Smooth estimate of quantiles under association

S'rATISllCS i PROBABILITY LETTERS ELSEVIER

Statistics & Probability Letters 36 (1997) 275-287

Smooth estimate of quantiles under association Zongwu

C a i a, G e o r g e

G . R o u s s a s b,*

a Department of Mathematics, Southwest Missouri State University, Springfield, MO 65804, USA b Division of Statistics, University of California, Davis, CA 95616, USA

Received 1 February 1997; received in revised form 1 April 1997; accepted 5 May 1997

Abstract Let XI,X2 .... be real-valued random variables forming a strictly stationary sequence, and satisfying the basic requirement of being positively or negatively associated. Let ~p denote the pth quantile of the marginal distribution function of the X,.'s, which is estimated by a smooth (kernel-type) estimate ~e,, on the basis of the segment X1,... ,Xn. The main results of this paper are those of establishing pointwise consistency, asymptotic normality with rates, and weak convergence of a stochastic process generated by ~'pn. (~) 1997 Published by Elsevier Science B.V. Keywords: Berry-Esseen bounds; Negative association; Positive association; Smooth estimate; Strict stationarity; Strong consistency; Weak convergence

1. Introduction Suppose that { X j ; j > ~ I ) is a strictly stationary sequence of random variables (r.v.s) with absolutely continuous distribution function (d.f.) F. Define the inverse of F as F - l ( p ) = i n f { x E ~; F(x)>~p}, where is the real line. The traditional estimate of F has been the empirical distribution function F n based on the first n r.v.s X1 . . . . . X,, while the estimate of the pth quantile ~ p = F - l ( p ) , 0 < p < l , is the sample quantile function ~pn = F n - I ( P ) - However, as stated in Falk (1983), Fn does not take into account the smoothness o f F , i.e., the existence of a probability density function (p.d.f.) f . In order to incorpo~te this characteristic, investigators proposed several smoothed quantile estimates, one of which is based on F~ obtained as a convolution between F, and a properly scaled kernel function K:

Fn(x)=

K

~-n

n j=l

~

hn

]

(1.1)

where K is a continuous d.f. and {hn;n>~l} is a sequence of bandwidths with 0
276

z. CaL G.G. Roussas Statistics & Probability Letters 36 (1997) 275-287

~pn, based on Fn, is defined by

"~pn=~ -l(p)= inf{x E ~; f ( x ) ~>p}; ~'pn is referred to in literature as the perturbed (smoothed) sample quantile. Asymptotic properties of fin, both under independence as well as under certain modes of dependence, have been investigated extensively in literature. These studies started with Nadaraya (1964) and continued in a series of papers by Winter (1973, 1979), Yamato (1973), Parzen (1979), Cai (1993), Cai and Roussas (1992, 1995a, b, 1997), and Roussas (1969, 1991, 1995). The kernel-type estimate of the quantile ~p, ~'pn, was proposed by Nadaraya (1964), and it was further studied in a number of articles, including those of Yamato (1973), Parzen (1979), Falk (1983, 1985), Falk and Reiss (1989), Ralescu (1992) and Ralescu and Sun (1993). The framework of all these investigations is that of independence. Finally, in the paper by Xiang (1995) the deficiency of the sample quantile estimator with respect to kernel quantile estimators is studied for i.i.d, observations which, however, may be subject to random fight-censoring. In this paper, some of the results pertaining to ~'pn under independence are extended to the case, where the r.v.s involved are subject to positive or negative dependence. More specifically, the paper is organized as follows. The remaining of this section is devoted to the presentation of necessary definitions and the formulation of the underlying assumptions. In Section 2, under either PPQD or PNQD (see Definition 1.1(ii) below), it is shown that ~pn is strongly consistent (the second convergence in Theorem 2.1(i)). Furthermore, under negative association (see Definition 1.2), a Hoeffding-type probability inequality is established for ~'p, (see Theorem 2.2). In Section 3, the asymptotic normality of a suitably normalized version of "~pnis established, under both positive association and negative association (see Theorem 3.1). It is also shown that the rate of convergence is O(n -1/2 log n) (see Theorem 3.2). Finally, in Section 4, a certain normalized smoothed sample quantile process, generated by "fpn (see (4.1)), is considered and its weak convergence is established (see Theorem 4.1 ). In order to avoid repetitions, it is stated once and for all that all limits are taken, as n ~ c~. We proceed with some definitions. Definition 1.1. (i) Two r.v.s X and Y are said to be positively-quadrant-dependent (PQD), if

P(X>x,Y>y)-P(X>x)P(Y>y)>>.O,

x, yfg~,

and negatively-quadrant-dependent (NQD), if the left-hand side above is nonpositive. (ii) The r.v.s {Xj;j/> 1} are said to be pairwise-positively-quadrant-dependent (PPQD) (pairwise-neoativelyquadrant-dependent (PNQD)), if, for any i ¢ j , the r.v.s X/ and Xj are PQD (respectively, NQD). (iii) The r.v.s {Xj;j~> 1} are said to be linearly-positive-quadrant-dependent (LPQD) (linearly-negativelyquadrant-dependent (LNQD)), if for any nonempty disjoint and finite subsets A and B of { 1,2 .... } and for any positive 2i's, the r.v.s )--~icA 2iX/ and ~ j ~ s 2jXj are PQD (respectively, NQD). Definition 1.2. The r.v.s {Xj; 1 <<.j<<.n} are said to be positively-associated (PA), if for every G,H: ff~n~ •, which are coordinatewise nondecreasing, and for which E G 2(Xj, 1 ~0. The above r.v.s are said to be negatively-associated (NA), if for every nonempty proper subset A of {1,2 .... ,n} and for every G : R #A---~R, H:R#Ac---~R, which are nondecreasing as above, and for which E G2(Xi, i E A ) < oo, E HZ(Xj, j E AC) < ~ , it holds that Cov[G(X/, i EA), H(Xj, j EAt)] ~<0;

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277

here #A denotes the cardinality of A. Infinitely many r.v.s are said to be PA(NA), if any finite subset of them is a set of PA(NA) r.v.s. Remark 1.1. From the above definitions, it is immediate that PA(NA) implies LPQD(LNQD) and PPQD (PNQD); neither converse is true. Examples to this effect are discussed in Joag-Dev (1983) and Newman (1984). The main results of this paper are derived under the following assumptions, although not all of them are used everywhere.

Assumptions. (A1) (i) Suppose {Xj;j>~ 1} is a stationary sequence of real-valued r.v.s having common d.f. F with p . d . f . f . (ii) Set u ( n ) - ~ [Cov(X1, X:)I 1/3, n~> 1, and suppose that u(1) 0 . (A5) nh 4 --* O, as n --+ c~. (A6) fR u dK(u) = 0 and fn u2 dK(u) < ~ . (A7) nh2-O(log n).

2. Strong consistency and a probability inequality Let ~p denote the pth quantile of the d.f. F(x), i.e., a root of the equation F(~p)= p, with 0 < p < 1, which A A is assumed to be unique. As an estimate of ~p, we take a root of Fn(~pn)= p, or, more precisely ~pn = F n - I ( P ) = inf{x C R; Fn(x)>/p},

(2.1)

where Fn(x) is defined in (1.1). It is first shown that ~'p~ is a consistent estimate of ~p.

Theorem 2.1. Let {Xj; j>~ 1} be a stationary sequence of r.v.s with continuous marginal d.f F(x), and let "~pn be given by (2.1). Then: (i) I f the r.v.s are PPQD and satisfy the condition o~

E j=2

j

j-2 E

[Cov(X,, X/)] 1/3
(2.2)

i=2

it follows that sup{lff~(x) - F(x)[; x E ~} a.s 0

and

"~p~~

~p.

(ii) The results in (2.3) hoM true, if the r.v.s are PNQD. The following lemma is needed in the proof of Theorem 2.1.

(2.3)

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278

Lemma 2.1. Let {Xj;j~> 1} be a stationary sequence of r.v.s. Then: (i) I f the r.v.s are PPQD, have finite second moment, and satisfy the condition j

E+j=2

E cov(.,, i=2

it follows that n

1 E ( X i _ exi) a.% 0. n

(2.4)

i=1

(ii) I f the r.v.s are PNQD and have finite first moment, the convergence in (2.4) holds true. Proof, Case (i) is discussed in Birkel (1989, Theorem 1 ), and case (ii) is found in Theorem 5.1 of Bozorgnia et al. (1993). [] We now embark on the proof of Theorem 2.1. Proof of Theorem 2.1. The assumption that ~p is unique implies that, for some 5 > 0, 5(5) = min{F(~p) - F(~p - e), F(~p + 5) - F(~p)} > 0.

(2.5)

It is easy to see that the following inequalities hold, where the sup is taken over N: P(l~fpn -

Cpl >~

i.o.) ~< P(IF(~pn) -

F(¢p)l >5(5)

i.o.)

= P(lF(~'pn)-/~n(~'pn)l > ~(5) i.o.) ~< e ( s u p Iff,(x)- F(x)l > 6 ( 5 ) i . o . ) .

(2.6)

Integration by parts in (1.1) gives ft,(x) = f R F n ( x - hnu)dK(u), so that

and /,

[fi, (x) - F(x)l ~ sup IF~(x) - F(x)[ + [ [F(x - h,u) - F(x)l dK(u). x

(2.7)

dR

It follows from Lemma 3.2 in Cai and Roussas (1995a), (2.2) and Lemma 2.1(i) that, for each x E R, Fn(x ) a.s. F(x).

It follows (see, e.g., Theorem 1 in Tucker, 1967, pp. 127-128) that sup ]Fn(x) - F(x)[ a.s. 0.

(2.8)

x

From (2.7) and (2.8), the Dominated Convergence Theorem, and for each x E N, we have ~ ( x ) a'%F(x),

and then as above sup [fin(x) - F(x)[ a'~', 0. x

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279

This last convergence implies that, for any 6 >0, P ( s u p I/~n(x) - F(x),>6 i . o . ) = 0 . The proof of part (i) follows from this result and (2.6). (ii) Follows by the same arguments as those used in the proof of part (i) and Lemma 2.1(ii).

[]

Theorem 2.2. Let {Xj;j~> 1} be a stationary sequence of NA r.v.s with continuous d.f F. Then, for any e and for all sufficiently large n, we have P(l~pn - ~p[ > e) <~2e-n62(~)/8,

(2.9)

where 6(e) is given in (2.5). In the proof of Theorem 2.2, we will make use of the following lemma which is the Hoeffding inequality for the case of LNQD r.v.s. The inequality for the NA case was established in Roussas (1996). Lemma 2.2. Let {Xs;j >~1} be a sequence of LNQD r.v.s with IXjl <~C a.s. for all j >11 and of zero-mean. Then, ./or all t > O,

Proof. Indeed,

P(~-~Xj>~tn)j=l

<~e-tnEe~=tXJ=e-tnE(e~="XJeX") = e_tn Cov (e)-]°-' J=' ~, eX,) +e-tnEe ~--]"-' ~=' xj .Ee x° =11 +I2

(say).

Next, we will show that 11 is nonpositive. To this effect, an application of the Hoeffding identity (see, for example, Lemma 2 in Lehmann, 1966) yields et~I1 -- /Jn

> log u,Xn > log v

- P

> log

P (Am> log v)

dF(u, v),

\/=1 where F(u, v) is the joint d.f. of e~-]~=-~'x~ and ex., and Rz+ = (0, cxz)× (0, c~). By the fact that ~-]~=11X/ and X, are NQD, it follows that, for all u, v > 0, P

Xj>logu, X~>logv \;=1

-P

Xj>logu \j=~

so that 11 is nonpositive. Therefore, P

>1tn

~<12= e-~Ee }--~"-~j=~XJEeX°.

P(X~>logv)<~O,

280

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CaL G.G. RoussasI Statistics& ProbabilityLetters36 (1997)275-287

A repetition of this argument leads to the inequality

P

Xj>~tn j=l

e-"IIEeX . j=l

The remaining arguments are the same as those used in Roussas (1996).

[]

We now turn to the proof of Theorem 2.2. Proof of Theorem 2.2. For ~ > 0,

P(~p,>~p+~)=P(?,(~p+e)
(Vj,-EVjn)>n[(1-p)-EV1,]

,

where Vj, = 1 - K((~p + ~ -Xj)/hn), j = 1,... ,n. Next, it is easily seen that 1 - p - E(VI,)---+F(~p + e) F(~p)>~6(e), so that 1 - p - E ( V l , ) > 6 ( e ) / 2 for all sufficiently large n. Furthermore, the r.v.s V:.,, j = 1..... n, are NA, since so are Xj, j = 1..... n, and Vj, is a nondecreasing function of Xj. Then, by Lemma 2.2, it follows that, for all sufficiently large n,

P( ~pn> ~p q- ~) ~ e-(n/8)62(t). In a similar fashion, and for all sufficiently large n, it follows that

P( ~p, < ~p - ~) ~ e-(~/8)6:(t). These results complete the proof of the theorem.

[]

3. Asymptotic normality with rates

It is the purpose of this section to show that the asymptotic distribution of v/-n(~'p, - ~p) is normal, and also to provide rates for the convergence. Regarding convergence to normality, we have the following result. Theorem 3.1. Let {Xj; j~> 1} be a sequence of NA or PA r.v.s, and suppose assumptions (A1)-(A6) are

satisfied, and let "~p, be given by (2.1). Then V~(~pn -- ~p) d - ~ N ( 0 , O'2(~P) '~

(3.1)

f- p)j'

oo where a2(x) =F(x)[1 - F(x)] + 2 ~-~j=I[P(X1 ~
Proof. Set

Gn(x; y) = P

\( ~n(x---~)-E~n(-x)V/Var(f~
and

H,(y)=p(

V~(~Pn - ~P)f(~P)-a-(~-~

<<.Y].

(3.2)

Observe that

Un(y)=P

~p,<~¢p + f (a(~p)y ~ p ) V ~ J~ =p(~n(xn)>~p ) = 1 - Gn(x,; tn),

(3.3)

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281

where

a(~p)y p Effn(Xn) x n = ~p "q- - f(~p)v/- ~ --~ ~p and tn - v/Var(ffn(x.) ). -

(3.4)

Convergence (3.1) asserts that H n ( y ) - ~ ( y ) ~ O , where ~(x) is the d.f. of the standard normal, which, in view of the above, becomes 1 - G.(x.; t.) - q~(y)~O. However, 1 - Gn(x.; t.) - ~ ( y ) = - [ G . ( x n ; t.) • (t.)] + [1 - ~(t.)] - ~(y), and G.(x.; t.) - ~(t.) ~ 0 , 1 - ~(t.) - ~ ( y ) ~ 1 - ~ ( - y ) - ~ ( y ) =0. The first convergence follows from Corollary 2.1 in Cai and Roussas (1995a), and the second is a consequence of the fact that tn ~ - y , as is explained below. Indeed, by (1.1) and integrating by parts, we have

EFn(xn)= ~ K ( x . - u'~ dF(u) = ~ F(xn - h.u)dK(u).

\h.)

Expanding F around ~p up to terms of second order, we obtain

F(xn - h.u)= F(~p) + : t ~ p ) [ f ( ~ p ) v ~

+

for some u* with lu* - ~pl <~I(a(~p)Y/f(~p)v/-~) - hnul. Integrating and taking into consideration our assumptions, we get ~P )~Y + 0 E F n ( x n ) = F ( ~ p ) + a( x/n

+ h2n = P - b - - +x/n O

p - EFn(xn)=

h

+h2n

'

or

a(~p)~yx/~ + 0 ( ~

.

(3.5)

By our assumptions and Lemma 3.3 (iii) in Cai and Roussas ( 1995 a), it follows that n Var(/~n(xn)) ~ a 2(~p), so that V/Var(/~ (x~))= (r(~p)x/~+ o ( - - ~ ) .

(3.6)

From the definition of t~ in (3.4) and relations (3.5) and (3.6), it follows that t~ ~ - y , as was to be seen.

[]

The theorem just proved is now refined as follows. Theorem 3.2. Let {Xj; j~>l} be a sequence of NA r.v.s and suppose assumptions (A1)-(A4), (A6) and (A7) are satisfied. Then, with u(n) and H~ as given in (A1) and (3.2), respectively, the following hold (i) I f u(n) = O(e -t~n) for some fl > 0, then

sup iH~(Y)- ~(Y)l<~C(~p)(1-~n), yER

where C(~p) is a positive constant depending only on ~p. (ii) I f u(n) = O(n -~) for some ct > O, then sup IH,(y) - ~(Y)I = O(n-~(logn)~),

yER

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282

where 7=~/2(3 + ~) and v = 0 , /f 0 < ~ < 1 ; 7 = 1 and v = ¼, /f ~ = 1; and 7 = ( ~ - 1 ) / 2 ( ~ + 1) and ~= l/(l+c

O, / f ~ > l .

Proof. In the first place, define A.(xn) by A.(xn)= SUpyeRIG.(x., y) - ~(Y)], where x. is given in (3.4). Then, under either NA or PA, it holds by Corollaries 2.3-2.5 in Cai and Roussas (1995a) that

An(x~)<~C(~p)(l~n)

if u(n) = O(e-/~"),

(3.7)

and

(3.8)

An(x.) = O(n-7(logn) ~) if u(n) = O(n-~). Next, IH.(y) - @(y)l ~
I@(-t.) -

@(y)l

~

-It.- +

y[

since Iq"(y)l ~ _ ~ 1

v%

Thus, It. + yl IH.(y) - @(y)l <~ A . ( x . ) + - -

(3.9)

From the definition of t. in (3.4), and relations (3.5) and (3.6), we obtain t. + y = O ( y 2 n -1/2 -]- x/~h2n). However, this is O(y2n -1/2 + n -1/2 l o g . ) due to the assumption (A7). Therefore, relation (3.9) becomes

IH,(y) -

~(Y)[ <~A.(x.) + O{/ y2 \

+ logn'~

(3.10)

In (3.10), we will take the sup over y c ~ by splitting ~ as follows: y; lyi~a, and y; lyi>a,, for some a . > 0 ; actually, a. is taken to be: an = d l ~ for some d > max{2/a(~p), 1}. Then, (3.10) yields

sup

+ ofl°gn'

(3.11)

Next, sup IHn(y) - ~(Y)I ~< 2 sup [1 - q~(y)] + sup [1 - Hn(y)] + sup H n ( - y ) lyl >a.

y>a. 2

__e-an~2 V~an

y>a~

y>an

2

q-

y>anSUp [1 -- Hn(y)] + y>anSUpHn(-y),

because 1 - ~ ( x ) < ~ ( 1 / ~ ) e -x2/2 for x > 0 and the right-hand side is a decreasing function of x. For the above choice of an, it is clear that (2/x/~an)e -a2./2 =o(n-l/2). Thus, fl\ sup IH.(y) - ~(y)l <<.o[--~__] + s u p [ 1 - Hn(y)] + sup Hn(-y). lYl > a .

\ ~/rl fl

y>a.

y>a~

(3.12)

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283

Next, let

]+EK

,

j---1 ..... n,

and observe that these r.v.s are NA, since so are Xj, j = 1. . . . . n, and the transformation involved is monotone. Furthermore, it follows by (3.3) that

An application of Lemma 2.2 to the right-hand side above yields, for all sufficiently large n,

P(~Yj(xn)>~V~ff(~p)y+O(y2+nh2n) ) j = l ~2exp{-l[x/na(~p)y+O(y 2 +nh2n)]2} ~<2expI-~o'2(~p)y2 ]

•

Therefore,

supaoll-- Hn(y)l<~2exp[-~a(~p)an] = 2 e x p [

d2°'2(~4)l°gn ]

o

1

.

In a similar fashion, and for all sufficiently large n, it follows that ySUpH n ( - y ) =

o ( ~ )1.

By means of these results, inequality (3.12) yields

,Hn(y)- 4~(y)l=O(~n ) .

sup

(3.14)

lYl>a.

Now suppose that u(n)=O(e-#n). Then relations (3.7), (3.11) and (3.14) establish the first part of the theorem. If u(n)= O(n-~), then relations (3.8), (3.11) and (3.14) imply part (ii) of the theorem. [] Theorem 3.3. Let {Xj; j>~l} be a sequence of PA r.v.s and suppose assumptions (A1)-(A4), (A6) (A7) are satisfied. Then the results in Theorem 3.2 hold true.

and

Proof. Observe that Yj(xn), j = 1.... , n, are PA, since so are Xj, j = 1. . . . . n, and the transformation is monotone. It is easily seen that the condition of Theorem 2 in Birkel (1988) is satisfied for r>~6(1 + a ) > 6 , under either one of the assumptions: u(n)=O(n-~) for some ~>0, or u(n)=O(e-#n) for some fl>0. Then, by (3.13), the Chebyshev inequality, Theorem 2 in Birkel (1988) and (3.5), we have, for any r > 2 ,

y>an

y>an

Likewise, sup

y>an

Hn(-y) = O ( a n r ) .

n [ E F n ( x n ) - p]

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By means of these results, inequality (3.12) becomes sup

[Hn(y) - ~(y)l <~o(-~n ) +O(anr),

[Yl>a.

which, in view of (3.11) and (3.10), is modified as follows: sup [y[>an

[Hn(y) - ~(y)[ <~o(--~n ) +O(anr) + O ,/;)

Take an to be sup

Iq,(y)

din 1/2~r+2)for - Hn(y)[

some dl >0. Then

<~O(n -~/2~+2))+ An(xn).

(3.15)

Y

Now suppose that u(n)=O(e-#n). Then relations (3.7) and (3.15) establish the first part of the theorem. If u(n)=O(n-~), then relations (3.8) and (3.15) yield part (ii) of the theorem. [] 4. Weak convergence

Let /~n(x) = v'~[/~n(x) - F(x)],

x E R,

and

yn(p)= v'~f(~p)(~pn - ~p)= x/nf(¢p)[Fn-l(p)- F-l(p)],

0
(4.1)

which is referred to as the normalized smoothed quantile process. We wish to show that 7n(P) converges to a Gaussian process. To this effect, by Taylor expansion, we have F(~p):

p = Fn(/~n-l(p)) =

ffn(~p) +

fn(~;n)('~pn -- ~p),

where ~p* is a random point between ~'pn and ~p, and of f(x), namely,

fn(x)

is the Rosenblatt-Parzen kernel density estimate

1 ~j n1 k ( ~ n ~ j )

f.(x) =

Therefore, f(~p)

~pn - ~p= fn(¢*)

-/~n(¢p)]

and

f(~p)

Vn(P)=-f--7-~-)

-

(4.2)

Let

qn(P) =

I

f(¢P) )' fn(¢~*

so that

7n(P) + [3n(¢p)= qn(p)[3n(¢p).

Under assumptions (A1)-(A6), Corollary 3.1 in Cai and Roussas (1995b) applies and gives that//n(') converges weakly to a Gaussian process B(.), namely, /~n(') = ~ B(.),

(4.3)

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285

where B(.) is a Gaussian process with zero-mean and covariance structure as follows: a2(x, y) = EB(x)B(y) = F(x A y ) -- F ( x ) F ( y ) o~

+ E

[Cov(I(X1 <~x),I(Xj+I <~y)) + Cov(I(Xj <<.y),l(Xj+l ~
(4.4)

j=l

for all x and y. Then, by the continuity mapping theorem (see, for example, Theorem 5.1 in Billingsley, 1968, p. 30), it follows that sup

d

Ifl,(~p)l

0
sup

IB(p)l,

0
which implies that SUPo< p< 1 [fln(~p)[ = Op(l). So, it suffices to show that sup

[ ? n ( P ) + fln(¢p)[ P

O,

(4.5)

0
and this would be implied by

I~Mp)IL 0.

sup 0
Observe that tl,(p) = f~(¢P~) - f(~P) = f~(~) - f(~p) = zn(p) f~(~) f~(4;~) -- f C p ) + f ( ~ p ) 1 + z~(p)' where zn(p) =

f~(¢p~) - f ( ~ p ) f(¢p)

Clearly, it is enough to show that sup Iz~(P)] P O. 0
Under the assumptions that 0 < inf f ( x ) <<.sup f ( x ) < cx~, x

x

(4.6)

and the first part of (A4), it suffices to show that sup If~(x) - f ( x ) I e 0

(4.7)

x

and sup O
I~pn - ~pl ~ O.

(4.8)

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Recall that if Fn is the empirical distribution function of X1 ..... Am, then 1

o~

1

°°

(~tl[f(t)_f(x)]dt

(4.9)

At this point, assume that k(u) ~ 0 as [u[ ---+~ , that the derivative k'(u) exists for u E • and that f-~o~ [k'(u)l du < cx~. It is further assumed that f satisfies a Lipschitz condition and that f _ ~ lulk(u)du < c~. Then, integrating by parts in (4.9), we are led to suP lfn(x) - f ( x ) l <~C [ ~nhn SUp x 'x/n[Fn(x) - F(x)]l +

for some constant C > 0. Utilizing Corollary 1 in Yu (1993), one has, under assumption (4.11) below, that sup v/-~lF, (x) - F(x)l -- Op(1 ),

(4.10)

x

so that sup If~(x) - f(x)l • 0, x

provided nh~--+ co; then (4.7) follows. Next, by (4.2), supx f ( x ) sup Iff,(x) - F(x)l. sup [~'p, - ~p] ~< infxf(X) - SUPxIf,(x) - f(x)[ x

0
The first convergence in (2.3) and relations (4.6) and (4.7) imply (4.8), and hence (4.5). We have established the following results. Theorem 4.1. Let assumptions (A1)-(A6) hold and suppose that the r.v.s {Xj; j~> 1} are either PA or NA. Also, assume that there exists a constant 0 < z < 1 such that n

j21Cov(X1,Xj+I )11/3 = O(n ~)

(4.11)

j=l

and that k(u)~O

as

lu[~,

and

Ik'(u)ldu
It is further assumed that f satisfies the Lipschitz condition: I f ( x ) - f ( Y ) l <<.Llx - Yl f o r any x, y E g~, as well as assumptions in (4.6). Finally, it is assumed that nh~27..., oo. Then the process Yn(P) defined in (4.1) converges weakly to a Gaussian process G(.) in C(0, 1); i.e., ~n(')

--~

a('),

where G(.) has zero-mean and covariance structure specified by

Cov(G(pl ), G(pe)) = a2(~p,, ~p2) for all Pl, p: e (0, 1); a2(x, y) is defined in (4.4).

Z. Cag G.G. Roussas / Statistics & Probability Letters 36 (1997) 275-287

287

Acknowledgements The authors are indebted to an anonymous referee whose truly constructive comments helped to improve an earlier version of this paper, including weakening of the conditions under which Theorem 4.1 holds. This research was supported in part by a research grant from the University of California, Davis.

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