..
SECOND-ORDER ASYMPTOTIC RELAnONS AND
GOODNESS-OF-FIT TESTS
by
11
/
JANA JURECKOVA
and
PRANAB K. SEN
Department of Biostatistics
University of North Carolina
Institute of Statistics
Mimeo Series No. 2189
May 1998
SECOND-ORDER ASYMPTOTIC RELATIONS AND
•
GOODNESS-OF-FIT TESTS*t
JANA JURECKOVA AND PRANAB KUMAR SENt
Charles University, Prague, and University of North Carolina at Chapel Hill
For goodness-of-fit tests for a model (distribution) admitting nuisance location or scale parameters,
second-order asymptotic distributional representations are exploited for some robust estimators that are
asymptotically first-order equivalent; their difference then provides a goodness-of-fit test criterion, whose
asymptotic properties are studied under the null hypothesis as well as under local alternatives.
··
1. Introduction. Let {Xi; i? I} be a sequence of independent, identically distributed (i. i. d.) random
variables with an absolutely continuous distribution function (d.f.) F((x - (h)/8 2 ), x E IR where 81 E IR,
and 82 E IR+ are unknown location and scale parameters; we let 0 = (8 1 ,8 2 ) and denote the parameter
space by 0. Let Fa be a hypothetical d.f., absolutely continuous with density fa, symmetric around 0.
We intend to test the null hypothesis of goodness-of-fit (GOF):
H a : F == Fa
(1.1 )
vs.
K: F =F Fa, 0 nuisance.
In the classical GOF testing problem with known 0, we may reduce the testing problem to the case of
uniform R(O, 1) d.f by means of the probability integral transformation; hence, the classical KolmogorovSmirnov, Cramer-von Mises and allied GOF tests, that are exact distribution-free (EDF), are applicable.
The situation is more complex when fJ is nuisance. In some specific cases, such as the normal Fa, we
could adapt a transformation on the Xi eliminating 0 and then apply the classical GOF tests. For
example, Durbin (1961) formed the residuals (Xi - Xn ) / Sn, i = 1, ... ,n by using the sample mean Xn
and sample standard deviation Sn, characterized the spherical-uniformity property for normal Fa and
used these residuals for GOF testing. However, the distribution theory is more complex in such a case.
Also for normal Fa, Shapiro and Wilk (1965) exploited the best linear unbiased estimators (BLUE) and,
they formulated a highly intuitive GOF test for normality.
contrasting with
Both of these tests seem to suffer from two drawbacks: (i) spherical uniformity or the specific distributional properties of the BLUE are are tied-down to the assumed normality of Fa; for non-normal Fa
we may need some other approach, and (ii) because these tests involve the highly nonrobust
they are
likely to be sensitive to the outliers and error contaminations (even to an infinitesimal extent).
Bhattacharyya and Sen (1977) considered an alternative approach to GOF testing, applicable not
only for testing the normality but any Fa admitting a minimal sufficient statistic for the associated
s;,
.
...
s;,
'Key words and phrases: Hadamard differentiability; logistic model; M-, L- and R-estimator; nuisance location and
scale; one-step version; SOADR; spherical uniformity.
tAMS 1991 subject classification: 60G99, 62G10, 62G20
tThis work was supported by the US-Czech Collaborative Research Grant (NSF INT-9600518 and ES 046/1996); the
first author's research was also supported by the Czech Republic Grant 201/96/0230
1
J. JURECKOV A AND
2
•
•
P. K.
SEN
parameters; their test criterion is based on the difference Fn - F~ of the empirical d.f Fn and its RaoBlackwell version F~. Though such test is exact (under Fa), its sampling distribution is usually quite
complicated. In asymptotic considerations we may use the dominance of the distribution of a functional
r.p(Fn - F~) by r.p(Fn - Fa), but it may result in considerable conservativeness. Moreover, robustness
aspects of such GOF tests merit careful examination.
There has been another line of attack for GOF tests: estimate 0 by 0, (asymptotically) optimal under
Fa, consider the d.f Fn
F(., B) and base a GOF test on Fn - Fn . Unfortunately, the asymptotic
1 2
representation for n / (Fn - F n ), consists of two terms, one weakly converging to a Brownian bridge
(under Fa), but the other one is generally a complex functional of a Gaussian function (and cannot be
treated as a drift function, linear or not, even under H o). For a broad overview of such GOF tests, we
may refer to Durbin (1973).
Motivated by these GOF tests (in the presence of nuisance parameters), and having the robustness
aspects in mind, we advocate the use of robust (L-, M -, R-) estimators of 0; we put special emphasis on
the location parameter in this context, and a similar case can be worked out for scale parameter oriented
GOF tests. We motivate our proposed testing procedure as follows. Let T 1 , T 2 be two estimators
of 01 , translation equivariant, such that each of them admits a second order asymptotic distributional
representation (SOADR)
=
(1.2)
·•
1
Tnj - 01 = ;;
n
L 'l/Jj,p(X; -
0d + ~n,j,P + op(n- 1),
j
= 1,2,
;=1
where the score function 'l/Jj,P generally depends on F and is so standardized that IEPo'l/Jj (X) = 0 and
~n,j,P are the second order terms (typically Op(n- 1)), j = 1,2. The ~n,j,P are typically functionals
reflecting quadratic variation of Fn - Fa, and hence they are natural candidates for GOF test statistics.
Even if T 1 , T 2 belong to different classes of estimators, they could be first order asymptotically equivalent
under Fa in the sense that
(1.3)
Or, in other words, the influence functions of Tn! and Tn2 coincide at and only at Fa. First order
asymptotically equivalent M-, L- and R-estimators were studied by a host of researchers and reported in
Chapter 7 of Jureckova. and Sen (1996). Generally, 'l/J1,P(X) == 'l/J2,P(X) holds in the family of logconcave
densities if and only if F == Fa. If T 1 , T2 are first order asymptotically equivalent under Fa, then we have
at Fa
(1.4)
...
Then n(T1 - T 2), independent of the nuisance 01, may be used as a test criterion. Whenever T 1,T 2 are
scale-equivariant, n(T1 - T 2) may as well be studentized by any (robust) translation-invariant estimator
of O2 . Under the null hypothesis, it has an asymptotic (non-normal) distribution. On the other hand, if
F i= Fa, then 'l/J1,P(X) and 'l/J2,P(X) are different and, by (1.2), n 1/ 2(T1 - T 2) has an asymptotic normal
distribution; hence, n(T1 - T 2) = Op(n 1/ 2), what insures a high order consistency of GOF test based on
(1.4) under non-local alternatives. Under contiguous alternatives, which in our case could be formulated
in a reasonably broad (non- or semi-)parametric way, (1.2) leads to a nondegenerate non-null distribution
of n(T1 - T 2 ).
The main tools used in the present paper are the SOADR of the type (1.2) supplemented with the
asymptotic distribution of ~n,p. The SOADR results, often, could be simplified with the aid of the
Hadamard expansions of the pertaining functionals, though SOADR is not restricted to only Hadamard
SOADR FOR GOODNESS-OF-FIT-TESTS
..
..
3
differentiable functionals. Remark that L-estimators are location-scale equivariant and also Hadamard
differentiable under fairly general regularity conditions. The R-estimators are location and scale equivariant and Hadamard differentiable for bounded score generating functions: this is the case of the Restimator based on the Wilcoxon scores (viz., Hodges and Lehmann (1963), Sen (1963)), and we shall
mostly apply this in our GOF tests. In Section 2, we incorporate the Wilcoxon score R-estimator and
its dual L-estimator for constructing a GOF test statistic that can be used when (J is nuisance (vector).
Since both of these estimators are Hadamard differentiable, by appealing to the allied SOADR results on
the difference of two Hadamard-differentiable functionals outlined in the Appendix, we are able to study
the asymptotic null distribution.
The M -estimators of location are generally not scale-equivariant. Nevertheless, if we have only a
nuisance location parameter in the formulation of H o, such M -estimators can be used to provide robust
GOF tests. Led by this motivation, in Section 3 we consider the pair of M - and L-estimators, say M n
and L n , first-order asymptotically equivalent under H o. Our GOF test criterion is based on the difference
MAl) - Ln where MAl) is the one-step version of M n , starting at Ln. We shall derive asymptotic null
distribution of the criterion, as well as the asymptotic distribution under the local alternatives.
2. GOF tests based on Wilcoxon scores R-estimator and its dual L-estimator. Let Xl, ... , X n
be i.i.d. observations with the distribution function F(x - (}) where F(x) + F( -x) = 1 "Ix E IR1 , F has
an absolutely continuous density f> 0 and finite Fisher's information I(F).
I. As an inversion of the Wilcoxon one-sample signed-rank test, we define the Wilcoxon score Restimator Tn of () as the solution of the estimating equation
(2.1)
where Fn (x) = ~ 2:7=1 I[Xi ~ x] is the empirical distribution function based on Xl, ... , X n . The solution
of (2.1) could be written in an explicit form
Tn=med{Xi~Xi :l~i~j~n}.
(2.2)
The corresponding functional T(F) = (} may be defined implicitly as the root of the equation
JR[F(X) - F(2t - x)]dF(x) =
(2.3)
o.
The following theorem presents the allied SOADR result; by virtue of location and scale equivariance,
0, (}2 1, for simplicity of notation.
without loss of generality, we let (}1
=
=
Theorem 2.1 Let Xl, . .. , X n be i. i. d. observations with the distribution function F where F (x) +
F(-x) 1, x E JR, and F has two bounded derivatives f, f'. Then the R-estimatorTn defined in (2.1)
admits the SOADR :
=
-I nL (F(Xi) n'"V .
I
(2.4)
.=1
_~U(2) +
2"1
n
1
-)
2
0 (n- 1 )
P
In
+ -n'"V
I
L (F(X
.
.=1
lIn
i) -
-) -
L (f(Xi) - "I)
2 wy.
.=1
J. JURECKOV A AND
4
P. K.
SEN
where
(2.5)
and
•
(2.6)
Proof. We shall use the notation OE(.) and OE(.) in the sense that An = OE(a n ) means IEIAnl = O(a n )
and An oE(a n ) means IEIAnl o(a n ) as n -+ 00. Denote
=
=
(2.7)
Then noting that Yn ( y'n"Tn ) = 0, we have
(2.8)
Then we could write
Yn(O)
L.
[Fn(x) - Fn(-x)]dFn(x)
n
n
L (I[Xj ~ X;] - I[Xj ~ -X;])
n- 2 L
;=1 j=l
n
(2n)-1 - (1- n- 1)(Un - IEUn ) - n- 1(n- 1 L I(X; ~ 0) -
··
1
"2)
;=1
(2.9)
where
(2.10)
1
Un
=
1
n
n(n -1) LLI[X; +Xj ~ 0],
IEUn ="2'
;=1 ji::;
Therefore, by the Hoeffding decomposition of U-statistics,
2 n
1
(IE{I[X; +Xj ~ 0] IX;} --)
2
= - L
n
;=1
1
+ n(n _ 1)
(I[X;
+ Xj ~
+ Xj ~
O]IXj}
L
0] - IE{I[X;
+ Xj ~
O]IX;}
l:5;¢j:5 n
-IE{I[X;
+ ~)
2 n
1
= - "(F(-X;) --)
n LJ
2
;=1
+ (
•
1
1
{I[X; + X j ~ 0] - F(-X;) - F(-Xj) +"2}
) L
n n - 1 .< ...
'_J,...:5 n
(2.11)
= 2U~1) + U~2)
(say)
where y'n"U~l) ~ N(O, 1/12) and (n -1)U~2) ~ I:~o )..,.(Z~ -1) where Z,.'s are i.i.d. standard normal
variables, and {A,.} is a sequence of eigenvalues of the functional U~2)(.) E L 2(F) corresponding to
SOADR
5
FOR GOODNESS-OF-FIT-TESTS
pertaining orthonomal functions. Moreover,
1
= n2
n
2u
n
~f;1[Xi
1
n
= n 2 (L1[O
+Xj
< Xi ~
i=l
~ .;n]
fi-] +
1
= n - 1 U~(u)
n
uniformly for
lui ~
K, 0
< K < 00
l~i;i:j~n
+ Xj ~
;D
2
+ -U~(u)
n
+ OE(n-7/4)
any fixed positive number, where
U~(u)= n(n~l)
(2,13)
1[Xi
n - 1
= -[Fn(u/vn) - Fn(O)]
n
(2.12)
L
L
1[O<Xi+Xj
l~i;i:j~n
~ ~],
and
IEU~(u)
=
L[F(~
=
2~
yn
- x) - F(-x)]dF(x)
2
r
f(-x)dF(x) + 2u r j'(-x)dF(x) + oE(n- 1)
l
n l
R
R
2u
-1
= .;n'+OE(n )
(2.14)
uniformly in
lui ~ I<.
Again, the Hoeffding decomposition to
U~ (u)
leads to
(2.15)
where we write U~l)(u) as
(1 )
n n- 1
=
L
'J,
'<
L[F(~
r:::
= 2u
yn
{IE(I[O
< Xi + X j ~ 2un-1/2IXi) - IEU~(u)}
1:5'",) _n
1
- x) - F(-x)]d(Fn(x) - F(x))
f(-x)d(Fn(x) - F(x))
R
+ -2u21
n
f ' (-x)d(Fn(x) - F(x)),
R
and hence,
(2.16)
as n -+
•
00,
uniformly in
lui ~
K, where
1 n
Z~ = -L(f(-Xi)-,)
(2.17)
.;n i=l
Similarly we could show that
(2.18)
IEU~2)(u) = 0,
2
IE(U~2)(u))2 = ~3 OE(l) 'iu E [-K, K].
J. JURECKOV A AND
6
P. K.
SEN
Combining (2.16), (2.17) and (2.15), we obtain
•
•
2
(2.19)
n IE[
U~(u) - ~,( 1 + ,~z~)] 2 = 0(:2)
for lui
~ f{
and this implies, by Theorem 12.1 of Billingsley (1968), that
(2.20)
hence
(2.21)
n (Yn(O) - Yn(u)) --1
lul$K n -
n sup
I
* I = op(I).
r:;:' ( 1 + 'v.1~Zn)
n
2u
vn
Inserting u -+ ..jiiTn into (2.21) and regarding (2.9) and (2.11) leads to the SOADR result:
(2.22)
.,
II. Let L n be the L-estimator based on the weight function J(t)
where under H a , F == Fa, J(t) is completely known. Then
= ,-1
o
f(F-1(t)), , = JR P(x)dx),
n
(2.23)
Ln =
L CniXn:i
i=1
with
(2.24)
Cni
=
l
i/n
J(t)dt, i
= 1, ... , n,
(i-1}/n
and we may set
(2.25)
Ln =
L
xJ(Fn(x))dFn(x).
L n admits a SOADR, characterized in the following theorem (where again, without loss of generality, we
set B1 = 0,B 2 = 1):
Theorem 2.2 Under the above conditions, L n admits the expansion
(2.26)
where B(F) = Bd= 0),
-(1)
(2.27)
•
Ln
and
(2.28)
Moreover, as n -+
(2.29)
L~2) = 2~
00,
L
1 ~
1
LJ(F(Xd - 2)
n,
= -
;=1
(-f'(x)/f(x))(Fn(x) - F(x))2dF(x).
SOADR FOR GOODNESS-OF-FIT-TESTS
7
and
(2.30)
as n
-l- 00,
where B = {B(t) : 0 :S t :S I} is the Brownian bridge and
<p(t) =
- f'(F- 1 (t)
f(F-1(t) , 0 < t < 1.
PROOF. For the sake of brevity, we let Bn(x) = Fn(x) - F(x), x E lR, and write
Ln
=
=
(2.31)
L
xJ(Fn(x))dFn(x)
L
xJ(F(x)
+ Bn(x))d(F(x) + Bn(x))
and this, after some arithmetics and using the special form of J(.), leads to (2.26) with
L~l)
(2.32)
n,
= -,11R F(x)dBn(x) = -1
1
n
:L)F(X;) - -2)
;:1
and
(2.33)
:
The remaining propositions follow from (2.32) and (2.33).
0
Let's now consider the GOF testing problem based on the pair of statistics (Tn, L n ), studied above.
Let Xl,' .. , X n be independent observations with dJ. F(x; 9), where 9 is a nuisance parmeter (vector).
Under the null hypothesis, F == Fo, so for the L-estimator, we put J(t) = ,0"1 fo(Fo- 1(t)), 0 < t < 1
with ,0 = f R f~(x)dx. Since both the estimators are location and scale equivariant, the distribution of
n(Tn - L n )/()2 is independent of 8, and without loss of generality (WLOG), we put ()1
0, ()2
1. It
follows from Theorems 2.1 and 2.2 that under H o,
=
(2.34) n(Tn - L n ) =
~
~
1
1 t
(Fo(X;) - -2 ). 1 t (fo(X;) ,oyn ;=1
,oyn ;=1
=
'0) - -.!!:.-U~2)
- nL~2) + op(l)
2'0
where U~2) and L~2) are defined as in before, but for the dJ. Fo. Since ()2 is unknown, we use any
consistent estimator of a scale parameter (viz., the inter-quartile range), say 02,n, and consider the test
statistic
(2.35)
which will have the same asymptotic distribution as of n(Tn - L n )/()2. Further note that by (2.34),
the asymptotic null distribution of Zn is nondegenerate, and hence the asymptotic critical levels are all
0(1). On the other hand, if F f. Fo, F symmetric, then the distribution of n 1/ 2 (Tn - L n ) will still be
independent of ()1, and
•
(2.36)
n 1/ 2 (Tn - L n ) = n- 1/ 2 ,-1 t[F(X;) -
where
1/>(x) = -
i:
;=1
~]- n- 1/ 2
t
1,b(X;)
;=1
J(F(y)){I[x :S y]- F(y)}dy,
x E lR
1
+ Op(l),
8
•
·•
J. JURECKOVA AND
SEN
and J(t) = 1'0 1 fo(Fo- 1 (t)), 0 < t < 1. Therefore, under such a fixed alternative, Zn is Op(n 1 / 2 ), so that
the test based on Zn will be consistent. When the unknown F is not symmetric, the centering constants
for Tn and L n may not be the same, so that Zn would be Op (n), and thereby, the consistency property
would remain in tact. Granted this consistency property, we could consider some local alternatives and
study the behavior of the proposed test. Toward this end, in the last section, we will consider some
general types of contiguous alternatives that are not necessarily of the parametric type, and indicate the
merits and demerits of the proposed GOF test relative to some other alternative ones proposed earlier.
In order to use the GOF test in practice, we need to have a handle over the null hypothesis distribution
of Zn, at least in the asymptotic case. Though (2.34) can be used to study the asymptotic null distribution,
2
it is not very simple. Recall that U~) is a U -stastistic that is stationary of order 1, while Lh ) can also
be expressed as a. U -statistic (or more precisely a von Mises (1947) functional) that is also stationary of
order 1; the first term on the right hand side of (2.34) is the product of two normal variables, and its
distribution theory can be handled in a more manageable way. Therefore, to study the asymptotic null
distribution of Zn, we need to study the asymptotic distributions of linear combinations of U-statistics
that are first order stationary. This has been accomplished in the Appendix by reference to some existing
SOADR results on statistical functionals that are Hadamard differentiable or differentiable in the von
Mises (1947) sense. In that way, the results would apply to a bigger class of GOF test statistics that
are characterizable in the same sense. Based on the advent of modern computational techniques and
algorithms, we may consider the following simple simulation (Monte Carlo) procedures for estimating the
(asymptotic) null distribution of Zn.
Generate {Y1 , .•• , Yn } as independent copies of a r.v. from the hypothesized distribution F o. Based on
this set, compute the corresponding Tn, L n , B2 ,n, and hence ZN. Repeat this random experiment M times
(with independent samples of size n), where M is a large positive integer, say 500. Plot these simulated
Zn values, and obtain the corresponding empirical distribution. By the Glivenko-Cantelli lemma, this
simulated empirical distribution converges almost surely (uniformly) to the true limiting null distribution
of Zn, the existence of which is ensured by its SOADR representation in (2.34). This simulation technique
also applies to other GOF test statistics that can be expressed in a form similar to Zn.
In the rest of this section, we illustrate the proposed GOF test for some specific Fa.
First, consider the classical Normal Fo. In this case, l'
f~oo f6(X)dx
(2V1i)-1 and fo(x) =
2
(21l')-1/2 exp{
}, so that for L n we can use the normal quantile densities as tabulated extensively in
the literature. In this case, we can use the interquartile range for the estimation of ()2. Moreover note
that J(t) = fo(Fo- 1 (t)) is bounded, and differentiable, so that all the regularity assumptions are satisfied
here. To obtain the critical levels for Zn (by simulation), all we need to generate random standard normal
deviates and proceed as sketeched in the preceding paragraph.
Next consider the case of a logistic Fa, so that
=
_!x
=
Fo(x) = {l+exp(-x)}-l, xEIR.
(2.37)
•
P. K.
For this model, fo(x) = Fo(x)(l - Fo(x)), so that for Ln , we have a simple J(t) = t(l - t), 0 ::; t ::; 1,
and 1'0 1/6. In addition, in this case, the Wilcoxon scores estimator Tn is asymptotically optimal (for
the location parameter ()1), so that we have a natural appeal for using this particular R-estimator for
this model.
Consider the case of a Laplace Fa where fo(x)
!exp{-Ixl}, x E IR, so that fo(x)
Fo(x) or
1 - Fo(x) according as x is negative or not. For Ln we have a simple J(t) = t or 1 - t, according as
t ::; 1/2 or not, so that 1'0
1/4. For the Laplace distribution, an asymptotically optimal (location-
=
=
=
=
SOADR
11
9
FOR GOODNESS-OF-FIT-TESTS
scale equivariant) estimator of the location parameter is the sample median which is simultaneously an
L-, M - and R-estimator. So it might sound more appealing to use the sample median instead of the
Wilcoxon scores estimator Tn. However, there is an awkward feature of the sample median: the SOADR
for the sample median (or any quantile) involves the second-order term of the order (n- 3/4 ), not (n- 1 )
[viz., Jureckova and Sen, 1996].This slower rate of convergence will naturally affect the performance
characteristics of the GOF tests that are based on them.
Finally, consider the case of the Cauchy Fa, where the density function fa is given by
(2.38)
This leads us to the quantile function Fo- 1 (t) = tan{1I'(2t - 1)/2}, t E [(0,1), so that
(2.39)
Therefore J(t) is bounded and continuous, 'Yo = (211')-1, and we have no difficulty in constructing a GOF
test statistic Zn based on Tn, its dual L n and the interquartile range {h,n'
Remark that the Shapiro-Wilk (1965) type of GOF tests may require considerable modifications for
Fa other than a normal d.f., and its distribution problem may have similar complexities. In that sense,
the proposed GOF tests have greater robustness, flexibility and adaptiveness for a larger class of Fa. A
similar lack of robustness property is ascribable to the other GOF tests based on the estimated dJ. Fn or
the Rao-Blackwell version, mentioned in Section 1. We shall make more comments on it from alternative
hypotheses perspectives in Section 4. Actually, as we shall see later in Section 5, the choice of a suitable
R-estimator may be linked to such plausible alternatives, particularly in the local case.
.•
3. GOF tests for the nuisance location parameter case. Primarily guided by robustness considerations, we consider here some M -estimators (of Bd that are not generally scale equivariant, and formulate
suitable GOF tests for H o : F(x) = Fo(x - B) where only the location parameter B is treated as nuisance.
Let Xl, ... , X n be i.i.d. observations with the distribution function F(x - B) where F(x) + F(-x) =
1 "'Ix E 1R1 , F has an absolutely continuous density f > and finite Fisher's information. Let L n be
1
the L-estimator generated by a smooth weight function J(u) : [0,1] f-t 1R~,
J(u)du = 1, J has an
absolutely continuous derivative J' in (0,1), i.e.
°
n
(3.1)
Ln
=
L CniXn:i
i=l
where
Cni,
i = 1, ... , n are generated by some I n
:
[0,1]
f-t
1R~ such that
in the following way:
•
(3.3)
eni
=
l
iln
(i-1)ln
Note that L n can be equivalently expressed as
(3.4)
I n (t)dt,
i=l, ... ,n.
fo
10
J. JURECKOVA AND P.
K.
SEN
where Gn(t) = Fn (F- 1(t)), t E (0,1), and we may set
1
1
"
Ln =
(3.5)
F- 1(t)J(G n (t))dG n(t)
+ op(n- 1).
Then L n admits an asymptotic expansion, characterized in the following theorem:
•
Theorem 3.3 Under the above conditions, L n admits the expansion
(3.6)
where
(3.7)
and
_2-1°1
(3.8)
t~2)
where Bn(t) = vn(Gn(t) - t), 0
< t < 1. Moreover,
=
2n
vnL~l) ~ N(O, 01); 0"1 = 2
(3.9)
and, provided
H2) E
J'(t)(B n (t))2dF- 1(t)
as n -+
J1
00,
J(s)J(t)s(l - t)dF- 1(s)dF- 1(t)
O<8<t<1
L 2(F), then
(3.10)
where B = {B (t) : 0 ~ t
·•
PROOF.
~
I} is the Brownian bridge.
We write
1
+
+
+
+ 11
°
+ 1
1°1
1
+
+
1
+ ~1
+ 2-1
°
°
1
11
+ -111
°
+ ~1
+ 1
°
°
1
2~ 1
1
Ln
=
F- 1(t)J(t
F- 1(t)J(t)dt
1
+
)nBn(t))d[t
)n
F- 1(t)J(t
1
T(F)
yn
+
~
yn
F- 1(t){J(t
1
T(F)
F- 1(t)J'(t)B n (t)dt
n
F- 1(t)d[J(t)B n (t)]
1
(3.11)
=
op(n- 1)
F- 1(t)(B n )2JII(t)dt
2n
F- 1(t)B n (t)J'(t)dB n (t) + op(n- 1)
1
1
-2
F- 1(t)d[J'(t)(B n (t))2] + op(n- 1)
n
yn
T(F) - )n
vnBn(t)) - J(t)}dt
)nBn(t))dBn(t)
F-1(t)J(t)dW~(t)
0
op(n- 1)
)nBn(t)]
J(t)B n (t)dF- 1(t) -
1
J'(t)(B n (t))2dF- 1(t)
+ op(n- 1)
T(F) + L~l) + t~2) + op(n- 1)
with L~l) and t~2) given in (3.7) and (3.8), respectively. This, in turn, implies (3.9) and (3.10).
Alternatively, we could express (3.6) in the following way:
(3.12)
0
SOADR
11
FOR GOODNESS-OF-FIT-TESTS
where
</J(x) = -
(3.13)
•
i:
J(P(y)){/[X
~ y] -
P(y)}dy,
x E IR 1 .
We shall refer to the representation (3.12) in the subsequent text .
•
Let us now proceed to the case of an M-estimator M n of B, generated by a smooth nondecreasing
skew-symmetric score function 1/;, 1/;( -x) = -1/;(x), as a solution of the equation
n
L 1/;(X
(3.14)
i -
t) = O.
i=1
The authors proved in (1990) that M n admits SOADR:
(3.15)
where
(3.17)
= n-l/2~{1/;'(X;) -,Il =
Unl
(3.16)
Un2=n- 1/ 2 t1/;(X;)
=
.,
1
B n (t)d1/;'(P- 1(t)),
J
1/J(x-B)d(y'n[Fn(x)-F(x)]=-1 1Bn (t)d1/J(F- 1(t))
and
(3.18)
-1
11
=
i:
1/;'(x)dF(x) = IE o1/;'(X;).
where WOLG we put B = 0, and assume the following regularity conditions on 1/; and F :
(M2) 1/J has an absolutely continuous derivative, 0 < 11 <
00
and there exist 6
> 0,
/{1, /{2
> 0 such
that
(M3) There exists a function H(x) such that IEoH(Xd < 00 and
11/J"(x - t) -1/;"(x)1 ~
for some a
It!'" H(x)
a.s.
[F] for
It I ~ 6
> 0, 6 > o.
Assume that the function ¢ in (3.13) is proportional to 1/;, more precisely.
(3.19)
and that the conditions (M1) - (M3) as well as the conditons of Theorem 3.1 are fullfilled. Then vn(Mn L n ) = op(1) and n(Mn - L n ) has a nondegerate (nonnormal) asymptotic distribution, characterized in
the folowing theorem:
J. JURECKOV A AND P.
12
"
K.
SEN
Theorem 3.4 Let L n be an L-estimator (2.23) satisfying (3.2) and M n be an M -estimator defined as a
solution of (3.14). Assume the conditions (M1) - (M3) and the conditions of Theorem 3.1. Then, under
(3.19), as n -r <Xl,
1
~
2
'
If (3.19) is not satisfied, then
I:
1
~1
+
(3.20)
00
-00
vn(Mn
-
'f/;"(x) [B(F(x)Wdx
f(x)
I:
'f/;'(x)B(F(x))dx
'f/;"(x)B(F(x))dx + op(l).
L n ) is asymptotically normally distributed
N(O, (12) with
(3.21)
Under general ¢, the representations (3.12) and (3.15) imply that vn(Mn
Under (3.19), combining (3.15) and (3.12), we obtain
PROOF.
(3.22)
n(Mn - L n ) = vn (Un 2
where
1
Un3 = vn
(3.23)
•
.
.t
-
II
tt
n
,1Un3 ) + (.2.-Un1Un2 II
¢(X; - 0) = -
-
Ln ) ~
nL~2)) + op(l)
r J(t)Bn(t)dF- 1(t).
io
1
Thus, under (3.19), Un2 - I1Un3 = 0, while nL~2) has the representation (3.8); this implies
(3.24)
Actually, by (3.23) we obtain
(3.25)
Un2 - ,1Un3
hence (3.19) is equivalent to
=
-1
1
B n (t)[d'f/;(F- 1(t)) - I1J(t)dF-1(t)]
1
I J(t) = d'f/;(F- (t)) = 'f/;'(F- 1(t))
1
dF-1(t)
.
(3.26)
Then, combining (3.16), (3.17) and (3.8), we get that, under (3.26),
Un1 Un2
-
n,l Lh2)
=
(1
+
~1 1\Bn(t))2 J'(t)dF- 1(t)
(1
1
1
B n (t)d'f/;'(F- 1(t)))
(1
1
B n (t)d'f/;(F- 1(t)))
B n (t)'f/;' (F- 1(t) )dF- 1(t))
(1
1
B n (t)'f/;" (F- 1(t) )dF- 1(t))
1
t
(B (t))2'f/;"(F- (t)) dF- 1(t)
2io n
f(F-1(t))
+ ~
(I:
(3.27)
11
+ '2
(Bn(F(x))'f/;'(x)dx)
00
-00
(Bn(F(x)))
(I:
2'f/;"(x)
f(x) dx.
Bn(F(x))'f/;"(x)dx)
N(o, (12).
SOADR FOR GOODNESS-OF-FIT-TESTS
.
13
Theorem 3.2 sparks an idea to use n(Mn - L n ) as an test criterion for H o. Indeed, we shall describe such
tests in the next section. However, while L n is computationally appealing, Mn may be awkward. We
eliminate this problem by replacing M n by its one-step version MJ1) starting with L n in the role of the
initial estimator.
The one-step M -estimator is defined as follows:
(3.28)
where
(3.29)
Under the conditions (M1) - (M3),
12
= IEutl'"(X1 -
8)
=0
IEutl'"(X1 - 8 - TJ) -+ 0 when TJ -+
and
o.
Then, by Theorem 2.2 of Jureckova. and Sen (1990), letting (WLOG) 8 = 0,
(3.30)
hence
..
.
l
(1) _
(3.31)
Mn
-
1
1
-1
r.;Un 2 + - Un1 Un 2 + op(n ).
11 yn
n,l
Put MAO) - L n ; then, using the representations (3.31), (3.12) and Un3 = n- 1j2 2:7=1 ¢l(Xi ), we have
(3.32)
n(Mn(
1)
vn
- L n ) = -(Un2 -/1 Un3)
/1
1
+ (-U
n1 Un 2 /1
'('»
nL n- ) + Op(l) .
If ¢l(x) == 11 1 t1'(x), i.e., the M-estimator and the initial L-estimator have the same influence functions,
then Un2 -/1Un3 = 0 and we get
(3.33)
n(MJ1) - L n )
= ~Un1Un2 - nL~2) + op(l)
/1
while H2) has the representation (3.8). This, in turn, implies, that n(M~l) - L n ) has the asymptotic
representation (3.20), identical with the representation of n(Mn - L n ).
Let now F o be a hypothetical distribution function, symmetric about 0 and we want to test the
hypothesis
H 1 : F == F o, 8 unspecified
(3.34)
F =f. F o. We propose a class of tests of H 1 based on the above
against the general alternative K 1
relations of the M-estimator and of its one-step version described above. Generally, the performance of
.
such a test consists in the following steps:
(i) Select a skew-symmetric function tI' : rn. 1 t-t rn.1 such that Fo and tI' fulfill the conditions (M 1) - (M3)
of Section 3 and calculate the L-estimator L n of 8 according to (3.4) with the coefficients eni, i = 1, ... , n
generated by the weight function
(3.35 )
according to (3.2) and (2.24).
14
J. JURECKOV A AND
P. K. SEN
(ii) Calculate the one-step M-estimator MAl) defined in (3.28), starting with Ln as the initial estima.
M(O)
tor, z.e.
n = L n.
•
•
(iii) Calculate the test criterion, which is equal to
(3.36)
and reject H o provided
(3.37)
where
U a /2
is the 100(aj2)% critical value of the distribution of lSI, and S is the functional
S
=
1
1
00
_1
2/1
-00
t/J"(x) [B(Fo(x))]2dx
fo(x)
00
+ /11
(3.38)
-00
1
00
t/J'(x)B(Fo(x))dx
-00
t/J"(x)B(Fo(x))dx
of the Brownian bridge B(.).
Actually, the weak convergence Sn ~ Sunder H 1 follows from Theorem 3.2 and from the considerations after. While the percentiles of lSI cannot be directly calculated, they should be approximated
either by the bootstrap or by the random walk.
On the other hand, if F
·•
(3.39)
.
with
j
-=1=
Fo but both Ln and MAl) admit the asymptotic representations, then
(3.40)
and
(3.41 )
Hence, the test based on Sn has asymptotic power 1 against every fixed alternative F
-=1=
Fo.
Under the special choice of M-estimator generated by t/J(x) = Fo(x) -~, x E JR. , the criterion (3.36)
partially simplifies. The corresponding asymptotically equivalent L-estimator is based on the weight
function
1
(3.42)
..
J(t)
= /1 1 fo(Fo-
1
(t)), 0
< t < 1,
/1
=
i:
f5(x)dx
and (3.38) takes on the form
(3.43)
..
This special case when used for GOF testing for the logistic distribution with nuisance location parameter
will be based on the criterion, relates to Sn ~ S where
SOADR
•
15
FOR GOODNESS-OF-FIT-TESTS
4. Contiguous alternatives and GOF tests. Consider the family of alternatives to Hi,
Hn
(4.1)
f(n) (x) = fa (x)(1
:
+ n -1/2 '\u( x)),
where u(x) is a fixed function satisfying
I:
(4.2)
I:
u(x)fo(x)dx = 0,
,\ unspecified
(u(x))2 fo(x)dx <
00.
By the third LeCam's lemma, the sequence {H n } is contiguous with respect to Hi. The corresponding
distribution function p(n) could be written in the form
p(n)(x) = Po(x)
(4.3)
A(x) =,\ l~ u(y)dPo(Y)·
+ n- 1/ 2A(x),
=
=
Notice that A is bounded, A( -00) A( 00) 0. It is also possible to consider general local nonparametric
alternatives, satisfing contiguity; the following theorem describing the asymptotic behavior of the test
criterion (3.36) under H n could be easily extended for such alternatives as well.
=
1
Theorem 4.1 Assume the conditions of Theorem 3.2. Then, under H n , the test criterion Sn
n(MA ) L n ), described in steps (i) - (iii) of Section 3, converges in distribution to the functional S* = S + Z,
where S is defined in (3.38) and
z
=
••
- 'Yl1
~1
+
(4.4)
PROOF.
~1
JOO tf(x)dA(x) JOO B(Po(x))tf"(x)dx
-00
I:
I:
-00
tf'(x)dA(x)
tf(x)dA(x)
I:
I:
B(Po(x))tf'(x)dx
tf'(x)dA(x).
WLOG, we put () = 0. Notice that
'Yin) =
(4.5)
I:
tf'(x)f(n)(x)dx = 'Yl
+ n- 1/ 2,\
I:
tf'(x)u(x)fo(x)dx.
Moreover,
(4.6)
and hence
(4.7)
as n -+
00
for the Brownian bridge B. By (3.8) and (3.35), under H n
nD2)
=
n
v
--+
(4.8)
•
as n -+
•
00.
2'Yl
-00
'Yl
-00
tf"(x) n(Fn(x) _ p(n)(x))2dx
fo(x)
tf"(x)
2
~(
) [B(Po(x))] dx,
JO X
Further, by (3.16) and (3.17), under H n , as n -+
Un1 Un2 =
(4.9)
JOO
1 JOO
--2
__1_
[I:
~ (i:
[I:
(vn(Fn(x) - p(n)(X))tf"(x)dx -
(vn(Fn(x) - p(n)(X))tf'(x)dx B(Po(x))tf"(x)dx -
i:
,
I:
00,
I:
tf'(x)dA(x)] .
tf(x)dA(x)]
tf'(x)dA(x))
(i:
B(Po(x))tf' (x)dx -
i:
tf(x)dA(x)).
J. JURECKOV A AND
16
•
P. K.
SEN
The asymptotic relation (3.24) remains true under the contiguous alternative; hence, combining (4.5) 0
(4.9), we arrive at the proposition of the theorem.
Denote, for the sake of brevity,
x=
_1
2/ 1
Joo
-00
W=
and
a = -1
/1
Y = -1
'ljJ"((X)) [B(Fo(x))fdx,
fa x
I:
Joo 'ljJ'(x)B(Fo(x))dx,
/1
-00
'ljJ"(x)B(Fo(x))dx
Joo 'ljJ(x)u(x)fo(x)dx,
b=
-00
I:
'ljJ'(x)u(x)fo(x).
Then the limiting functional under H 1 is
S=X+YW
while that under H n is
s + Z = X + (Y - Aa)(W - Ab).
We could easily verify that lEX = lEY = lEW = 0 and that X, Y and Ware uncorrelated.
The function u(.) in (4.1) is typically either even or odd. For instance, the function u(x) = -l-x~~I:l,
•.
corresponding to the scale alternative, is even for symmetric fa.
If u is even, then a 0, hence, denoting 17 2 var S,
=
=
(4.10)
and
(4.11)
where f{ > 0 is a constant dependent on the distribution of S.
Similar conclusion we obtain in the case of odd u (local alternative of skewness etc.), when b
= O.
Appendix. Let Xl, ... , X n be i.i.d random variable with a d.f F and let Pn be the corresponding
sample d.!. A natural estimator for a functional T(F) is T(Pn ). 1fT is the second order differentiable at
F in the Hadamard sense (see Jureckova and Sen (1996) for details) then we could write an expansion
T(F
..
+ (Pn
T(F)
T(F)
(5.1)
where
F))
+ ~ tT(1)(F;Xi ) + 2\
n
•
-
n
i=l
+ t~l) + t~2) + op(n-
1
).
LLT(2)(F; Xi, Xj)
i
j
+ op(n- 1 )
SOADR
17
FOR GOODNESS-OF-FIT-TESTS
and
2nt~2)
•
=
(5.2)
1
1
n
T(2)(p; p-l(s), P-l(t))dG n (s)dG n (t)
1
1
T(2)(p; p-l(s), P-l(t))dBn (s)dB n (t)
and, provided T2(.) E L 2 (P) (see .Jureckova and Sen (1996), Chapter 4),
00
2nt.(2)
-.!!....,. ~ Ak Z2k
n
------r L.J
k=O
(5.3)
where {Zdk"=o are i.i.d standard normal random variables and {Ak} is a sequence of the eigenvalues of
T(2) (.) with respect to orthonormal functions {Tk (.): k;::: O} such that
and
•..
•
!
Tj(X)Tk(X)dP(x) = iSjk .
A similar expansion holds for U -statistics and related von Mises' functionals that need not be Hadamard
differentiable.
Suppose now that T 1 (.) and T 2 (.) are two Hadamard differentiable functionals, estimating the same
parameter (i.e., T 1 (P) = T 2 (P)) and such that
a.e.
(5.4)
I
Under (1.3), Td.) and T 2(.) are the first order asymptotically equivalent. Then, denoting T o(.) = Td·) T 2 (.),
(5.5)
where
(5.6)
and
(5.7)
Then (5.6) has an asymptotic (nonnormal) distribution which follows from the asymptotic theory of
Hoeffding's U -statistics. More precisely, we could write
(5.8)
•
A(2)A(2) _ 1 ~ (2)
. .
.
2n(T1n - T 2n ) - ; L.J To (F, X" X,)
i=1
where
UOn
=(
n )
2
-1
+ (n -
(2)
1)UOn
L TJ~)(F; Xi, Xj).
19<j$n
Then, by the Khintchine law of large numbers,
(5.9)
_!
).
a.s
;1 ~
~To( 2(F,Xi,Xd
-+
To(2)( F ) -
,=1
To(2)(F',X,X )dF()
X
18
J. JURECKOV A AND
P. K.
SEN
and
00
(5.10)
(2)
V
'"'
(n - l)UOn -+ LJ AOk(Zk2 - 1).
k=O
REFERENCES
•..
Bhattacharyya, B. B. and Sen, P.K. (1977). Weak convergence of the Rao-Blackwell estimator of a distribution function. Ann. Probability 5, 500-510.
Billingsley, P. (1968). Convergence of Probability Measures. John Wiley, New York.
Durbin, J. (1961). Some methods of constructing exact tests. Biometrika 48,41-55.
Durbin, J. (1973). Distribution theory for Tests Based on the Sample distribution Function, SIAM,
Phiadelphia.
Hodges, J. L. and Lehmann, E.L. (1963). Estimates of location based on rank tests. Ann. Math. Statist.
34,598-61l.
Hoeffding, W. (1948). A class of statistics with asymptotically normal distribution. Ann. Math. Statist.
18, 293-325.
Jaeckel, L.A. (1971). Robust estimation of location: symmetric and asymmetric contamination. Ann.
Math. Statist. 42, 1020-1034.
Jureckova, J. (1977). Asymptotic relations of M-estimates and R-estimates in linear regression models.
Ann. Statist. 5, 664-672.
Jureckova, J. (1995). Trimmed mean and Huber's estimator: Their difference as a goodness-of-fit
criterion. J. Statist. Res. 29, 31-35.
Jureckova, J. and Sen, P.K. (1990). Effect of the initial estimator on the asymptotic behavior of onestep M -estimator. Ann. Inst. Statist. Math. 42, 345-357.
Jureckova, J. and Sen P.K. (1996). Robust Statistical Procedures: Asymptotics and Interrelations. John
Wiley, New York.
Sen, P. K. (1963). On the estimation of relative potency in dilution (-direct) assays by distribution-free
methods. Biometrics 19, 532-552.
Sen, P.K. (1996). Statistical functionals, Hadamard differentiability and martingales. Probability
Models and Statistics. A J.Medhi Festschrift. New Age Publishers, New Delhi, pp. 29-47.
Shapiro, S. S. and Wilk, M. B. (1965). An analysis of variance test for normality (complete samples).
Biometrika 52,591-611.
Shorack, G. and Wellner, J. (1986). Empirical Processes with Applications to Statistics. John Wiley,
New York.
von Mises, R. (1947). On the asymptotic distribution of differentiable statistical functions. Ann. Math.
\
Statist. 18, 309-348.
•
.
Charles University
Department of Probability and Statistics
Sokolovska 83
CZ-186 75 Prague 8
Czech Republic
jurecko@karlin. mff. cuni. cz
University of North Carolina
Departments of Biostatistics and Statistics,
3101 MacGavran-Greenberg Hall
Chapel Hill, NC 27599-7400
U.S.A.
[email protected]