
Q. E. D.
APPENDIX B
PROOFS
OF
STATISTICAL PROPERTIES
This Appendix gives the proofs of Theorem 2, Propositions 2 and 3, and Theorem 5. Lemmas are
proved in Appendix C. Throughout < ⭈ < r, ) denotes the supnorm of the r th derivatives of ⭈ on the
set ).
556
E. GUERRE, I. PERRIGNE, AND Q. VUONG
B.1. Optimal Uniform Con¨ ergence Rate
PROOF OF THEOREM 2: The proof relies on the argument of Khas’minskii Ž1976., though it is
somewhat more complicated due to the indirect nature of our statistical problem. The proof is
divided in 4 steps, where the fourth step uses Fano’s lemma. Let rLU ' Ž Lrlog L. R rŽ2 Rqdq3..
Step 1: We show that the rate rLU is given by the estimation of the joint density f Ž ¨ , x .. Let fˆŽ⭈.
be Žsay. the kernel density estimator Ž21. of the density f Ž⭈. of X. From the triangular inequality, for
any estimator fˆŽ⭈< ⭈ . of the conditional density, we have
rLU < fˆŽ ¨ < x . y f Ž ¨ < x . < G
rLU
fˆŽ x .
Ž < fˆŽ ¨ < x . fˆŽ x . y f Ž ¨ < x . f Ž x . < y f Ž ¨ < x . < fˆŽ x . y f Ž x . <.
G C1 rLU < fˆŽ ¨ < x . fˆŽ x . y f Ž ¨ , x . < y C0 rLU < fˆŽ x . y f Ž x . < , a.s.
for some constants C1 and C0 since f Ž⭈. and f Ž⭈< ⭈ . are uniformly bounded above on C Ž V . by
definition of U⑀ Ž f 0 ., and since f Ž⭈. converges uniformly to f Ž⭈.. In addition, it is well-known Žsee
Hardle Ž1991.. that the uniform rate of convergence of fˆŽ⭈. to f Ž⭈. is Ž Lrlog L.Ž Rq1.rŽ2 Rqdq2.,
which is faster than rLU . Hence the above second term converges uniformly to zero. Now, fˆŽ⭈< ⭈ . fˆŽ⭈. is
an estimator of the joint density f Ž⭈, ⭈ .. It is easy to show that, if Theorem 2 holds for some positive
X
constant ␬ where fˆŽ ¨ , x . and f Ž ¨ , x . replace fˆŽ ¨ < x . and f Ž ¨ < x ., respectively, then Theorem 2
would hold for some ␬ .
Step 2: The set U⑀ Ž f 0 . can be replaced by any subset U; U⑀ Ž f 0 . since
sup
fgU⑀ Ž f 0 .
Pr g Ž rLU < fˆŽ ⭈, ⭈ . y f Ž ⭈, ⭈ . < 0 , C ŽV . ) ␬ . G sup Pr g Ž rLU < fˆŽ ⭈, ⭈ . y f Ž ⭈, ⭈ . < 0 , C ŽV . ) ␬ . .
fgU
The subset U will be discrete and of the form f m k Ž⭈, ⭈ , ⭈ ., k s 1, . . . , m dq 1 4, where m is increasing
with the sample size. To construct the joint density f m k Ž⭈, ⭈ , ⭈ . for Ž V, X, I ., we consider a
nonconstant and odd C⬁ -function ␾ , with support wy1, 1x dq 1 , such that
Ž B.1.
Hwy1 , 0x␾ Ž b, x . dbs 0,
␾ Ž 0, 0. s 0,
␾ X Ž 0, 0 . / 0.
Let Ci Ž B . ' sŽ C Ž V ., F0 , i . be the image of C Ž V . by the bidding function associated to f 0 when
I s i. Without loss of generality, assume hereafter that Pr 0 Ž I s 2. / 0 so that C2 Ž B . is a nonempty
inner compact subset of S2 Ž G 0 .. Let Ž bk , x k ., k s 1, . . . , m dq 1 , be distinct points in the interior
C2o Ž B . such that the distance between Ž bk , x k . and Ž bj , x j ., j / k, is larger than ␭1rm. Thus, one can
choose a constant ␭2 such that the m dq 1 functions
␾m k Ž b, x . '
1
m Rq1
␾ Ž m ␭2 Ž by bk . , m ␭2 Ž x y x k ..
Ž k s 1, . . . , m dq 1 .
have disjoint hypercube supports SŽ ␾m k . centered at Ž bk , x k . with side equal to 2rŽ m ␭2 .. Hence
H␾m j Ž b, x . ␾m k Ž b, x . db dxs 0 for j / k.
Let C3 be a positive constant Žchosen below., and for each k s 1, . . . , m dq 1 define
g m k Ž b, x, i . '
½
g 0 Ž b, x, 2 . y C3 ␾m k Ž b, x .
g 0 Ž b, x, i .
if i s 2,
if i / 2.
That is, g m k Ž⭈, ⭈ , i . is identical to g 0 Ž⭈, ⭈ , i . except when i s 2, in which case it differs from g 0 Ž⭈, ⭈ , 2.
only on the set S Ž ␾m k .. For m large enough, S Ž ␾m k . is in S2o Ž G 0 ., while g Ž⭈, ⭈ , 2. is bounded away
from zero on S2 Ž G 0 . and integrates to Pr 0 Ž I s 2. as g 0 Ž⭈, ⭈ , 2. is bounded away from zero ŽProposition 1-Žii. and A2Žii.. and Hwy 1, q1x ␾ Ž b, x . dbs 0.
557
FIRST PRICE AUCTIONS
Now consider the function ␰ m k Ž b, x, 2. s b q Gm k Ž b, x, 2.rg m k Ž b, x, 2.. It is easy to see that
X
Gm k Ž⭈, ⭈ , 2., g m k Ž⭈, ⭈ , 2., and g m k Ž⭈, ⭈ , 2. converge uniformly on S2 Ž G 0 . to G 0 Ž⭈, ⭈ , 2., g 0 Ž⭈, ⭈ , 2., and
X
Ž
.
g 0 ⭈, ⭈ , 2 , respectively. Thus ␰ m k Ž⭈, ⭈ , 2. is strictly increasing in b with a differentiable inverse. From
Theorem 1, it follows that g m k Ž⭈, ⭈ , 2.rPr 0 Ž I s 2. can be interpreted as the bid density associated
with a unique density f m k Ž⭈, ⭈ , 2.rPr 0 Ž I s 2. of Ž V, X . when I s 2, namely,
Ž B.2.
X
Ž
.
. Ž
Ž y1 Ž
.
..
f m k Ž ¨ , x, 2 . s g m k Ž ␰y1
m k ¨ , x, 2 , x, 2 r ␰ m k ␰ m k ¨ , x, 2 , x, 2
s
Ž
.
.
g m3 k Ž ␰y1
m k ¨ , x, 2 , x, 2
Ž
2 g m2 k Ž ␰y1
mk ¨ ,
X
Ž
.
.
Ž y1 Ž
.
.
x, 2 . , x, 2 . y Gm k Ž ␰y1
m k ¨ , x, 2 , x, 2 g m k ␰ m k ¨ , x, 2 , x, 2
.
As g m k Ž⭈, ⭈ , i . s g 0 Ž⭈, ⭈ , i . for i / 2, we have f m k Ž⭈, ⭈ , i . s f 0 Ž⭈, ⭈ , i . for those i’s. This completes the
construction of the densities f m k Ž⭈, ⭈ , ⭈ ., m s 1, . . . , m dq 1 , which compose the set U.
It remains to show that U is a subset of U⑀ Ž f 0 ., i.e., that all f m k Ž⭈, ⭈ , ⭈ .’s belong to U⑀ Ž f 0 .. For this,
we use the following lemma.
LEMMA B1: Gi¨ en A1᎐A2, the following properties hold for m large enough:
Ži. For any k s 1, . . . , m dq 1 , the supports of Gm k and Fm k are SŽ G 0 . and SŽ F0 ..
Žii. There is a positi¨ e constant C4 depending upon ␾ , G 0 , and C Ž V . such that for j / k,
< f m k y f m j < 0 , C ŽV . G C4
C3 ␭2
mR
.
Žiii. Uniformly in k s 1, . . . , m dq 1 , we ha¨ e
< f m k y f 0 < r , SŽ F 0 . s C3 ␭2rq 1 O Ž 1rm Ry r .
<
Ž r s 0, . . . , R y 1 . ,
f m k y f 0 < R , SŽ F 0 . s C3 ␭ 2Rq 1 O Ž 1 . q o Ž 1 . ,
where the big O Ž⭈. only depends upon ␾ and g 0 .
As 5 f 0 5 R - M by assumption, there exists a positive constant ␭3 such that < f 0 < r, SŽ F 0 . - My ␭ 3 for
all r s 1, . . . , R. As < f m k < r, SŽ F 0 . F < f m k y f 0 < r, SŽ F 0 . q < f 0 < r, SŽ F 0 . - < f m k y f 0 < r, SŽ F 0 . q My ␭ 3 , it follows
from Lemma B1-Žiii. that the first term can be made smaller than ␭3 as soon as m G m 0 Ž ⑀ . and C3
is small enough. Hence, f m k is in U⑀ Ž f 0 . so that U is a subset of U⑀ Ž f 0 ., as desired. As m is
increasing with L, we have for L G L0 Ž ⑀ .:
sup
fgU⑀ Ž f 0 .
Pr g Ž rLU < fˆŽ ⭈, ⭈ . y f Ž ⭈, ⭈ . < 0 , C ŽV . ) ␬ .
G
max
ks1 , . . . , m dq 1
Pr g mkŽ rLU < fˆŽ ⭈, ⭈ . y f m k Ž ⭈, ⭈ . < 0 , C ŽV . ) ␬ . .
Step 3: We now reduce our problem to the model selection problem of deciding which of the
hypothesis Hm k : f s f m k 4 holds. Consider the decision rule that selects Hm k whenever f m k is
closest to fˆ in the supnorm < ⭈ < 0, C ŽV . . Let Pr g mkŽ Hm k . be the probability of error of this decision
when Hm k : f s f m k 4 holds. Choose m and ␬ such that
Ž B.3.
m s Ž rLU .
1r R
and
␬ s C4 C3 ␭2r2.
From Lemma B1-Žii., we have rLU < f m k y f m j < 0, C ŽV . G 2 ␬ uniformly in j / k. Thus, from the triangular
inequality, if rLU < f y f m k < 0, C ŽV . F ␬ , then rLU < f y f m j < 0, C ŽV . G ␬ , for any j / k, i.e., the decision rule of
ˆ
ˆ
the closest accepts Hm k . Hence, for any estimator fˆŽ⭈, ⭈ . we have
max
ks1 , . . . , m dq 1
Pr g mkŽ rLU < fˆŽ ¨ , x . y f m k Ž ¨ , x . < 0 , C ŽV . ) ␬ . G
G
max
ks1 , . . . , m dq 1
1
m dq1
Pr g mkŽ Hm k .
m dq 1
Ý
ks1
Pr g mkŽ Hm k . ' Pr e .
558
E. GUERRE, I. PERRIGNE, AND Q. VUONG
Step 4: Assume that the parameter k in g m k is uniformly distributed over 1, . . . , m dq 1. Thus Pr e
is the Bayesian posterior probability of misclassification of the above decision rule based on the
ˆ We can bound Pr e from below using Fano’s Lemma Žsee Khas’minskii Ž1976.,
closest f m k to f.
Ibragimov and Has’minskii Ž1981, p. 323... Indeed, this lemma gives a lower bound on the probability
of misclassification for any decision rule that selects a value among a finite number of equally
probable values. In our case, because of i.i.d. sampling, we obtain
Pr e G 1 y
L K ŽŽ B, X , I . , k . q log 2
log Ž m dq1 y 1 .
,
where KŽŽ B, X, I ., k . is the amount of Shannon information in Ž X, I, Bp , p s 1, . . . , I . relative to
the parameter k, i.e.,
K ŽŽ B, X , I . , k . s E Ž log Ž g Ž k < B, X , I . rg Ž k ... s E Ž log Ž g Ž B, X , I < k . rg Ž B, X , I ...
ž
s E log
sy
g m k Ž B, X , I .
Ž 1rm
dq 1
Ž
.
Ým
js 1 g m j B, X , I
m dq 1
1
m
dq 1 .
dq1
Ý ÝH
log
ks1 ig I
ž
1
m
/
m dq 1
dq1
Ý
js1
g m j Ž b, x, i .
g m k Ž b, x, i .
/
g m k Ž b, x, i . db dx.
We now bound KŽŽ B, X, I ., k . from above. Using the concavity of the logarithm function, a simple
second-order expansion, the definition of the g m k ’s, ŽB.1., the fact that g m k is bounded away from 0,
the orthogonality of the ␾m k ’s, and a change of variables, we get:
K ŽŽ B, X , I . , k . F y
F
s
s
s
m dq 1
1
m
k , js1
m2Ž dq1.
C5
m dq1
C5
dq1
g m k Ž b, x, 2 .
/
g m k Ž b, x, 2 . db dx
Ý HŽ ␾m j Ž b, x . y ␾m k Ž b, x .. 2 db dx
k , js1
m dq 1
Ý H␾m2 k Ž b, x . db dx
ks1
m dq 1
C2
3
Ý H m2 Rq2 ␾ 2 Ž ␭2 mŽ by bk . , ␭2 mŽ xy x k .. db dx
ks1
C5
␭2dq 1
s C6
ž
g m j Ž b, x, 2 .
m dq 1
C5
m
Ý Hlog
2Ž dq1.
m
C32
2 Rqdq3
H␾ 2 Ž b, x . db dx
␬2
Ž rLU . Ž2 Rqdq3.rR
s C6
␬ 2 log L
L
by ŽB.3. and the definition of rLU . Thus Fano’s Lemma gives
Pr e G 1 y
C6 ␬ 2 log L q log 2
log Ž m dq1 y 1 .
.
As m s Ž rLU .1r R s Ž Lrlog L.1r Ž2 Rqdq3., taking ␬ Ži.e., C3 . small enough gives Pr e G 1r2 for
L G L0 Ž ⑀ .. As this can be done for any value of ⑀ , this completes the proof of Theorem 2.
Q. E. D.
559
FIRST PRICE AUCTIONS
B.2. Uniform Consistency of Boundary Estimators
PROOF OF PROPOSITION 2: Because the proof for ˆ
bŽ x . is similar, we consider only the upper
ˆ
Ž
.
boundary estimator b x, i . The proof is divided into two steps.
Step 1: First, we establish an exponential type inequality. Assume that x lies in ␲ k 1, . . . , k d ' ␲ .
Define bU Ž x, i . s sup y g ␲ bŽ y, i .. Using Ž16., the triangular inequality, and Proposition 1-Ži., we have
<ˆ
bŽ x, i . y bŽ x, i .< F bU Ž x, i . y ˆ
bŽ x, i . q < b < 1 h⭸ with < b < 1 s < b < 1, w x, x x=II . Thus
Ž B.4.
Pr Ž < ˆ
b Ž x, i . y b Ž x, i . < ) t . F Pr Ž ˆ
b Ž x, i . - bU Ž x, i . y t q < b < 1 h⭸ . .
Let RHS be the right-hand side of ŽB.4.. Using A1 we have
RHS s Pr L
ž
sup
ps1 , . . . , i
/
Bp| Ž X g ␲ , I s i . - bU Ž x, i . y t q < b < 1 h⭸ .
Provided t and h⭸ are sufficiently small so that bU Ž x, i . y t q < b < 1 h⭸ ) 0, then Ž RHS .1r L equals
Ž B.5.
w 1 y Pr Ž I s i and X g ␲ .x
H
q Pr i Ž B - bU Ž x, i . y t q < b < 1 h⭸ < X s y, I s i . |␲ Ž y . f Ž y, i . dy.
Using Proposition 1-Žii., an upper bound for the second term is
HPr i Ž B - b Ž y, i . y t q 2 < b < 1 h⭸ < X s y, I s i .|␲ Ž y . f Ž y, i . dy
F 1y
i
inf
Ž b , x , i .gS Ž G .
g Ž b < x, i .Ž t y 2 < b < 1 h⭸ . Pr Ž I s i and X g ␲ . ,
provided t y 2 < b < 1 h⭸ G 0 Žwhich is satisfied by our choice of t in Step 2., and provided t and h⭸ are
sufficiently small to ensure that bŽ y, i . y t q 2 < b < 1 h⭸ G bŽ y, i . and that the term in brackets is strictly
between 0 and 1. Note that we have used inf Ž x, i.g w x, x x=II Ž bŽ x, i . y bŽ x, i .. ) 0, which is ensured by
Proposition 1-Ži.. Moreover we have h⭸d inf x g ␲ f Ž x, i . F PrŽ I s i and X g ␲ . F h⭸d sup x g ␲ f Ž x, i ..
Hence, because i ) 1, then ŽB.5. is bounded above by
1 y h⭸d inf f Ž x, i . q 1 y
xg ␲
s 1 q h⭸d
y h⭸d
inf
Ž b , x , i .gS Ž G .
g Ž b < x, i .Ž t y 2 < b < 1 h⭸ . h⭸d sup f Ž x, i .
xg ␲
ž sup f Ž x, i . y inf f Ž x, i . /
xg ␲
sup
Ž x , i .gS Ž X .
xg ␲
f Ž x, i .
inf
Ž b , x , i .gS Ž G .
g Ž b < x, i .Ž t y 2 < b < 1 h⭸ .
F 1 q c1 h⭸dq 1 y c 2 h⭸d Ž t y 2 < b < 1 h⭸ .
ž
F Ž 1 q Ž c1 q 2 c 2 < b < 1 . h⭸dq1 . 1 y
c2
1 q Ž c1 q 2 c2 < b < 1 . h⭸dq1
h⭸d t
/
F Ž 1 q c 3 h⭸dq 1 . Ž 1 y c4 h⭸d t . ,
for h⭸ and t sufficiently small, where c i ’s are strictly positive constants independent of Ž x, i ., and
where we have used sup x g ␲ f Ž x, i . y inf x g ␲ f Ž x, i . F supŽ x, y .g ␲ 2 < f Ž x, i . y f Ž y, i .< F c1 h⭸ , for Ž␲ , i .
; SŽ f Ž x, i .. by A2-Žiii.. Therefore ŽB.4. and A4-Žiii. give
L
dr Ž dq1 .
.
Pr Ž < ˆ
b Ž x, i . y b Ž x, i . < ) t . F Ž 1 q c5 log LrL . Ž 1 y c6 t Ž log LrL .
F Lc 5 Ž 1 y c6 t Ž log LrL .
for t sufficiently small and L sufficiently large.
dr Ž dq1 .
L
. ,
L
560
E. GUERRE, I. PERRIGNE, AND Q. VUONG
Step 2: Let x␲ g ␲ . Note that ˆ
bŽ x, i . s ˆ
bŽ x␲ , i . whenever xg ␲ by Ž16.. Thus, using the
triangular inequality and letting t s ␮rŽ2 r⭸ . in the above inequality, we have for each i g I
ž
Pr r⭸ sup < ˆ
b Ž x, i . y b Ž x, i . < ) ␮
xg w x , x x
ž
/
F Pr sup < ˆ
b Ž x␲ , i . y b Ž x␲ , i . < ) ␮r Ž 2 r⭸ .
␲
ž
/
q | sup sup < b Ž x, i . y b Ž x␲ , i . < ) ␮r Ž 2 r⭸ .
F
␲
xg ␲
/
Ý Pr Ž < ˆb Ž x␲ , i . y b Ž x␲ , i . < ) ␮rŽ2 r⭸ ..
␲
q
Ý | ž sup < b Ž x, i . y b Ž x␲ , i . < ) ␮rŽ2 r⭸ . /
xg ␲
␲
L
F card Ž ␲ . Lc 5 Ž 1 y c6 ␮ Ž log Lr2 L .. q card Ž ␲ . | Ž < b < 1 r⭸ h⭸ ) ␮r2. ,
where cardŽ␲ . is the number of elements in the partition of w x, x x.
Now, take ␮ large enough. The second term vanishes, as r⭸ h⭸ s ␭⭸ from A4-Žiii.. Moreover, the
first term gives a converging series since
L
log Ž 1 y c6 t Ž log Lr2 L .. s L log Ž 1 y c6 ␮ Ž log Lr2 L .. ; y Ž c6 r2. ␮ log L,
while cardŽ␲ . s O Ž h⭸yd . s O ŽŽ Lrlog L. d rŽ dq1. . by A4-Žiii.. Hence, by the Borel-Cantelli Lemma,
we have shown that there exists ␮ Žsufficiently large. such that
ž ½
Pr lim sup r⭸ sup < ˆ
b Ž x, i . y b Ž x, i . < ) ␮
xg w x , x x
5/
s 0.
This establishes the desired result for each i. Since I is finite, the desired result follows.
Q. E. D.
B.3. Uniform Consistency of Pseudo Pri¨ ate Values
˜ ˜g, fˆŽ x ., and
To prove Proposition 3 we need a lemma that gives uniform convergence rates of G,
f˜Ž ¨ , x ., where the latter is the infeasible estimator defined in Ž22.. We consider uniform convergence
on a fixed subset as well as on subsets expanding to the supports. Specifically, let
CLŽ B . ' Ž b, x, i . g S Ž G . ; Ž b, x . q S Ž h G . j S Ž h g .4 ; Si Ž G .4 ,
CLŽ V . ' Ž ¨ , x . g S Ž f Ž ¨ , x .. ; Ž ¨ , x . q S Ž h f .4 ; S Ž f Ž ¨ , x ..4 ,
CLŽ X . ' xg S Ž f Ž x .. ; x q S Ž h X .4 ; S Ž f Ž x ..4 ,
where SŽ h f . and SŽ h X . are defined similarly to SŽ h G . and SŽ h g . in the text, i.e., are the supports of
the kernels K f Ž⭈rh f , ⭈ rh f . and K X Ž⭈rh X ., respectively. Let
Ž B.6.
ž /
ž /
rG s r X s
rU
g s
L
Ž Rq1 .r Ž2 Rqdq2 .
,
log L
L
log L
rg s rf s
ž /
L
log L
Ž Rq1 .r Ž2 Rqdq3 .
,
which give the rates of uniform consistency of our estimators.
Rr Ž 2 Rqdq3 .
,
FIRST PRICE AUCTIONS
561
LEMMA B2: Gi¨ en A1᎐A4, we ha¨ e almost surely
Ži.
˜Ž b, x, i . y G Ž b, x, i . < 0 , C LŽ B . s O Ž1rrG . ,
<G
< f˜Ž ¨ , x . y f Ž ¨ , x . < 0 , C LŽV . s O Ž 1rr f . ,
Ž ii .
<˜
g Ž b, x, i . y g Ž b, x, i . < 0 , C LŽ B . s O Ž 1rr g . ,
< fˆŽ x . y f Ž x . < 0 , C LŽ X . s O Ž 1rr X . ,
<˜
.
g Ž b, x, i . y g Ž b, x, i . < 0 , C Ž B . s O Ž 1rr U
g ,
where CLŽ B ., CLŽ V . and CLŽ X . are expanding subsets defined abo¨ e, while C Ž B . is a Ž fixed . arbitrary
inner subset of SŽ G ..
˜p l s G˜Ž Bp l , X l , Il ., g p l s g Ž Bp l , X l , Il ., and
PROOF OF PROPOSITION 3: Let Gp l s GŽ Bp l , X l , Il ., G
g˜p l s ˜
g Ž Bp l , X l , Il .. Define ˜
c g s min< ˜
g p l <; l s 1, . . . , L, p s 1, . . . , Il , Vˆp l / q⬁4.
Ži. Because Il G 2, it follows from Ž12. and Ž18. that
| Ž Vˆp l / q⬁. < Vˆp l y Vp l < F | Ž Vˆp l / q⬁. < ␺˜ Ž Bp l , X l , Il . y ␺ Ž Bp l , X l , Il . < .
Hence, using the set CLŽ B . introduced in Lemma B2, we obtain
Ž B.7.
| Ž Vˆp l / q⬁. |C LŽ B . Ž Bp l , X l , Il . < ␺˜ Ž Bp l , X l , Il . y ␺ Ž Bp l , X l , Il . <
s
F
| Ž Vˆp l / q⬁. |C LŽ B . Ž Bp l , X l , Il .
g p l < g̃ p l <
| Ž Vˆp l / q⬁. |C LŽ B . Ž Bp l , X l , Il .
c f c g c̃ g
˜p l y Gp l . g p l q Ž g p l y ˜g p l . Gp l <
<Ž G
˜p l y Gp l < q < f m < 0 < ˜g p l y g p l < ,
< g < 0 <G
because GŽ b, x, i . F GŽ bŽ x, i ., x, i . s g Ž x, i . s f m Ž x, i . and g p l s g Ž Bp l < X l , Il . f m Ž X l , Il . G c f c g by A2Žii. and Proposition 1-Žii.. Thus
Ž B.8.
| Ž Vˆp l / q⬁. < Vˆp l y Vp l <
s | Ž Vˆp l / q⬁.Ž |C LŽ B . Ž Bp l , X l , Il . q 1 y |C LŽ B . Ž Bp l , X l , Il .. < Vˆp l y Vp l <
F
| Ž Vˆp l / q⬁. |C LŽ B . Ž Bp l , X l , Il .
c f c g c̃ g
˜p l y Gp l < q < f m < 0 < ˜g p l y g p l <
< g < 0 <G
q | Ž Vˆp l / q⬁. Ž 1 y |C LŽ B . Ž Bp l , X l , Il .. < Vˆp l y Vp l < .
Now, from A3-Ži. S Ž2 h g . Ž S Ž2 h G .. is a hypercube centered at 0 with edge 2 ␳ g h g Ž2 ␳G h G .. Thus, if
Vˆp l / q⬁, Proposition 2 and Ž19. imply that the distance in supnorm of Ž Bp l , X l . to the frontier of
SI l Ž G . is at least ␳ g h g y t Žlog LrL.1r Ž dq1., for some t and L large enough. But, for L large enough,
␳ g h g y t Žlog LrL.1r Ž dq1. G ␳ g h gr2 because Žlog LrL.1rŽ dq1. s oŽ h g . from A4-Žii.. Hence, if Vˆp l /
q⬁, the distance in supnorm of Ž Bp l , X l . to the frontier of SI l Ž G . is at least ␳ g h gr2. Therefore
Ž Bp l , X l , Il . g CLŽ B .. Hence we have shown that for any p, l, |Ž Vˆp l / q⬁.Ž1 y |C Ž B .Ž Bp l , X l , Il .. s 0
L
almost surely as L ª ⬁. It follows that the second term in ŽB.8. vanishes. Moreover, from Lemma
B2, we have ˜
c g ª c g ) 0 by Proposition 1-Žii. and the first term is O Žmax1rrG , 1rr g 4. s O Ž1rr g ..
The desired result follows.
Žii. Define C Ž B . s Di g I Ci Ž B . where Ci Ž B . s Ž b, x, i . g Si Ž G . : Ž ␰ Ž b, x, i ., x, i . g C Ž V .4. Because ␰ is a strictly increasing continuous function and C Ž V . is an inner compact subset of
SŽ f Ž ¨ , x .., then Ci Ž B . is a Žfixed. inner compact subset of Si Ž G .. Hence, from Ž19. and Proposition
562
E. GUERRE, I. PERRIGNE, AND Q. VUONG
2, it is easy to see that Vˆp l / q⬁ if Ž Bp l , X l , Il . g C Ž B . for L large enough. Thus
|C ŽV . Ž Vp l , X l . < Vˆp l y Vp l < s |C Ž B . Ž Bp l , X l , Il . < Vˆp l y Vp l <
s |C Ž B . Ž Bp l , X l , Il . | Ž Vˆp l / q⬁. < Vˆp l y Vp l <
almost surely for L large enough. With C Ž B . replacing CLŽ B . in the RHS of ŽB.7., the desired
result follows applying Lemma B2-Žii. instead, and noting that 1rrG - 1rr U
Q. E. D.
g .
B.4. Optimality and Uniform Consistency With a Binding Reser¨ ation Price
PROOF OF THEOREM 5: Given A2 and A5, it is easy to see that the truncated conditional
distribution F U Ž⭈< x . s w F Ž⭈< x . y F Ž hŽ x .< x .xrw1 y F Ž hŽ x .< x .x satisfies the requirements on F Ž⭈< ⭈ , i . in
A2, where the support is now SU . Hence the proof of Theorem 2 with f Ž⭈< ⭈ , ⭈ . replaced by the
truncated conditional density f U Ž⭈< ⭈ . establishes that the best rate for estimating uniformly the latter
on inner compact subsets of SU is bounded above by rLU s Ž Lrlog L. R rŽ2 Rqdq3.. On the other
hand, we have Ew I U < X x s I w1 y ⌽ Ž X .x. Because Iˆs I almost surely and ⌽ Ž⭈. s F Ž hŽ⭈.< ⭈ . is Ž R q 1.continuously differentiable on w x, x x by A2 and A5, it follows from Stone Ž1982. that the best rate for
estimating uniformly ⌽ Ž⭈. is Ž Lrlog L.Ž Rq1.rŽ2 Rqdq2., which is larger than rLU . Hence, because
f Ž ¨ < x . s w1 y ⌽ Ž x .x f U Ž ¨ < x ., the best rate for estimating uniformly f Ž⭈< ⭈ . on inner compact subsets of
SU is bounded above by rLU .
ˆ Ž x .x fˆU Ž ¨ < x ., consider fˆU Ž⭈< ⭈ ..
Turning to uniform consistency of our estimator fˆŽ ¨ < x . s w1 y ⌽
U
U
Ž
.
From Theorem 4- ii , B and I are conditionally independent given X. Using a result analogous to
Lemma B2 with B† instead B, it can be shown that Proposition 3 still holds using a similar proof,
where the pseudo private values are now given by Ž27. and the new trimming is needed because of
the presence of fˆŽ X l . in ␰ˆ† Ž⭈, ⭈ .. It follows that Theorem 3 applies with f Ž⭈< ⭈ . replaced by f U Ž⭈< ⭈ ..
That is, fˆU Ž⭈< ⭈ . converges uniformly to f U Ž⭈< ⭈ . on inner compact subsets of SU at the rate rLU . On the
ˆ Ž⭈.
other hand, because Iˆs I almost surely, we know from A4-Žii. and Hardle Ž1991. that ⌽
converges uniformly on inner compact subsets of w x, x x at the rate Ž Lrlog L.Ž Rq1.rŽ2 Rqdq2.. The
desired result follows.
Q. E. D.
APPENDIX C
PROOFS
OF
LEMMAS
This Appendix gives the proofs of all lemmas in Appendices A and B.
C.1. Proofs of Lemmas in Appendix A
To prove Lemmas A1 and A2 we need a version of the Implicit Function Theorem.
LEMMA C1: For e¨ ery i g I , let ␣ Ž⭈, ⭈ , i . be a function on Si Ž ␣ .. Let ␤ Ž⭈, ⭈ , i . be a function on
Si Ž ␤ . such that ␤ Ž⭈, ⭈ , i . g Si Ž ␣ .. For some nonnegati¨ e real number ␥ , assume that
Ž C.1.
␣ Ž ␤ Ž b, x, i . , x, i . s ␥ b ᭙ Ž b, x . g Si Ž ␤ . and ᭙ i g I .
Moreo¨ er, for each i g I , assume that the following conditions hold:
Ži. The mapping ␣ Ž⭈, ⭈ , i . admits up to Ž R q 1. continuous bounded partial deri¨ ati¨ es on S i Ž ␣ .,
X
with ␣ Ž ¨ , x, i . G c␣ ) 0 for all Ž ¨ , x . g Si Ž ␣ ..
Žii. Si Ž ␤ . s Ž b, x . : x g w x, x x, bg w bŽ x, i ., bŽ x, i .x4, where the boundaries bŽ⭈, i . and bŽ⭈, i . are
continuous on w x, x x and inf x g w x, x xŽ bŽ x, i . y bŽ x, i .. ) 0.
Then ␤ Ž⭈, ⭈ , i . admits up to R q 1 continuous bounded partial deri¨ ati¨ es on Si Ž ␤ ..
563
FIRST PRICE AUCTIONS
PROOF
Ž C.2.
Ž C.3.
OF
LEMMA C1: For Ž b, x . in Sio Ž ␤ ., ŽC.1. and the Implicit Function Theorem yield:
␤ X Ž b, x, i . s
⭸␤ Ž b, x, i .
⭸ xk
␥
␣ X Ž ␤ Ž b, x, i . , x, i .
sy
,
1
⭸␣ Ž ␤ Ž b, x, i . , x, i .
␣ Ž ␤ Ž b, x, i . , x, i .
⭸ xk
X
.
Hence, in view of Ži., ␤ Ž⭈, ⭈ , i . is a Lipschitz function on Sio Ž ␤ .. Thus, by continuity ␤ Ž⭈, ⭈ , i . can be
extended to Sio Ž ␤ . , which is equal to S i Ž ␤ . by Žii.. Let ␤˜Ž⭈, ⭈ , i . denote this extension, and note that
␤˜Ž⭈, ⭈ , i . solves ŽC.1.. But, because ␤ Ž b, x, i . is the unique solution of ŽC.1., then ␤˜Ž⭈, ⭈ , i . s ␤ Ž⭈, ⭈ , i ..
Therefore ␤ Ž⭈, ⭈ , i . is continuous and hence bounded on Si Ž ␤ ..
X
Moreover, since ␣ is bounded away from 0, ŽC.2. ᎐ ŽC.3. show that the first partial derivatives of
␤ Ž⭈, ⭈ , i . are continuous and bounded on Si Ž ␤ .. Proceeding by induction yields that ␤ Ž⭈, ⭈ , i . admits
up to R q 1 continuous partial derivatives over Si Ž ␤ ..
Q. E. D.
PROOF OF LEMMA A1: Ži. Because F Ž⭈< x, i . is strictly increasing by A2-Žii., ¨ Ž x . and ¨ Ž x . are the
unique solutions in Si Ž F . of F Ž ¨ Ž x .< x, i . s 0 and F Ž ¨ Ž x .< x, i . y 1 s 0, respectively. Hence, given
A2, Lemma C1 applies with ␣ Ž⭈, ⭈ , ⭈ . s F Ž⭈< ⭈ , ⭈ . or F Ž⭈< ⭈ , ⭈ . y 1, ␤ Ž b, x, i . s ¨ Ž x . or ¨ Ž x ., ␥ s 0,
and S i Ž ␤ . s wy1, 1x = w x, x x Žsay.. Moreover, 1 s F Ž ¨ Ž x .< x, i . y F Ž ¨ Ž x .< x, i . F Ž ¨ Ž x . y
¨ Ž x .. supŽ ¨ , x .g S iŽ F . < f Ž ¨ < x, i .<. Because f Ž⭈< ⭈ , i . is bounded above, then inf x g w x, x xŽ ¨ Ž x . y ¨ Ž x .. ) 0.
Žii. For the sake of simplicity, we assume that x g ⺢. From Ž8. and the Lebesgue Dominated
Convergence Theorem it is immediate that sŽ⭈, ⭈ , i . is Ž R q 1.-times continuously differentiable on
Ž ¨ , x . : ¨ g Ž ¨ Ž x ., ¨ Ž x .x, x g w x, x x4. Thus it remains to show that the derivatives of sŽ⭈, ⭈ , i . up to
order R q 1 are bounded near the lower boundary ¨ Ž⭈..
For each xg w x, x x, an Ž R q 1.th order Taylor series expansion of F Ž ¨ < x . at Ž ¨ Ž x ., x . gives
F Ž¨ < x. s
f Ž¨ Ž x.< x.
1!
t q ⭈⭈⭈ q
f Ž R. Ž ¨ Ž x . < x .
Ž R q 1. !
t Rq 1 q o Ž t Rq1 . ,
uniformly in x, where t s ¨ y ¨ Ž x .. Hence
F iy 1 Ž ¨ < x . s t iy 1 Ž a0 Ž x . q ⭈⭈⭈ qa R Ž x . t R q o Ž t R .. ,
uniformly in x, for each k s 0, . . . , R, a k Ž x . is a homogeneous polynomial of degree Ž i y 1. in
f Ž ¨ Ž x .< x . and its derivatives f Ž1. Ž ¨ Ž x .< x ., . . . , f Ž k . Ž ¨ Ž x .< x . with respect to ¨ . In particular, a0 Ž x . s
f Ž ¨ Ž x .< x . iy 1 , which is nonzero by A2-Žii.. It also follows that
¨
H¨ Ž x . F iy1 Ž u < x . dus t i
ž
a0 Ž x .
i
q ⭈⭈⭈ q
aR Ž x .
Rqi
/
t R q oŽ t R . ,
uniformly in x. Therefore
1
F iy1 Ž ¨ < x .
¨
H¨ Ž x . F iy 1 Ž u < x . dus t Ž b0 Ž x . q ⭈⭈⭈ qbR Ž x . t R q o Ž t R .. ,
where bk Ž x ., k s 0, . . . , R, is a polynomial in a1Ž x .ra0 Ž x ., . . . , a k Ž x .ra0 Ž x . and hence in
f Ž0. Ž ¨ Ž x .< x .rf Ž ¨ Ž x .< x ., . . . , f Ž k . Ž ¨ Ž x .< x .rf Ž ¨ Ž x .< x .. In particular, it is easy to see that b 0 Ž x . s 1ri.
Thus we obtain an Ž R q 1.th order Taylor series expansion of sŽ ¨ , x, i . around Ž ¨ Ž x ., x . as
s Ž ¨ , x, i . s ¨ Ž x . q Ž 1 y b 0 Ž x .. t y b1Ž x . t 2 y ⭈⭈⭈ ybR Ž x . t Rq 1 y o Ž t Rq 1 . ,
uniformly in x. Hence sŽ ¨ , x, i . is continuous in ¨ at ¨ Ž x . with R q 1 bounded derivatives with
respect to ¨ in the neighborhood of ¨ Ž x .. In particular, ⭸ sŽ ¨ Ž x ., x, i .r⭸ ¨ s Ž i y 1.ri / 0.
For each r s 1, . . . , R q 1, it remains to prove that the derivatives ⭸ r sŽ⭈, x, i .r⭸ x p ⭸ ¨ ryp , p s
1, . . . , r are bounded in the neighborhood of ¨ Ž x . for xg w x, x x. The proof is by induction on r. For
564
E. GUERRE, I. PERRIGNE, AND Q. VUONG
r s 1, differentiating the boundary condition sŽ ¨ Ž x ., x, i . s ¨ Ž x . gives
⭸ s Ž ¨ Ž x . , x, i . d¨ Ž x .
⭸¨
dx
q
⭸ s Ž ¨ Ž x . , x, i .
⭸x
s
d¨ Ž x .
dx
.
But ⭸ sŽ ¨ Ž x ., x, i .r⭸ ¨ is bounded, as shown above, while d¨ Ž x .rdx is bounded from Part Ži.. Hence
⭸ sŽ ¨ Ž x ., x, i .r⭸ x is bounded.
Suppose next that all the partial derivatives up to order r of sŽ⭈, ⭈ , i . are bounded, 1 F r F R.
From the above Taylor series expansion of sŽ⭈, x, i . we obtain the system
s Ž ¨ Ž x . , x, i . s ¨ Ž x . ,
⭸ s Ž ¨ Ž x . , x, i .
⭸¨
s 1 y b0 Ž x . , . . . ,
⭸ r s Ž ¨ Ž x . , x, i .
⭸¨r
s ybry1Ž x . .
Now, differentiate the first equation r q 1 times, the second equation r times, . . . , the last equation
one time. This is possible because bk Ž x . is R y k times differentiable Žwith bounded derivatives..
Because ⭸ rq 1 sŽ ¨ Ž x ., x, i .r⭸ ¨ rq 1 and d¨ Ž x .rdx are bounded, differentiation of the last equation
shows that ⭸ rq 1 sŽ ¨ Ž x ., x, i .r⭸ ¨ r ⭸ x is bounded. Combining this with the twice-differentiation of the
next to the last equation shows that ⭸ rq 1 sŽ ¨ Ž x ., x, i .r⭸ ¨ ry 1 ⭸ x 2 is bounded. Continuing up to the
first equation shows that ⭸ rq 1 sŽ ¨ Ž x ., x, i .r⭸ x rq 1 is bounded. This completes the proof that sŽ⭈, ⭈ , i .
admits R q 1 bounded continuous derivatives on its support.
X
Lastly, s Ž⭈, x, i . can only vanish at ¨ Ž x . because
X
s Ž ¨ , x, i . s
Ž i y 1. f Ž ¨ < x .
H
F iy1 Ž u < x . du.
F iŽ¨ < x.
¨ Ž x.
X
X
But, as noted above, s Ž ¨ Ž x ., x, i . s Ž i y 1.ri. Thus s Ž⭈, ⭈ , i . is bounded away from zero.
Q. E. D.
PROOF OF LEMMA A2: Because bŽ x, i . s sŽ ¨ Ž x ., x, i . and bŽ x, i . s sŽ ¨ Ž x, i ., x, i ., then Ži. follows
immediately from Lemma A1. To prove Žii., we note that the function ␰ solves
s Ž ␰ Ž b, x, i . , x, i . s b ᭙ Ž b, x . g Si Ž G . .
X
The desired result follows from Part Ži., Lemma A1-Žii., and Lemma C1. Moreover, ␰ Ž b, x, i . s
X
1rŽ s Ž ␰ Ž b, x, i ., b, x, i .. is bounded away from 0 by Lemma A1-Žii..
Q. E. D.
PROOF OF LEMMA A3: Ži. directly follows from Lemma A1-Ži., A5-Žii., and A5-Žiii.. Regarding Žii.,
because F Ž ¨ < x . G F Ž p 0 < x ., which is bounded away from zero by A5-Žiii., it is easy to see that
s†2 Ž⭈, ⭈ , ⭈ . and hence s† Ž⭈, ⭈ , ⭈ . have R q 1 continuous partial derivatives on SŽ F U ., except possibly
near ¨ s p 0 as s† Ž p 0 , ⭈ , ⭈ . s 0. However, letting ¨ † s ¨ y p 0 , we have
F Ž ¨ † q p0 < x . s F Ž p0 < x .
ž
f Ž p0 < x . ¨ †
F Ž p 0 < x . 1!
q ⭈⭈⭈ q
f Ž R. Ž p 0 < x . ¨ †Ž Rq1.
F Ž p0 < x .
Ž R q 1. !
q o Ž ¨ †Ž Rq1. .
uniformly in Ž x, p 0 .. Similarly to the proof of Lemma A1-Žii., it can be shown that
/
s†2 Ž ¨ , x . s a2 Ž x, p 0 . ¨ †2 q ⭈⭈⭈ qaRq2 Ž x, p 0 . ¨ †Rq 2 q o Ž ¨ †Rq 2 .
uniformly in Ž x, p 0 ., where a k Ž x, p 0 . is a polynomial in f Ž p 0 < x .rF Ž p 0 < x ., . . . , f Ž ky2. Ž p 0 < x .rF Ž p 0 < x .
for k s 2, . . . , R q 2. In particular, a2 Ž x, p 0 . s Ž I y 1. f Ž p 0 < x .r Ž2 F Ž p 0 < x ... Hence
s† Ž ¨ , x . s b1Ž x, p 0 . ¨ † q ⭈⭈⭈ qbRq1Ž x, p 0 . ¨ †Rq 1 q o Ž ¨ †Rq 1 .
uniformly in Ž x, p 0 ., where b1Ž x, p 0 . bk Ž x, p 0 . is a polynomial in f Ž p 0 < x .rF Ž p 0 < x ., . . . ,
f Ž ky2. Ž p 0 < x .rF Ž p 0 < x . for k s 1, . . . , R q 1. In particular, b1Ž x, p 0 . s wŽ I y 1. f Ž p 0 < x .r Ž2 F Ž p 0 < x ..x1r 2 .
Because f Ž p 0 < x . is bounded away from zero and infinity on S Ž F U . by A2, and because 1 G F Ž p 0 < x .,
which is bounded away from zero by A5-Žiii., then s† Ž⭈, x . admits R q 1 bounded derivatives at
X
¨ s p 0 and hence on S Ž F U . with s† Ž⭈, ⭈ . bounded away from zero. It remains to show that the cross
partial derivatives up to order R q 1 are also bounded in the neighborhood of ¨ s p 0 . This is proved
as in the end of the proof of Lemma A1-Žii..
Q. E. D.
PROOF OF LEMMA A4: The proof is similar to that of Lemma A.2
Q. E. D.
565
FIRST PRICE AUCTIONS
C.2. Proofs of Lemmae in Appendix B
PROOF OF LEMMA B1: Ži. The Ž bk , x k .’s are bounded away from the boundaries of S2 Ž G 0 .. Thus
for m large enough, g m k Ž⭈, x, 2. s g 0 Ž⭈, x, 2., Gm k Ž⭈, x, 2. s G 0 Ž⭈, x, 2., and ␰ m k Ž⭈, x, 2. s ␰ 0 Ž⭈, x, 2. in
the neighborhood of b 0 Ž x, 2. or b 0 Ž x, 2.. Because g 0 Ž⭈, ⭈ , 2. is bounded away from zero on S2 Ž G 0 .,
then the support of Gm k is S Ž G 0 . for m large enough. Moreover, ŽB.2. implies that the support of
f m k Ž⭈, ⭈ , 2. is S2 Ž F0 .. The desired result follows as f m k Ž⭈, ⭈ , i . s f 0 Ž⭈, ⭈ , i . for i / 2.
To prove Žii., note that f m k Ž⭈, ⭈ . y f m j Ž⭈, ⭈ . s Ý i g I w f m k Ž⭈, ⭈ , i . y f m j Ž⭈, ⭈ , i .x s f m k Ž⭈, ⭈ , 2. y f m j Ž⭈,
⭈, 2.. Note also that Ž bk , x k . g C2 Ž B . implies Ž ¨ k , x k . g C Ž V ., where ¨ k ' ␰ 0 Ž bk , x k , 2.. Thus it
suffices to prove that < f m k Ž ¨ k , x k , 2. y f m j Ž ¨ k , x k , 2.< s C4 C3 ␭ 2rm R . Consider f m k Ž ¨ k , x k , 2.. Note
that ŽB.1. implies Gm k Ž bk , x k , 2. s G 0 Ž bk , x k , 2. and g m k Ž bk , x k , 2. s g 0 Ž bk , x k , 2.. Hence ¨ k '
X
Ž
.
Ž
.
␰ 0 Ž bk , x k , 2. s ␰ m k Ž bk , x k , 2., which implies ␰y1
m k ¨ k , x k , 2 s b k . Therefore, computing g m k b k , x k , 2
and using ŽB.2. give
Ž C.4.
f m k Ž ¨ k , x k , 2. s
g 03 Ž bk , x k , 2 .
X
X
2 g 02 Ž bk , x k , 2 . y G 0 Ž bk , x k , 2 .Ž g 0 Ž bk , x k , 2 . q ␭ 2 C3 ␾ Ž 0, 0 . rm R .
.
Consider next f m j Ž ¨ k , x k , 2.. From ŽB.1., we have g m j Ž⭈, ⭈ , 2. s g 0 Ž⭈, ⭈ , 2., Gm j Ž⭈, ⭈ , 2. s G 0 Ž⭈, ⭈ , 2., and
hence ␰ m j Ž⭈, ⭈ , 2. s ␰ 0 Ž⭈, ⭈ , 2. except on S Ž ␾m j ., which is a hypercube centered at Ž bj , x j .. Moreover,
for m sufficiently large SŽ ␾m k . and SŽ ␾m j . are disjoint so that Ž bk , x k . f SŽ ␾m j .. Hence
Ž
.
Ž .
␰ m j Ž bk , x k , 2. s ␰ 0 Ž bk , x k , 2. s ¨ k so that ␰y1
m j ¨ k , x k , 2 s b k . Thus B.2 gives
Ž C.5.
f m j Ž ¨ k , x k , 2. s
g 03 Ž bk , x k , 2 .
X
2 g 02 Ž bk , x k , 2 . y G 0 Ž bk , x k , 2 . g 0 Ž bk , x k , 2 .
.
X
Now compare ŽC.4. and ŽC.5.. As ␾ Ž0, 0. / 0 and G 0 Ž bk , x k , 2. ) c G 0 Žbecause the Ž bk , x k .’s are
far enough from the boundaries., the desired result follows.
The proof of Žiii. is more involved and is divided in three steps of which the first two establish
properties similar to Žiii. for ␰ m k and sm k , respectively.
Step 1: As g m k Ž⭈, ⭈ , i . s g 0 Ž⭈, ⭈ , i . except possibly when i s 2 and Ž b, x . g C2 Ž B ., then < g m k Ž⭈, ⭈ , ⭈ .
y g 0 Ž⭈, ⭈ , ⭈ .< r, S 2 ŽG 0 . s < g m k Ž⭈, ⭈ , 2. y g 0 Ž⭈, ⭈ , 2.< r, C 2 Ž B . and < Gm k Ž⭈, ⭈ , ⭈ . y G 0 Ž⭈, ⭈ , ⭈ .< r, S 2 ŽG 0 . s < Gm k Ž⭈, ⭈ , 2.
y G 0 Ž⭈, ⭈ , 2.< r, C 2 Ž B . for all r G 0. Now, using a change of variable, we have
Gm k Ž b, x, 2 . s G 0 Ž b, x, 2 . q
s G 0 Ž b, x, 2 . q
C3
␭2 m Rq2
C3
␭2 m Rq2
.
␾ Ž u, m ␭2 Ž x y x k .. du
Hmm␭␭Ž Žbbyb
Ž x .yb .
2
2
k
k
⌽ Ž m ␭2 Ž by bk . , m ␭2 Ž x y x k .. ,
b
where ⌽ Ž b, x . ' Hy1
␾ Ž u, x . du. Since g 0 admits up to R q 1 bounded continuous derivatives on
C2 Ž B . by Proposition 1-Živ., we have for any n 0 q n1 q ⭈⭈⭈ qn d s r, 0 F r F R q 1,
Ž C.6.
⭸ r Gm k Ž b, x, 2 .
⭸ n 0 b ⭸ n 1 x 1 ⭈⭈⭈ ⭸ n d x d
y
⭸ r G 0 Ž b, x, 2 .
⭸ n 0 b ⭸ n 1 x 1 ⭈⭈⭈ ⭸ n d x d
r
s
C3 Ž ␭2 m . ⭸ r⌽ Ž ␭2 m Ž by bk . , ␭2 m Ž x y x d ..
⭸ r g m k Ž b, x, 2 .
⭸
n0
b⭸
n1
x 1 ⭈⭈⭈ ⭸
n2
,
⭸ n 0 b ⭸ n 1 x 1 ⭈⭈⭈ ⭸ n d x d
␭2 m Rq2
xd
y
⭸ r g 0 Ž b, x, 2 .
⭸
n0
b ⭸ n 1 x 1 ⭈⭈⭈ ⭸ n 2 x d
r
s
C3 Ž ␭2 m . ⭸ r␾ Ž ␭2 m Ž by bk . , ␭2 m Ž x y x d ..
m Rq1
⭸ n 0 b ⭸ n 1 x 1 ⭈⭈⭈ ⭸ n d x d
,
566
E. GUERRE, I. PERRIGNE, AND Q. VUONG
for all Ž b, x . g C2 Ž B .. Thus ŽC.6. immediately gives < Gm k y G 0 < r, SŽG 0 . s C3 ␭2ry 1 O Ž1rm Rq 2yr . and
< g m k y g 0 < r, SŽG 0 . s C3 ␭2r O Ž1rm Rq 1yr .. Since ␰ m k Ž b, x, i . s 1 q Gm k Ž b, x, i .rŽ i y 1.Ž g m k Ž b, x, i .., the
r th partial derivatives of ␰ m k Ž⭈, ⭈ , i . are some polynomial functions in the partial derivatives of order
less than or equal to r of Gm k and g m k , divided by g mrqk 1. Since g m k is bounded away from 0, we
have uniformly in k
Ž C.7.
< ␰ m k y ␰ 0 < r , SŽG 0 . s C3 ␭2r O Ž 1rm Rq 1yr .
Ž r s 0, . . . , R q 1 . .
Step 2: We have ␰ m k Ž sm k Ž ¨ , x, i ., x, i . s ¨ for Ž ¨ , x, i . g S Ž F0 .. Lemma C1 and ŽC.7. show that
sm k admits up to R q 1 bounded continuous derivatives on SŽ F0 .. First we consider < sm k y s0 < 0, SŽ F 0 .
and show that
Ž C.8.
< sm k y s0 < 0 , SŽ F 0 . s C3 ␭2 O Ž 1rm Rq 1 . .
X
Since ␰ m k s ␰ 0 q C3 ␭2 O Ž1rm Rq 1 . uniformly by ŽC.7. and ␰ 0 G c␰ ) 0 by Lemma A2, we get
< sm k Ž ¨ , x, i . y s0 Ž ¨ , x, i . <
F
F
1
c␰
1
c␰
␰ 0 Ž sm k Ž ¨ , x, i . , x, i . y ␰ 0 Ž x 0 Ž ¨ , x, i . , x, i .
␰ m k Ž sm k Ž ¨ , x, i . , x, i . y ␰ 0 Ž s0 Ž ¨ , x, i . , x, i . q C3 ␭2 O Ž 1rm Rq 1 .
s C3 ␭2 O Ž 1rm Rq 1 . .
Next, letting sm k s sm k Ž ¨ , x, i ., ŽC.2. ᎐ ŽC.3. give
X
sm k s
1
␰ mX k Ž sm k , x, i .
⭸ sm k
and
⭸ xp
sy
⭸␰ m k Ž sm k , x, i . r⭸ x p
␰ mX k Ž sm k , x, i .
.
Differentiating one more time gives
Y
sm k s y
⭸ sXm k
⭸ xq
␰ mY k Ž sm k , x, i . sXm k
Ž ␰ mX k Ž sm k , x, i .. 2
sy
⭸ 2 sm k
⭸ xp ⭸ xq
,
1
Ž ␰ mX k Ž sm k ,
sy
x, i ..
2
ž
⭸ sm k
⭸ xq
q
⭸␰ mX k Ž sm k , x, i .
⭸ xq
/
,
1
Ž ␰ mX k Ž sm k , x, i .. 2
ž
X
= ␰ m k Ž sm k , x, i .
y
␰ mY k Ž sm k , x, i .
⭸␰ m k Ž sm k , x, i .
⭸ xp
½
½
⭸␰ mX k Ž sm k , x, i . ⭸ sm k
⭸ xp
␰ mY k Ž sm k , x, i .
⭸ xq
⭸ sm k
⭸ xq
q
q
⭸ 2␰ m k Ž sm k , x, i .
⭸ xp ⭸ xq
⭸␰ mX k Ž sm k , x, i .
⭸ xq
5/
5
.
X
An induction argument shows that any r th partial derivative of sm k times Ž ␰ m k Ž sm k , x, i .. r is a
polynomial function of the Ž1, . . . , r .th’s partial derivatives of ␰ m k taken at Ž sm k , x, i ., and of the
Ž1, . . . , r y 1.th’s partial derivatives of sm k . Thus, a bound for < ␰ mŽ rk. Ž sm k , x, i . y ␰ 0Ž r . Ž s0 , x, i .< 0, SŽ F 0 .
Žwhere ␰ mŽ rk. is any r th partial derivatives of ␰ m k . shall give a similar bound for < sm k y s0 < r, SŽ F 0 . ,
567
FIRST PRICE AUCTIONS
r s 1, . . . , R q 1. Thus, using equations ŽC.7. and ŽC.8. gives
< ␰ mŽ rk. Ž sm k , x, i . y ␰ 0Ž r . Ž s0 , x, i . < 0 , SŽ F 0 .
F < ␰ mŽ rk. Ž sm k , x, i . y ␰ 0Ž r . Ž sm k , x, i . < 0 , SŽ F 0 . q < ␰ 0Ž r . Ž sm k , x, i . y ␰ 0Ž r . Ž s0 , x, i . < 0 , SŽ F 0 .
F C3 ␭2r O Ž 1rm Rq 1yr .
q
½
< ␰ 0 < rq 1 , SŽG 0 . C3 ␭2 O Ž 1rm Rq 1 .
supŽ ¨ , x , i.g SŽ F 0 . < ␰ 0Ž r . Ž sm k ,
x,
Ž r s 0, . . . , R . ,
i . y ␰ 0Ž r . Ž s0 ,
x, i . <
Ž r s R q 1. .
Because the sup is oŽ1., we obtain
½
Ž C.9.
< sm k y s0 < r , SŽ F 0 . s C3 ␭2r O Ž 1rm Rq 1yr .
Ž r s 0, . . . , R . ,
< sm k y s0 < Rq 1 , SŽ F 0 . s C3 ␭2Rq 1 O Ž 1 . q o Ž 1 . .
X
Step 3: By construction, we have f m k Ž ¨ , x, i . s sm k Ž ¨ , x, i . g m k Ž sm k Ž ¨ , x, i ., x, i .. Thus the r th
X
partial derivative of f m k is a polynomial function of the Ž0, . . . , r .th derivatives of sm k , g m k , and sm k .
By the same argument as in Step 2, Žiii. follows from ŽC.6. and ŽC.9..
Q. E. D.
Further Lemmas: To prove Lemma B2 we need three lemmas. Lemma C2 studies the uniform
˜ ˜g, fˆŽ x ., and f˜Ž ¨ , x ., Lemma C3 studies their variances, and Lemma C4 establishes
bias of G,
exponential-type inequalities. Hereafter, the condition ‘‘for L sufficiently large’’ means:
Ži. for any Ž b, x ., K G Ž x, Ž i y j .rh g I . s 0 and K g Ž b, x, Ž i y j .rh g I . s 0 if j / i, and
Žii. for inner closed possibly expanding subsets C Ž B . s Di g I Ci Ž B ., C Ž V . and C Ž X . of S Ž G .,
SŽ f Ž ¨ , x .. and S Ž f Ž x .., respectively, we have
D
Ž ¨ , x . q S Ž h f .4 ; S Ž f Ž ¨ , x .. ,
D
D
Ž b, x . q S Ž h G . j S Ž h G .4 ; Ci X Ž B . ,
Ž ¨ , x .g C Ž V .
x q S Ž h X .4 ; S Ž f Ž x .. ,
xg C Ž X .
Ž b , x .g Ci Ž B .
X
X
where C Ž B . s Di g I Ci Ž B . is an inner closed subset of SŽ G ..
Let 5 ⭈ 5 a be the absolute norm for vectors, and
M gr s
M fr s
1
H
r!
1
r!
H
Ž u, x .
r
a
K g Ž u, x, 0 . du dx,
MGr s
Ž u, x .
r
a
K f Ž u, x . du dx,
M Xr s
1
H5 x 5 ar
K G Ž x, 0 . dx,
H5 x 5 ar
K X Ž x . du dx,
r!
1
r!
where x s Ž x 1 , . . . , x d .. The next lemma on uniform bias considers two cases whether the subset
C Ž B . expands to S Ž G . Žpart Ži.. or is fixed Žpart Žii... In particular, given A4, < Egy
˜ g < 0, C Ž B . is of
order Žlog LrL. R rŽ2 Rqdq3. in Ži., and has the optimal magnitude Žlog LrL.Ž Rq1.rŽ2 Rqdq3. in Žii..
The reason is that C Ž B . can expand to SŽ G . in Ži., while C Ž B . is fixed in Žii. so that g admits up to
R q 1 continuous bounded derivatives on C Ž B . from Proposition 1.
LEMMA C2: Gi¨ en A1᎐A4, we ha¨ e for L sufficiently large
Ži.
˜ G < 0 , C Ž B . F ␭GRq 1 MGRq1 < G < Rq1 , SŽG .rrG ,
< EGy
< Egy
˜ g < 0 , C Ž B . F ␭ Rg M gR < g < R , SŽ g .rr g ,
< Ef˜Ž ¨ , x . y f Ž ¨ , x . < 0 , C ŽV . F ␭ Rf M fr f Ž ¨ , x .
< EfˆŽ x . y f Ž
Ž ii .
1
x . < 0 , C Ž X . F ␭ Rq
M XRq1
X
f Ž x.
R , S Ž f .rr f
Rq1 , S Ž f .rr X
1
< Egy g < 0 , C Ž B . F ␭ Rq
M gRq1 < g < Rq1 , C X Ž B .rr U
g
g .
˜
,
,
568
PROOF
E. GUERRE, I. PERRIGNE, AND Q. VUONG
OF
LEMMA C2: Because the proofs are similar, we prove Žii. only. Using A1, we have
Eg˜Ž b, x, i . s E
s
1
h dq1
g
Kg
ž
b y Bp
hg
,
xy X
hg
/
, 0 |Ž I s i .
HHK g Ž u, y, 0. g Ž b y h g u, x y h g y, i . du dy.
Define ␥ Ž t . s g Ž b y th g u, x y th g y, i . y g Ž b, x, i . for t g w0, 1x. Since the supports of kernels are
X
intervals, Ž by th g u, x y th g y . g Ž b, x . q SŽ h g . ; Ci Ž B . for Ž b, x, i . g C Ž B . and t g w0, 1x. Moreover,
using Proposition 1, ␥ Ž t . admits up to R q 1 continuous bounded derivatives with
Ž u, y .
sup < ␥ Ž Rq1. Ž t . < F h Rq1
g
tg w0 , 1 x
Ž Rq1 .
< g < Rq 1 , C X Ž B . .
a
Thus a Taylor expansion with integral remainder gives
1
␥ Ž 1 . y ␥ Ž 0 . s ␥ Ž1. Ž 0 . q ⭈⭈⭈ q
R!
␥ Ž R. Ž 0 . q
H01
Ž1 y t . R
R!
␥ Ž Rq1. Ž t . dt,
where ␥ Ž r . Ž0. is a polynomial of order r in Ž u, y .. Using A3 it follows that
< Eg˜Ž b, x, i . y g Ž b, x, i . < 0 , C Ž B .
s
HK g Ž u, y, 0. Ž ␥ Ž1. y ␥ Ž0.. du dy
< g < Rq 1 , C X Ž B .
F h Rq1
g
ž
H01
Ž1 y t . R
R!
/ žH
dt =
Ž u, y .
Rq 1
K g Ž u,
a
y, 0 . du dy
/
1
s h Rq
M gRq1 < g < Rq 1 , C X Ž B . .
g
The desired bound follows from A4-Žii. and ŽB.6..
Q. E. D.
The next lemma obtains uniform variance bounds. Let
H
Q g s K g2 Ž u, x, 0 . du dx,
H
H
Q X s K X2 Ž x . dx.
QG s K G2 Ž x, 0 . dx,
Q f s K f2 Ž u, x . du dx,
H
LEMMA C3: Gi¨ en A1᎐A4, we ha¨ e for L sufficiently large
˜. < 0 , C Ž B . F
<Var Ž G
QG < G < 0 , SŽG .
d 2
␭G
rG
<Var Ž f˜Ž ¨ , x .. < 0 , C ŽV . F
log L
<Var Ž ˜
g . < 0, C Ž B. F
,
h 2f Q f < f Ž ¨ , x . < 0 , SŽ f .
␭ dq3
r f2
f
log L
,
Q g < g < 0 , SŽ g .
Ž rU
. log L
␭ dq1
g
g
<Var Ž f˜Ž x .. < 0 , C Ž X . F
2
,
Q f < f Ž x . < 0 , SŽ f .
␭ dX r X2 log L
.
569
FIRST PRICE AUCTIONS
˜ only, since the other bounds are proved similarly. For
PROOF OF LEMMA C3: We consider G
Ž b, x, i . g C Ž B ., we have by A1 and the convexity of the square function,
˜Ž b, x, i .. F
Var Ž G
F
s
1
L
ž
E
|Ž I s i .
i
Lih2Gd
1
d
Lh G
Ý
Ý
| Ž Bp F b .
ps1
i
1
i
1
d
hG
KG
ž
E | Ž I s i . | Ž Bp F b . K G
ps1
ž
ž
xyX
hG
xy X
hG
//
//
2
,0
2
,0
HK G2 Ž y, 0. G Ž b, xy hG y, i . dy.
The desired bound follows from A4-Žii. and ŽB.6.. Note the somewhat different bound for VarŽ f˜Ž ¨ , x ..
because of the suboptimality of h f .
Q. E. D.
The next lemma derives some exponential-type inequalities for the probabilities of deviations of
˜Ž ¨ , x, i . y GŽ b, x, i ., ˜g Ž b, x, i . y g Ž b, x, i ., f˜Ž ¨ , x . y f Ž ¨ , x ., and fˆŽ x . y f Ž x . in supnorm over Ci Ž B .,
G
Ci Ž B ., C Ž V ., and C Ž X ., respectively. To this end, we need to introduce coverings of these sets. For
instance, Ci Ž B . is covered by N inner ‘‘balls’’ of the form
Bi n ' Bi ŽŽ bn , x n . ; ⌬ . s Ž b, x . g Si Ž G . : b g w bn y ⌬ , bn q ⌬ x , xg w x n y ⌬ , x n q ⌬ x4 ,
where ⌬ ) 0, and Ž bn , x n . g Ci Ž B . for n s 1, . . . , N. Moreover, we consider minimal coverings, i.e.,
coverings for which N is the smallest number denoted N Ž Ci Ž B ., ⌬ .. Similar notions apply to the sets
C Ž V . and C Ž X ..
U
Hereafter, when there is no possible confusion, we simplify the notation by omitting the in
< ⭈ < r, ) when the supnorm is taken over the whole support of the function. Let
eG Ž t , ␶ . s t q
rG < K G < 0
d
hG
(
t
log L
L
Rq1
q ␭G
MGRq1 < G < Rq 1
q ␶ Ž 2 d < K G < 1 q 4 h G < K G < 0 g Ž b, i . 0 . ,
e g Ž t , ␶ . s t q 2 d < K g < 1␶ q ␭ Rg M gR < g < R ,
e f Ž t , ␶ . s t q 2 d < K f < 1␶ q ␭ Rf M fR f Ž ¨ , x .
R,
1
e X Ž t , ␶ . s t q 2 d < K X < 1␶ q ␭ Rq
M XRq1 f Ž x .
X
Rq1 ,
1
eUg Ž t , ␶ . s t q 2 d < K g < 1␶ q ␭ Rq
M gRq1 < g < Rq1 , C X Ž B . ,
g
where t and ␶ are arbitrary strictly positive numbers. Define
ž
dq1
PG Ž t , ␶ . s 2 N Ž C Ž B . , ␶ h G
rrG . exp y
X
ž
ž
dq1
PG Ž t , ␶ . s 2 N Ž C Ž B . , ␶ h G
rrG . exp y
Pg Ž t , ␶ . s 2 N Ž C Ž B . , ␶ h dq2
rr g . exp y
g
/
d 2
␭G
t log L
,
2 QG < G < 0 q 4 t < K G < 0 r Ž 3rG .
t 2 log L
'
dq1
4 g Ž b, i . 0 ␶ h G
rrG q 2 t log LrL r3
3 2
Ž ␭ dq
t log L . rh2g
g
2 Q g < g < 0 q 4 t < K g < 0r Ž 3r g .
/
,
/
,
570
E. GUERRE, I. PERRIGNE, AND Q. VUONG
ž
3 2
Ž ␭ dq
t log L . rh2f
f
Pf Ž t , ␶ . s 2 N Ž C Ž V . , ␶ h dq2
rr f . exp y
f
2 Q f f Ž ¨ , x . 0 q 4 t < K f < 0 r Ž 3r f .
ž
ž
.
PX Ž t , ␶ . s 2 N Ž C Ž X . , ␶ h dq1
X rr X exp y
.
PgU Ž t , ␶ . s 2 N Ž C Ž B . , ␶ h dq2
rr U
g
g exp y
/
/
,
␭ dX t 2 log L
,
f Ž x . 0 q 4 t < K X < 0r Ž 3r X .
2 QX
1 2
␭ dq
t log L
g
.
2 Q g < g < 0 q 4 t < K g < 0r Ž 3r U
g
/
.
LEMMA C4: Gi¨ en A1᎐A4, for any t ) 0, ␶ ) 0 and i g I , we ha¨ e for L sufficiently large
Ži.
X
˜ G < 0 , C iŽ B . ) eG Ž t , ␶ .. F PG Ž t , ␶ . q PG Ž t , ␶ . ,
Pr Ž rG < Gy
Pr Ž r g < ˜
g y g < 0 , C iŽ B . ) e g Ž t , ␶ .. F Pg Ž t , ␶ . ,
Pr Ž r f < f˜y f < 0 , C ŽV . ) e f Ž t , ␶ .. F Pf Ž t , ␶ . ,
Pr Ž r X < fˆy f < 0 , C Ž X . ) e f Ž t , ␶ .. F PX Ž t , ␶ . ,
Ž ii .
PROOF
< g y g < 0 , C iŽ B . ) eUg Ž t , ␶ .. F PgU Ž t , ␶ . .
Pr Ž r U
g ˜
OF
˜ as it is the most involved.
LEMMA C4: We detail the proof for G,
Step 1: From Lemma C2 and the triangular inequality, we obtain
Ž C.10.
˜ G < 0 , C iŽ B . ) eG Ž t , ␶ ..
Pr Ž rG < Gy
Ž
˜ EG˜< 0 , C iŽ B . q rG < EGy
˜ G < 0 , C i Ž B . ) eG Ž t , ␶ .
F Pr rG < Gy
.
˜ EG˜< 0 , C iŽ B . ) eG Ž t , ␶ . y ␭GRq 1 MGRq1 < G < Rq1 . .
F Pr Ž rG < Gy
L
˜Ž b, x, i . y EG˜Ž b, x, i . s Ž1rL.Ý ms
Ž
.
Note that, for L sufficiently large, G
1 ␨ m L b, x, i , where
␨m LŽ b, x, i . s
i
1
Ý
d
ih G
ps1
ž
ž
| Ž Bp m - b . K G
yE | Ž B - b . K G
ž
xyX
hG
ž
x y Xm
hG
/
/
//
, 0 | Ž Im s i .
, 0 |Ž I s i .
.
The ␨m L ’s, m s 1, . . . , L, are i.i.d. centered variables. Moreover, using Lemma C3, we have
rG ␨m LŽ b, x, i . F
2 < K G < 0 rG
d
hG
s
2 L< KG < 0
d
␭G
rG log L
,
˜Ž b, x, i .. F
Var Ž rG ␨m LŽ b, x, i .. s LrG2 Var Ž G
LQG < G < 0
d
␭G
log L
for any Ž b, x, i . g C Ž B .. Hence, Bernstein inequality Žsee Serfling Ž1980, p. 95.. gives
Ž C.11.
ž
˜Ž b, x, i . y EG˜Ž b, x, i . < ) t . F 2 exp y
Pr Ž rG < G
s
t 2 log L
d
d
2 Q G < G < 0 r␭ G
q Ž 4 t < K G < 0 . r Ž 3 ␭G
rG .
PG Ž t , ␶ .
dq 1
N Ž C Ž B . , ␶ hG
rrG .
for any Ž b, x, i . g C Ž B ., t, and L, where the equality follows from the definition of PG Ž t, ␶ ..
/
571
FIRST PRICE AUCTIONS
Step 2: Consider a minimal covering of Ci Ž B . for some ⌬ ) 0. For any Ž b, x . g Bi n , we have
˜Ž b, x, i . y EG˜Ž b, x, i . <
rG < G
L
rG
F sup
Ý
L
1FnFN
q sup
1FnFN
␨m LŽ bn , x n , i .
ms1
rG
sup
Ž b , x .g Bi n L
L
Ý
Ž ␨m LŽ bn , x n , i . y ␨m LŽ b, x, i .. .
ms1
This gives
Ž C.12.
˜ EG˜< 0 , C iŽ B . ) eG Ž t , ␶ . y ␭GRq 1 MGRq1 < G < Rq1 .
Pr Ž rG < Gy
F Pr
ž
˜Ž bn , x n , i . y EG˜Ž bn , x n , i . < ) t
sup rG < G
1FnFN
q Pr
ž
sup
1FnFN
rG
sup
Ž b , x .g Bi n L
/
L
Ý
Ž ␨m LŽ bn , x n , i . y ␨m LŽ b, x, i ..
ms1
/
Rq 1
) eG Ž t , ␶ . y t y ␭G
MGRq1 < G < Rq 1 .
We now give some bounds for the increments at Ž bn , x n , i .. For any Ž b, x . g Bi n , we have
Ž C.13.
| Ž B F bn .
d
hG
F
KG
ž
d ⌬< KG <1
dq1
hG
xn y X
hG
q
/
,0 y
< KG < 0
d
hG
|Ž B F b .
d
hG
KG
ž
xy X
hG
,0
/
| Ž bn y ⌬ F B F bn q ⌬ . .
Using ŽC.13. twice in the definition of ␨m LŽ b, x, i . and the triangular inequality, we obtain
␨m LŽ bn , x i , i . y ␨m LŽ b, x, i .
F
i
1
Ý
i
ps1
qE
F
ž
ž
d ⌬< KG <1
dq1
hG
d ⌬< KG <1
dq1
hG
2 d ⌬< KG <1
dq1
hG
q
q
q
< KG < 0
d
hG
< KG < 0
d
hG
< KG < 0
d
hG
1
i
/
| Ž bn y ⌬ F B F bn q ⌬ . | Ž I s i .
Ž 2 E w | Ž bn y ⌬ F B F bn q ⌬ . | Ž I s i .x q Zm LŽ bn ..
where we have used Ž1ri .Ýips 1|Ž Im s i . F 1, Ew|Ž I s i .x F 1, and
Zm LŽ b . '
/
| Ž bn y ⌬ F Bp m F bn q ⌬ . | Ž Im s i .
i
Ý Ž|Ž by ⌬ F Bp m F b q ⌬ .|Ž Im s i .
ps1
yE w | Ž by ⌬ F B F b q ⌬ . | Ž I s i .x. .
572
E. GUERRE, I. PERRIGNE, AND Q. VUONG
Because Ew|Ž bn y ⌬ F B F bn q ⌬ .|Ž I s i .x F 2 ⌬ < g Ž b, i .< 0 it follows that
Ž C.14.
sup
1FnFN
rG
sup
Ž b , x .g Bi n L
F
2 drG ⌬ < K G < 1
dq1
hG
L
Ý
Ž ␨m LŽ bn , x n , i . y ␨m LŽ b, x, i ..
ms1
q
4 rG ⌬ < K G < 0 g Ž b, i .
0
d
hG
rG < K G < 0
q sup
1FnFN
d
Lh G
L
Ý
Zm LŽ bn . .
ms1
L
Step 3: We study sup1 F n F N Ž1rL.Ý ms1
Zm LŽ bn .. We have < Zm LŽ bn .< F 1 while VarŽ Zm LŽ bn .. F
Varw|Ž bn y ⌬ F B F bn q ⌬ .|Ž I s i .x F 2 ⌬ < g Ž b, i .< 0 . Thus Bernstein inequality gives
Ž C.15.
Pr
ž
Ý
L
1FnFN
N
F
L
1
sup
Ý Pr
ns1
'
Zm LŽ bn . ) t log LrL
ms1
ž
L
1
L
Ý
/
'
Zm LŽ bn . ) t log LrL
ms1
ž
/
t 2 log L
F 2 N Ž C Ž B . , ⌬ . exp y
'
4 g Ž b, i . 0 ⌬ q 2 t log LrL r3
/
.
dq 1
Step 4: Let ⌬ s ␶ h G
rrG . Hence, ŽC.14. and the definition of eG Ž t, ␶ . imply
Pr
ž
rG
sup
Ž b , x .g Bi n L
sup
1FnFN
L
Ý
Ž ␨m l Ž bn , x n , i . y ␨m l Ž b, x, i ..
ms1
Rq 1
) eG Ž t , ␶ . y t y ␭G
MGRq1 < G < Rq 1
F Pr
ž
rG < K G < 0
sup
d
Lh G
1FnFN
/
L
Ý
Zm LŽ bn . )
ms1
ž
dq 1
F 2 N Ž C Ž B . , ␶ hG
rrG . exp y
X
rG < K G < 0
d
hG
'
t log LrL
t 2 log L
'
/
dq1
4 g Ž b, i . 0 ␶ h G
rrG q 2 t log LrL r3
/
s PG Ž t , ␶ . ,
using ŽC.15.. The desired result now follows from ŽC.10., ŽC.12., and ŽC.11. as
˜ G < 0 , C iŽ B . ) eG Ž t , ␶ ..
Pr Ž rG < Gy
N
F
Ý Pr Ž rG < G˜Ž bn , x n , i . y EG˜Ž bn , x n , i . < ) t . q PGX Ž t , ␶ .
ns1
X
s PG Ž t , ␶ . q PG Ž t , ␶ . .
Proofs of the other inequalities of the lemma are simpler because Step 3 can be dropped. Indeed,
the corresponding bounds ŽC.13. only involve the derivative of the kernel. Thus the new bounds in
2
ŽC.14. only depend upon the first term. For instance, we choose ⌬ s ␶ h dq
rr g to prove the second
g
bound, etc.
Q. E. D.
PROOF OF LEMMA B2: The covering number N is of order ⌬y␦ , where ␦ is the dimension of the
covered set. Hence, in view of A4-Žii. and ŽB.6., the various covering numbers in the upper bounds of
FIRST PRICE AUCTIONS
573
dq 1
Lemma C4 are of the order Ž Lrlog L.␩ for some ␩ ) 0. For instance, N Ž C Ž B ., ␶ h G
rrG . s
O Ž Lrlog L.Ž dq1.Ž Rqdq2.rŽ2 Rqdq2.. Thus, by taking t sufficiently large, it is easy to see that the
series PG Ž t, ␶ . Žsay. can be made convergent as L ª ⬁. The desired result follows from the
Q. E. D.
Borel-Cantelli Lemma and the fact that eG Ž t, ␶ . s O Ž1..
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