(#2014)
THE QUANTILE-TRANSFORM APPROACH TO THE
ASYMPTOTIC
DISTRIBUTION OF MODULUS TRIMMED SUMS
,
.."
SANDOR CSORGO *
1.
DAVID M. lvIASON t
ERICH HAEUSLER
Introduction and statement of results.
Let X, XI, X 2 , ••• , be independent ran-
dom variables with a common non-degenerate distribution function F which is symmetric
about zero. For 1 ::; j ::; n, set
and let
(k) X n
= Xj if mn(j) = k, 1 ::; k ::; n. Thus
(k) X n
is the k th largest random variable
in absolute value among XI, ... , X n , with ties broken according to the order in which the
sample occurs. Consider the 'modulus' trimmed sums defined for 0 ::; k < n - 1 as
Following a preliminary investigation by Pruitt [7], Griffin and Pruitt [4] undertook a detailed
study of the asymptotic distribution of (k n)Sn, when {k n } is a sequence of integers such that
o::; kn
(1.1)
< n - 1 for n > 1 and both
k n --+
00
and kn/n --+ 0 as n --+
00.
The aim of the present paper is to illuminate their main results by means of our quantiletransform-weak-approximation approach.
·Partially supported by the Hungarian National Foundation for Scientific Research, Grants 1808/86 and
457/88
tpartially supported by a Fulbright Grant and NSF Grant #DMS-8803209.
1
Let
G(x) = P{IXI::; x},
be the distribution function of
lXI, with
-00
<x<
00,
corresponding quantile function
I«s) = inf{x: G(x) 2 s}, 0 < s ~ 1, I«O)
Moreover, for each integer n
2 1 and
-00
T~(X)
<x<
rn(x)
= n Jo
00
= I«O+).
set
I<2(u)du,
1
where un(x) = {(n-k n+xkJ)n- 1 YO}/\1 with Y and /\ standing for maximum and minimum.
The following theorem yields the direct halves of Theorems 3.2, 3.4, 3.5, 3.7, 3.9 and
3.10 of Griffin and Pruitt [4] under the cleaner forms of their conditions, which correspond
to those given in their paper when the underlying distribution function is continuous. In
what follows, the symbols =v and -tv will mean equality and convergence in distribution,
respectively.
THEOREM.
Let 0 ::; k n
::;
n - 1 for n 2 1 be a sequence of positive integers satisfying
(1.1). Assume that for a non-degenerate distribution function F, symmetric about zero, and
a subsequence {n'} C {n} there exist An'> 0 and Bnl such that for some
(1.2)
and for all
- BnljAnl
-00
<x <
-t
C as n'
-t
-00
< C < 00,
00
00
(1.3)
for a (necessarily non-decreasing convex) function v 2 which is not identically equal to zero.
Then
(1.4)
where the limiting random variable V is the variance mixture of normals with characteristic
function
E{exp(itV)} = exp(iCt)
1
00
exp(
-00
2
V2(X)t2
)d~(x),
2
-00
<t <
00,
where ep is the standard normal distribution function.
The proof, postponed until the next section, is based on a weighted approximation to a
'signed' empirical process, which is likely to be of separate interest. See the Proposition in
Section 2.
From our theorem it is easy to infer that a sufficient condition for the asymptotic normality of (knt) Sn is that for all
l
-00
<
x
<
00,
,
as n -+
(1.5)
00,
implying that
where V has the characteristic function for
-00
1
-00
00,
t2
00
E{exp(itV)} =
<t <
ex p(-"2)dep(x)
2
= e- T,
saying that V is standard normal. (Arguing as in Griffin and Pruitt [4], it can be shown
that (1.5) is also necessary for the asymptotic non-degenerate normality of (kn/)Sn " )
It is elementary to show that (1.5) is equivalent to
(1.6)
for all
-00
<x <
00.
From this form of the normality condition it can be seen that stochastic
compactness of the whole sums Sn, which in the present symmetric case is equivalent to
(1.7)
implies that
(kn)Sn/Tn(O)
converges in distribution to a standard normal random variable for
all sequences kn satisfying (1.1) as n -+
00.
(Refer to Corollary 10 and (3.13) in [1].) That
stochastic compactness of the whole sums Sn implies asymptotic normality of
sequences kn as in (1.1) was first pointed out by Pruitt [7].
3
(k
n)Sn for all
2.
Proof.
In the proof of the theorem we work on a probability space that carnes a
sequence U, U1 , U2 , • •• , of independent random variables uniformly distributed on (0,1) and,
independent of this sequence, a sequence
8,81, 82, ... ,
= I} =
tributed random signs, that is, P{s
P{s
of independent and identically dis-
= -I}
=~. Since I«U)
=1) lXI,
it is
elementary to show using symmetry that
(X,
IXI) =1)
(sI«U), I«U)),
from which we have immediately that
For each n ;::: 1, let U1,n ::; ... ::; Un,n denote the order statistics of U}, ... , Un'
D1,n,' .. ,Dn,n denote the antiranks of U}, . .. , Un, i.e. UDi n
= Ui,n
for 1 ::; i ::; n. Then
from the above distributional equality we easily obtain that for each fixed n ;::: 1,
so that
n-kn
=1) 2:
(kn)Sn
sDi,nI«Ui,n).
i=l
For each n ;::: 1, we define the 'signed' empirical process
n
Hn(t) =
n- 1
2: sJ( {Ui ::; t}),
0 ::; t ::; 1,
i=l
where I(A) denotes the indicator of a set A. Notice that almost surely for n ;::: 1,
n
Hn(t) =
n-
1
2: sDi.n I ({Ui,n ::; t}),
0 =::; t =::; 1.
i=l
Therefore we can write for each integer n ;::: 1,
(2.1)
We now introduce the uniform empirical distribution function
4
Let
and the empirical quantile function
k-l < s <
k
k. =
--;_;,
1, ... , n,
s = O.
The following proposition provides a joint weighted approximation of H n, Gn and Un' which
will be essential to our approach to the asymptotic distribution of
PROPOSITION.
((k n)Sn
- B n)/ An.
On a rich enough probability space there exists a sequence of proba-
bilistically equivalent versions (Hn, On, Un) of (Hn, G n, Un), meaning that for each integer
n
~
1,
such that for a Brownian bridge B and a standard Wiener process lV, with Band W independent, for all 0 <
II
<
!'
(2.2)
(2.3)
and for all 0 < 8 <~,
(2.4)
Proof.
The proof will be a consequence of the following lemmas. Our first lemma can be
derived easily from results and a starting element of the proof in Mason and van Zwet [6].
LEMMA 1.
On a rich enough probability space there exist independent Brownian bridges
B 1 , B 2 and B a and a triangular array of row-wise independent uniform (0,1) random variables
~in) (1), ... , ~in) (1),
ein ) (2), ... ,~in) (2), ein ) (3), ... , ~in) (3), n 2:: 1, such that for all 0 <
(2.5)
5
II
< ~,
and
(2.6)
where for i = 1,2,3, n
~
1,
G~)(t) -
n
n-
1
I: I( {~)n)(i) ~ t}),
0 ~ t ~ l.
i==l
Set now for n
~
1,
tI({~)n)(3) ~ ~})
Nn =
and M n = n - N n.
i=l
Let for n
~
1 and 0
~ t ~
1,
and
nHn(t) = NnG~~(t) - .MnG~Jt).
(We define G~i)
=0 for i = 1,2.) Also define
and
Wet)
= 2-!{B1 (t) -
B 2 (t)}
+ 2tB3(~)' 0 ~ t ~ l.
It is simple to verify that Band Ware independent processes with B being a Brownian
bridge and W a standard \Viener process.
LEMMA 2.
On the probability space of Lemma 1, for all 0 < v
<
!'
and
(2.8)
sup
In! Hn(t) - W(t)I/(l - t)!-v = Open-V).
0;5t;51-n- 1
6
Proof.
Obviously, since both
1
1
1
1
n- Nn -+p 2" and n- Mn -+p 2" as n
we have by (2.5), for all 0 < v
<!
-+ 00,
'
(2.9)
and
(2.10)
Also, by (2.6),
(2.11)
t 1 12Nn 1- n
sup
09$1-n- 1 (1 - t)2- 11
m
-
2B3 (1)1
= Op (n -II) .
2
Moreover, it is easily checked that
In addition, by well-known properties of the Brownian bridge, for i = 1,2,
Thus,
(2.12)
and
sup I(Mn )tB2(t) - 2- t B 2 (t)I/(1- t)t- II = Op(n- t ).
09:9
n
Using (2.9), (2.10), (2.11), (2.12) and (2.13), a little algebra now gives (2.7) and (2.8). 0
(2.13)
For each integer n ~ 1, let f)l,n =5 '"
=5 f)n,n be the order statistics of the random
variables e~Nn)(l), ... , e~n)(l), e~Mn)(2), ... , d~n)(2). Define the empirical quantile function
Un(t) based on these order statistics in the usual way. The following lemma is a consequence
of the fact that for all 0 < S < ~,
sup
n6IGn(t) + Un(t) - 2tl/(1 - t)t- 6 = Op(l),
°StSl-n-1
which can be readily inferred from Corollary 2.3 in [2] (also see the Proposition in Mason
[5]).
7
LEMMA 3.
On the probability space of Lemma 1, for all 0 < 8 < ~,
(2.14)
sup
In! {t - Un(t)} - B(t)I/(1 - t)!-S
O:5t~l-n-l
= Op(n- s).
Since it is routine to verify that for each integer n ;::: 1, (.ifn, an, Un) =v (Hn, Gn, Un),
assertions (2.2), (2.3) and (2.4) follow from (2.7), (2.8) and (2.14). This completes the proof
of the Proposition. 0
Returning to the proof of the Theorem, from now on, since we are only concerned with
distributional results, on account of (2.1) for the sake of the proof we can and do identify
(.ifn, an, Un) with (Hn, Gn, Un); that is, we work on the probability space of the Proposition.
Also for notational convenience we drop the prime on the n'. The proof of the Theorem
requires a number of lemmas.
LEMMA 4.
Under (1.3), for each 0 < Al
<
00,
!
limsupk~J«(un(M))/An
(2.15)
n-oo
Proof.
Fix any finite M
> M. Then for each n > 1,
-
A-n 2rn2(M) = A-n 2rn2(M)
from which by letting n
implying (2.15).
< 00.
~ 00
l
+ nA-n 2
un (M)
un(M)
and using monotonicity of
0
For each n ;::: 1, set
8
J(
J(2(u)du ,
we get
where
Clearly, we have
P{D n} -+ 1 as n
(2.16)
LEMMA 5.
Under (l.3) we have as n -+
/Un-kn,n
I«u)dHn(u)
-+ 00.
00,
/un(-Zn)
= nA;l Jo
I«u)dHn(u)I(D n) + op(l).
(2.17)
nA;l J
o
Proof.
Because of (2.16) the assertion will follow from
(2.18)
Towards a proof of (2.18) notice that from the weighted approximation (2.4) for Un (·),
for any 0 < 8 < ~,
(2.19)
which in combination with the fact that Zn = Op(l) gives
kn
Un-len ,n = 1- -n
(2.20)
Next, for fixed 0 < 8 < ~ and 0 < M <
00,
1.
+ Op(kJjn).
set
and
Then (2.19) and (2.20) imply
(2.21)
lim liminf P{An(M) n Bn,s(M)} = 1.
M-oo n_oo
9
On the event Dn n An(M)
n Bn,s(M)
=:
En' we have for all large n,
(2.22)
and we get the estimate after integrating by parts
InA~llUn-kn,n I«u)dHn(u)I(En)l
Un(-Zn)
=
nA~l I(En)II«Un-kn,n){Hn(Un-kn,n)
- Hn ( Un (- Zn) )}
Un kn
+ l - ,n {Hn(u n(-Zn)) - Hn(u)}dK(u)1
Un(-Zn)
< 2kt A~lI«un(2M))nk~i~n(.Al),
where
~n(M) =
sup
IHn(b + h) - Hn(b)/.
sup
un(-M)~b~un(M) O~lhl~Md-6
In
In view of Lemma 4, (2.16) and (2.21), the proof of (2.18) and hence of (2.17) will be complete
if we show that for all 0 < S < ~ and 0 < M <
_1
nkn 4 ~n(M)
(2.23)
-+p
00,
0 as n
-+ 00.
Consider a grid
such that
bl
-
bl - 1
1_6
= MkJ In
for l
= 1, ... , L -
1
and
For each bE [u n(-M), un(M)) there exists exactly one l E {I, ... ,L} such that bl This fact gives the bounds
10
1
~
b < bl •
e'
where
wn(a, b) =
1
sup n2"IHn(b + h) - Hn(b)l.
°SlhlSa
To finish the proof of (2.23) it suffices to show that for each fixed 0 < M < 00 and 0 < 8 < ~
the right side of the last inequality converges to zero in probability. For this, fix
€
> 0 and
note that
1
_1
1-5
P{n 2 kn 4 max wn (2MkJ
ISlSL
L
1_5
In, bl - 1 ) ~ €} ~ :LP{wn(2MkJ
l=1
1
1
In, bl-d ~ ekri n- 2 },
which by Theorem 1.1 of Einmahl and Ruymgaart [3] is
A€2
Be 5-1
~ CL exp{ - 2M k~1/J(4M kn 4)} =: I n( €, A1),
where A, Band C are universal positive constants and 1/J is an increasing function on (0,00)
such that t/J(z)
I n ( €, M)
--+
11
0 as n
as z
--+
1 O.
This when combined with the easy bound L ~ 2k~
+ 1 yields
00, assuming that 0 < 8 < ~. This completes the proof of (2.23) from
which (2.17) follows. 0
Finally, we require one more lemma.
11
so that to finish the proof we must verify that
rUn(-Zn)
1
n2A~1 Jo
(2.24)
1
{W(U) - n 2Hn(u)}dI«u)I(D n) ~p 0 as n
~
00
and
For the proof of (2.24) fix 0 < v <
!,O < 6 < ~ and 0 < M
<
00.
In view of (2.22) we
obtain on the event En
which by (2.3) is equal to
(2.26)
Now by integrating by parts and some elementary bounds,
which, by the Cauchy-Schwarz inequality applied to the second term above and by (1.3) and
(2.15),
< n ll k!-II {k! A~l I«u n(2M))} + 0(1 )A~lTn(2M)(kn/ntll
-
O(n ll k!-II).
This bound when combined with (2.26), (2.16) and (2.21) yields (2.24) as long as ~ < v < ~.
To establish (2.25), we fix ~ < v <
! and 0 < M <
00.
On the event En (recall (2.22)),
InlA~l I«u n(-Zn)){nlHn(un(-Zn)) - W(u n(-Zn))}/
<
sup
O~u~l-n-l
IntHn(u) -1 W(u)lntA~1I«Un(2M))(1-un(-2M))t- lI ,
(1 -U)2- 11
12
which by (2.3), (2.15) and ~ < v < ~,
1_ v
= Op(l)k~
1
1
kri A~ I«u n(2.M)) = op(l),
proving (2.25). This completes the proof of Lemma 6.0
According to Lemmas 5 and 6 combined with (2.1), the assertion of the Theorem will
follow if we show that
1
r1n(-Zn)
Vn .- A~l{ -n 2 Jo
--+
1> V
as n
W(u)dI«u)
+n
1
2
I«u n(-Zn))W(u n(-Zn)) - Bn}I(Dn )
--+ 00.
We prove this using characteristic functions. Since
we have
E{exp(itVn)}
Notice that for x
~
= E{exp(itVn)I(Dn )} + 0(1)
as n
--+ 00.
_1
-(1 - kn/n)nk n
1
rUn (x)
-n 2 Jo
2
the random variable
1
W(u)dI«u)
+ n 2 I«u n(x))l¥(u n(x))
is normally distributed with mean zero and variance T~(X). Therefore by the independence
of Zn and W, for any t,
From
-Zn
--+1>
N(O, 1) and - (1 -
~) ~ --+ -00
as n
-+ 00
it follows by conditions (1.2) and (1.3) that for any t,
.
E{exp(itVn)I(Dn )}
--+
1
00
exp(iCt)
completing the proof of the Theorem. 0
13
-00
v2 (x)
exp( --2-t)d~(x),
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