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A remainder estimate for the normal approximation of perturbed sample quantiles
Statistics & Probability North-Holland
Letters
17 July 1992
14 (1992) 293-298
A remainder estimate for the normal approximation of perturbed sample quantiles Stefan
S. Ralescu
Department of Mathematics, Queens College, City Uniuersi@ of New York, Flushing, NY, USA Received March Revised October
1991 1991
Abstract: For a smooth statistical normal law is considered. Under Keywords: Perturbed
1. Introduction
sample
model, suitable
quantile,
the rate of convergence in distribution of the kernel-smoothed quantile estimator assumptions, a Berry-Es&en type result with a rate O(n ‘/* log n) is established.
kernel
estimate,
central
limit theorem,
convergence
to the
rate.
and main result
The study of perturbed empirical distributions and associated quantiles has received considerable attention, due in large part to the work of M. Falk and R.D. Reiss. The monograph by Reiss (1989) gives a particularly lucid exposition of the mathematical attractions of smoothing the empirical distribution function (e.d.f.): it is intuitively appealing, it is easy to compute, it provides increased second order efficiency and it improves the speed of convergence in the bootstrap. Let X,, X2,... be independent and identically distributed (i.i.d.1 observations drawn according to a cumulative distribution function (c.d.f.) F. A fundamental approach to statistical estimation and hypothesis testing is to estimate directly F by a random distribution. In many statistical procedures, however, it is more convenient to formulate hypotheses on F in terms of F-’ rather than F itself. Here F-‘(p) = infix: F(x) >p}, 0
Here, (a,> is a sequence of (positive) ‘window-width’ tending to zero as n + m and k satisfies: k > 0, dx = 1. In the context of a smooth statistical model, it is more appropriate to use a smooth
/k(x)
Correspondence Blvd., Flushing, Research
to: Stefan S. Ralescu, Department NY 11367-1597, USA.
supported
0167-7152/92/$05.00
in part by the Research 0 1992 - Elsevier
of Mathematics,
Foundation
Science
Publishers
of CUNY,
Queens
Grant
College,
PSC-CUNY
B.V. All rights reserved
City University
of New York,
65-30 Kissena
661374.
293
Volume 14, Number 4
STATISTICS & PROBABILITY LETTERS
estimator gn rather than corresponding to f:, via &x)
=lX
K(x)
=
A(t) --m
the
step
dt=n-I
function
F,,. A suitable
method
17 July 1992 to obtain
&((~-&),a,) i=l
g,, is to use the
d.f.
(1.2)
where
lx
k(t)
dt.
-cc
The research of the large sample properties of F,, was initiated by Nadaraya (1964). A detailed listing of work on (1.2) is given in the references. It is noteworthy that Reiss (1981) and Falk (1983) have shown that for appropriately chosen kernels and sufficiently smooth distributions, the asymptotic performance of i,, is better than that of F,, in terms of relative deficiency. In the present paper we work directly with the perturbed kernel sample quantile gn-‘(p)
= inf{x:
Fn(.x) >p},
0
(1.3)
The statistical motivation for using (1.3) relies on the considerable amount of empirical evidence available to support the superiority of &l(p) over the classical sample quantile FnP*(p) for a variety of smooth distributions (cf. Azzalini, 1981). Indeed, a large number of simulation studies (cf. Kappenman, 1987) underlined this aspect by revealing that kernel quantile estimators perform substantially better than the sample quantile when they are used to estimate quantiles in moderate to heavy tails of the sample distribution. The limiting behavior of @nP1(p) was first elucidated by Nadaraya (1964), who obtained - under appropriate conditions - the central limit theorem A,= where
Q(x)
supIH,(x)
denotes
-@(~)I=o(l)
the standard
normal
(1.4) d.f. and H,(x)
is the c.d.f. of
For related investigations in this direction under different sets of conditions we refer to Mack (1987) and Ralescu and Sun (1992). However, the problem of determining a convergence rate for A, has remained open. The study of the accuracy of the normal approximation is a very important problem. Indeed, though sometimes usual statistical practice ignores this fact, such limit theorems are useless for applications, unless one is willing to derive an accompanying inequality providing control over the accuracy of the approximation. Estimating rates of convergence in the central limit theorem for various nonparametric estimators has been an important research topic over the last twenty-five years. From a statistical point of view, such a study has proved extremely fruitful by yielding much that is significant for statistical theory as well as useful in practical applications. The characterization of the error of approximation in the central limit theorem may be used to establish large deviation approximations, which in turn are useful in asymptotic efficiency considerations. Recently, attention has been focused to this problem in connection with classical sample quantiles for which Reiss (1974) has obtained a rate Berry-Es&en bound of order n-‘12, while Puri and Ralescu (1986) have validated this convergence for the generalized class of random central order statistics. The purpose of the present paper is to go beyond the limiting result (1.4) and investigate the speed of convergence. We produce a remainder term estimate by specifically showing, under appropriate regularity conditions, that A,, = O(n- ‘I2 log n). We do not know whether the Berry-Esseen rate 0(n-‘j2) is available. Our theorem is the first general estimation establishing a bound in the asymptotic normality for perturbed kernel quantiles. It is worth noting that our theorem requires a smooth statistical model. 294
Volume
14, Number
Throughout
4
STATISTICS
the paper,
we shall make the following
F is twice differentiable _/xk(x)
& PROBABILITY
dx=O
and
on its support /x’k(x)
17 July 1992
LETTERS
assumptions:
with F” bounded
and F’( F-‘( p)) > 0,
dx
(1.5) (1.6)
n”2Un = O( 1).
(1.7)
Remark 1.1. A word of comment about the above assumptions is in order: in practice, the kernel weight a, are the two items that the user must select in order to determine F,-‘(p). Now, while the choice of k will not significantly affect the asymptotic behavior of pnp l(p), it is clear that since the parameter a, determines the degree of smoothing, one may expect that the closeness of the normal approximation tends to depend more crucially on the choice of a, and thus, assumption (1.7) should not be viewed as an undue handicap, but rather as a technical requirement to force a better rate of convergence. A closer examination of the proof presented below shows that one could obtain a slower rate for the convergence to zero of A, by relaxing the requirement (1.7) to allow a wider range of variability for the bandwidth.
k and the bandwidth
Theorem. Under the assumptions (1.5)~(1.71, there exists a constant A > 0 not depending on n, such that for all n > 1,
A,
log n.
(1.8)
Remark 1.2. In the case of classical sample quantiles, a practically useful estimate of the constant corresponding to the constant A in (1.8) in the Theorem, was given by Reiss (1974) (see also Puri and Ralescu, 1986). Unfortunately, the present proof of (1.8) does not lead easily to an exact value of the constant A in (1.81, although we expect that the actual error terms are small enough to get approximations of practical relevance. There is, however, not much hope to find reasonable estimates for this constant for small sample sizes but this might be possible under more restrictive conditions on f.
2. Proof of the Theorem Let K,(x)
=jx a,’ k(t/a,) -CC
We begin by noting that H,(x)
dt,
pL,(x) =E{K,(x-X1)}
= P(@~W’(p)
and a:(x)
=Var(K,(x-XI)}.
+ c~n-“~ ) >p) where c = (~(1 -~>>“~/f(F-‘(p)).
Then,
if we set
Z,i=K,(F-l(p) v,(x)
+c~n-‘/~-X~),
=E(Zni)
=/_L,(F-‘(p)
l
p) + cxn-1/2),
we have with S,(x)
= n-1/2 I? [Z,, i=l
4x)l/~n(x)
and
w,J x) = n112( p - vn( x))/T,(
x)
295
Volume 14, Number 4
STATISTICS
& PROBABILITY
LETTERS
17 July 1992
that K(x)
Q%(411+ [@(b%(x)) -@WI =rln(x) +rzn(x>, say.
-Q(x)
= [qJ,(~))
-PI&z(x)
(2.1)
We now use (2.1) to estimate sup, x, ~ d, I H,,(x) - Q(x) I where d, = d(log n)‘/*, for some d > 0 to be specified later. By the Berry-Es&en theorem for independent r.v.‘s, there exists C, > 0 (universal constant) such that for all n > 1 and all x, Irl.(x)I~C,h,(x)n-“* where h,(x) = E I K,(F-l(p) uniformly for I x I < d,, IF(F’(p)
(2.2) + cxn -l/2 -Xl)
- vn(x)1 3/ri(x).
i/2)(1 -F(P(p)
+cxn-“2))
+cxn-
< cd(log n)l’*
sup
IlJ’(t)l=
In order to asses h,(x),
-p(l
we note that
-p)]
0(n-“2(log-n)1’2)
(2.3)
It I
where e(t) = F(F-l(p) + t) -F*@‘-‘(p) + t). Combining (2.3) and Lemma 2.1 of Reiss (1981), we find that sup (r~(X)-_P(1-p)~=O(a,+n-“2logn). 1x1
(2.4)
From (2.4) it is clear that suplX, Gd,hn(~) = O(1) and thus sup ITln(X)I = O(n-“2). Ixlcd,
(2.5)
On the other hand, by using a Taylor expansion and the Mean Value Theorem qiJ= @‘, Ir2n(X)I =]n”Q,(x)/r&)
+ ((P(I
-P))“‘/rAx)
- I)”
+c*X*12-1’*F”(5px)/2Tn(
x) I+(p)
for some intermediate points tPpxand p where e,(x) = v,(x) - F@-‘(P) + cxn-“*). (1.7), Lemma 2.1 of Reiss (1981) and (2.4) we have uniformly for I x 1 < d,, that d’“e,(x)/7,(x)
[(p(l-p))1’2/r,(a)-l]x=O(n-i’210g
= O(n-“2),
we can write with
(2.6) Applying condition
n)
and X2.-1’2F”(5px)/T,(X)
= 0(n-“*
log n).
On combining these estimates we find via (2.6), sup ITZn(X)) = O(K”2 Ixl
(2.7)
log n).
Thus, from (2.1), (2.5) and (2.7) we obtain sup I&(x) Ixlcd,, 296
-@(x)l=O(K”*
log n).
(2.8)
Volume
14, Number
4
STATISTICS
& PROBABILITY
17 July 1992
LETTERS
Next we consider the range I x I a d, for which we use the estimate sup [H,(x) lx/ >d,
-@(x)1~P{[;~--‘(p)
r~)“~}+ (1
-F-1(p)l&cdn-1/2(log
- 0( d,)). (2.9)
The second term on the right hand side of (2.9) is estimated by Mill’s ratio l-@(d,)
-d2/2/(2T)1’2dn
= O(K”~)
if d 2 1.
(2.10)
For the first term, it suffices to consider Z@-‘(p) > F-‘(p) + cdn-1/2(log r~)‘/~j = T,, since a similar estimate holds for the probability of the reversed inequality. To obtain an upper bound for r,,, note that if d is chosen sufficiently large we have for n large, n) r/2
m, >p + +n-1/2(log
where m, = p,jF-l(p) since
(2.11)
+ dn- 1/200g n)‘/‘).
-K,(F-l(p)
7=rn=P k(rnn 1 i=l
We establish this inequality at the end of the proof. Then,
+cdn-1/2(log
an(m,
n)““-Xi))
-p) 1
an application of Hoeffding’s inequality (Hoeffding, -p)‘)
rn G exp( -2n(m,
1963) and (2.11) gives
= O(K”2).
(2.12)
Using simliar arguments we obtain P{F’,I-‘(p)
-cdn-“2(log
n)“‘}
=O(n-1’2).
(2.13)
From (2.9), (2.10), (2.12) and (2.13) we get (2.14)
sup IH,(x)-@(x)l=O(C”2). lxl>d,
The statement (1.8) follows from (2.8) and (2.14). This completes the proof of the theorem. Finally we show the validity of (2.11). We write nr/2
(+I -p)
=n ‘12(m, -F(F-l(p) +nii2(F(F-‘(p)
+d,(log +d,(log
n)1’2)) n)1’2) -p).
(2.15)
By Lemma 2.1 of Reiss (1981) and (1.7), the first term on the right hand side of (2.15) is seen to be O(1). On the other hand, since as IZ+ 03, n1’2( log n) -“*(F(F-‘( for every 0 < E < F’(F-l(p)) n1’2 F F-‘(p) I(
p) + dn-“2(log
n)“2) -p)
+ dF’(F-‘(
p)) > 0
(2.16)
n)1’2.
(2.17)
we have for n large, +dn-
1’2(log n)1’2) -p> >d{F’(F-l(p))
For (2.15)-(2.17), the inequality (2.11) easily follows.
-~}(log
q 297
Volume
14, Number
4
STATISTICS
& PROBABILITY
17 July 1992
LETTERS
Acknowledgement The author
is grateful
to the referee
for a careful
reading
of the paper
and for helpful
comments.
References Azzalini, A. (1981), A note on the estimation of a distribution function and quantiles by a kernel method, Biometrika 68, 326-328. Falk, M. (1983), Relative efficiency and deficiency of kernel type estimators of smooth distribution functions, Statist. Neerlandica 37, 73-83. Hoeffding, W. (1963), Probability inequalities for sums of bounded random variables, J. Amer. Statist. Assoc. 58, 13-30. Kappenman, R.F. (19871, Improved distribution quantile estimation, Comm. Statist. - Simulation Comput. 16,307-320. Mack, Y.P. (1987), Bahadur’s representation of sample quantiles based on smooth estimates of a distribution function, Probab. Math. Statist. 8, 183-189. Nadaraya, E.A. (19641, Some new estimates for distribution functions, Theory Probab. Appl. 9, 497-500. Puri, M.L. and S.S. Ralescu (19861, Limit theorems for ran-
298
dom central order statistics, in: Adaptive Statistical Procedures and Related Topics, IMS Lecture Notes No. 8, 467475. Ralescu, S.S. and S.Sun (19921, Necessary and sufficient conditions for the asymptotic normality of perturbed sample quantiles, to appear in: J. Statist. Plann. Inference. Reiss, R.D. (19741, On the accuracy of the normal approximation for quantiles, Ann. Probab. 2, 741-744. Reiss, R.D. (19811, Nonparametric estimation of smooth distribution functions, Stand. J. Statist. 8, 116-119. Reiss, R.D. (19891, Approximate Distributions of Order Statistics (Wiley, New York). Rosenblatt, M. (1956), Remarks on some nonparametric estimates of a density function, Ann. Math. Statist. 27, 832837.
Letters
17 July 1992
14 (1992) 293-298
A remainder estimate for the normal approximation of perturbed sample quantiles Stefan
S. Ralescu
Department of Mathematics, Queens College, City Uniuersi@ of New York, Flushing, NY, USA Received March Revised October
1991 1991
Abstract: For a smooth statistical normal law is considered. Under Keywords: Perturbed
1. Introduction
sample
model, suitable
quantile,
the rate of convergence in distribution of the kernel-smoothed quantile estimator assumptions, a Berry-Es&en type result with a rate O(n ‘/* log n) is established.
kernel
estimate,
central
limit theorem,
convergence
to the
rate.
and main result
The study of perturbed empirical distributions and associated quantiles has received considerable attention, due in large part to the work of M. Falk and R.D. Reiss. The monograph by Reiss (1989) gives a particularly lucid exposition of the mathematical attractions of smoothing the empirical distribution function (e.d.f.): it is intuitively appealing, it is easy to compute, it provides increased second order efficiency and it improves the speed of convergence in the bootstrap. Let X,, X2,... be independent and identically distributed (i.i.d.1 observations drawn according to a cumulative distribution function (c.d.f.) F. A fundamental approach to statistical estimation and hypothesis testing is to estimate directly F by a random distribution. In many statistical procedures, however, it is more convenient to formulate hypotheses on F in terms of F-’ rather than F itself. Here F-‘(p) = infix: F(x) >p}, 0
Here, (a,> is a sequence of (positive) ‘window-width’ tending to zero as n + m and k satisfies: k > 0, dx = 1. In the context of a smooth statistical model, it is more appropriate to use a smooth
/k(x)
Correspondence Blvd., Flushing, Research
to: Stefan S. Ralescu, Department NY 11367-1597, USA.
supported
0167-7152/92/$05.00
in part by the Research 0 1992 - Elsevier
of Mathematics,
Foundation
Science
Publishers
of CUNY,
Queens
Grant
College,
PSC-CUNY
B.V. All rights reserved
City University
of New York,
65-30 Kissena
661374.
293
Volume 14, Number 4
STATISTICS & PROBABILITY LETTERS
estimator gn rather than corresponding to f:, via &x)
=lX
K(x)
=
A(t) --m
the
step
dt=n-I
function
F,,. A suitable
method
17 July 1992 to obtain
&((~-&),a,) i=l
g,, is to use the
d.f.
(1.2)
where
lx
k(t)
dt.
-cc
The research of the large sample properties of F,, was initiated by Nadaraya (1964). A detailed listing of work on (1.2) is given in the references. It is noteworthy that Reiss (1981) and Falk (1983) have shown that for appropriately chosen kernels and sufficiently smooth distributions, the asymptotic performance of i,, is better than that of F,, in terms of relative deficiency. In the present paper we work directly with the perturbed kernel sample quantile gn-‘(p)
= inf{x:
Fn(.x) >p},
0
(1.3)
The statistical motivation for using (1.3) relies on the considerable amount of empirical evidence available to support the superiority of &l(p) over the classical sample quantile FnP*(p) for a variety of smooth distributions (cf. Azzalini, 1981). Indeed, a large number of simulation studies (cf. Kappenman, 1987) underlined this aspect by revealing that kernel quantile estimators perform substantially better than the sample quantile when they are used to estimate quantiles in moderate to heavy tails of the sample distribution. The limiting behavior of @nP1(p) was first elucidated by Nadaraya (1964), who obtained - under appropriate conditions - the central limit theorem A,= where
Q(x)
supIH,(x)
denotes
-@(~)I=o(l)
the standard
normal
(1.4) d.f. and H,(x)
is the c.d.f. of
For related investigations in this direction under different sets of conditions we refer to Mack (1987) and Ralescu and Sun (1992). However, the problem of determining a convergence rate for A, has remained open. The study of the accuracy of the normal approximation is a very important problem. Indeed, though sometimes usual statistical practice ignores this fact, such limit theorems are useless for applications, unless one is willing to derive an accompanying inequality providing control over the accuracy of the approximation. Estimating rates of convergence in the central limit theorem for various nonparametric estimators has been an important research topic over the last twenty-five years. From a statistical point of view, such a study has proved extremely fruitful by yielding much that is significant for statistical theory as well as useful in practical applications. The characterization of the error of approximation in the central limit theorem may be used to establish large deviation approximations, which in turn are useful in asymptotic efficiency considerations. Recently, attention has been focused to this problem in connection with classical sample quantiles for which Reiss (1974) has obtained a rate Berry-Es&en bound of order n-‘12, while Puri and Ralescu (1986) have validated this convergence for the generalized class of random central order statistics. The purpose of the present paper is to go beyond the limiting result (1.4) and investigate the speed of convergence. We produce a remainder term estimate by specifically showing, under appropriate regularity conditions, that A,, = O(n- ‘I2 log n). We do not know whether the Berry-Esseen rate 0(n-‘j2) is available. Our theorem is the first general estimation establishing a bound in the asymptotic normality for perturbed kernel quantiles. It is worth noting that our theorem requires a smooth statistical model. 294
Volume
14, Number
Throughout
4
STATISTICS
the paper,
we shall make the following
F is twice differentiable _/xk(x)
& PROBABILITY
dx=O
and
on its support /x’k(x)
17 July 1992
LETTERS
assumptions:
with F” bounded
and F’( F-‘( p)) > 0,
dx
(1.5) (1.6)
n”2Un = O( 1).
(1.7)
Remark 1.1. A word of comment about the above assumptions is in order: in practice, the kernel weight a, are the two items that the user must select in order to determine F,-‘(p). Now, while the choice of k will not significantly affect the asymptotic behavior of pnp l(p), it is clear that since the parameter a, determines the degree of smoothing, one may expect that the closeness of the normal approximation tends to depend more crucially on the choice of a, and thus, assumption (1.7) should not be viewed as an undue handicap, but rather as a technical requirement to force a better rate of convergence. A closer examination of the proof presented below shows that one could obtain a slower rate for the convergence to zero of A, by relaxing the requirement (1.7) to allow a wider range of variability for the bandwidth.
k and the bandwidth
Theorem. Under the assumptions (1.5)~(1.71, there exists a constant A > 0 not depending on n, such that for all n > 1,
A,
log n.
(1.8)
Remark 1.2. In the case of classical sample quantiles, a practically useful estimate of the constant corresponding to the constant A in (1.8) in the Theorem, was given by Reiss (1974) (see also Puri and Ralescu, 1986). Unfortunately, the present proof of (1.8) does not lead easily to an exact value of the constant A in (1.81, although we expect that the actual error terms are small enough to get approximations of practical relevance. There is, however, not much hope to find reasonable estimates for this constant for small sample sizes but this might be possible under more restrictive conditions on f.
2. Proof of the Theorem Let K,(x)
=jx a,’ k(t/a,) -CC
We begin by noting that H,(x)
dt,
pL,(x) =E{K,(x-X1)}
= P(@~W’(p)
and a:(x)
=Var(K,(x-XI)}.
+ c~n-“~ ) >p) where c = (~(1 -~>>“~/f(F-‘(p)).
Then,
if we set
Z,i=K,(F-l(p) v,(x)
+c~n-‘/~-X~),
=E(Zni)
=/_L,(F-‘(p)
l
p) + cxn-1/2),
we have with S,(x)
= n-1/2 I? [Z,, i=l
4x)l/~n(x)
and
w,J x) = n112( p - vn( x))/T,(
x)
295
Volume 14, Number 4
STATISTICS
& PROBABILITY
LETTERS
17 July 1992
that K(x)
Q%(411+ [@(b%(x)) -@WI =rln(x) +rzn(x>, say.
-Q(x)
= [qJ,(~))
-PI&z(x)
(2.1)
We now use (2.1) to estimate sup, x, ~ d, I H,,(x) - Q(x) I where d, = d(log n)‘/*, for some d > 0 to be specified later. By the Berry-Es&en theorem for independent r.v.‘s, there exists C, > 0 (universal constant) such that for all n > 1 and all x, Irl.(x)I~C,h,(x)n-“* where h,(x) = E I K,(F-l(p) uniformly for I x I < d,, IF(F’(p)
(2.2) + cxn -l/2 -Xl)
- vn(x)1 3/ri(x).
i/2)(1 -F(P(p)
+cxn-“2))
+cxn-
< cd(log n)l’*
sup
IlJ’(t)l=
In order to asses h,(x),
-p(l
we note that
-p)]
0(n-“2(log-n)1’2)
(2.3)
It I
where e(t) = F(F-l(p) + t) -F*@‘-‘(p) + t). Combining (2.3) and Lemma 2.1 of Reiss (1981), we find that sup (r~(X)-_P(1-p)~=O(a,+n-“2logn). 1x1
(2.4)
From (2.4) it is clear that suplX, Gd,hn(~) = O(1) and thus sup ITln(X)I = O(n-“2). Ixlcd,
(2.5)
On the other hand, by using a Taylor expansion and the Mean Value Theorem qiJ= @‘, Ir2n(X)I =]n”Q,(x)/r&)
+ ((P(I
-P))“‘/rAx)
- I)”
+c*X*12-1’*F”(5px)/2Tn(
x) I+(p)
for some intermediate points tPpxand p where e,(x) = v,(x) - F@-‘(P) + cxn-“*). (1.7), Lemma 2.1 of Reiss (1981) and (2.4) we have uniformly for I x 1 < d,, that d’“e,(x)/7,(x)
[(p(l-p))1’2/r,(a)-l]x=O(n-i’210g
= O(n-“2),
we can write with
(2.6) Applying condition
n)
and X2.-1’2F”(5px)/T,(X)
= 0(n-“*
log n).
On combining these estimates we find via (2.6), sup ITZn(X)) = O(K”2 Ixl
(2.7)
log n).
Thus, from (2.1), (2.5) and (2.7) we obtain sup I&(x) Ixlcd,, 296
-@(x)l=O(K”*
log n).
(2.8)
Volume
14, Number
4
STATISTICS
& PROBABILITY
17 July 1992
LETTERS
Next we consider the range I x I a d, for which we use the estimate sup [H,(x) lx/ >d,
-@(x)1~P{[;~--‘(p)
r~)“~}+ (1
-F-1(p)l&cdn-1/2(log
- 0( d,)). (2.9)
The second term on the right hand side of (2.9) is estimated by Mill’s ratio l-@(d,)
-d2/2/(2T)1’2dn
= O(K”~)
if d 2 1.
(2.10)
For the first term, it suffices to consider Z@-‘(p) > F-‘(p) + cdn-1/2(log r~)‘/~j = T,, since a similar estimate holds for the probability of the reversed inequality. To obtain an upper bound for r,,, note that if d is chosen sufficiently large we have for n large, n) r/2
m, >p + +n-1/2(log
where m, = p,jF-l(p) since
(2.11)
+ dn- 1/200g n)‘/‘).
-K,(F-l(p)
7=rn=P k(rnn 1 i=l
We establish this inequality at the end of the proof. Then,
+cdn-1/2(log
an(m,
n)““-Xi))
-p) 1
an application of Hoeffding’s inequality (Hoeffding, -p)‘)
rn G exp( -2n(m,
1963) and (2.11) gives
= O(K”2).
(2.12)
Using simliar arguments we obtain P{F’,I-‘(p)
-cdn-“2(log
n)“‘}
=O(n-1’2).
(2.13)
From (2.9), (2.10), (2.12) and (2.13) we get (2.14)
sup IH,(x)-@(x)l=O(C”2). lxl>d,
The statement (1.8) follows from (2.8) and (2.14). This completes the proof of the theorem. Finally we show the validity of (2.11). We write nr/2
(+I -p)
=n ‘12(m, -F(F-l(p) +nii2(F(F-‘(p)
+d,(log +d,(log
n)1’2)) n)1’2) -p).
(2.15)
By Lemma 2.1 of Reiss (1981) and (1.7), the first term on the right hand side of (2.15) is seen to be O(1). On the other hand, since as IZ+ 03, n1’2( log n) -“*(F(F-‘( for every 0 < E < F’(F-l(p)) n1’2 F F-‘(p) I(
p) + dn-“2(log
n)“2) -p)
+ dF’(F-‘(
p)) > 0
(2.16)
n)1’2.
(2.17)
we have for n large, +dn-
1’2(log n)1’2) -p> >d{F’(F-l(p))
For (2.15)-(2.17), the inequality (2.11) easily follows.
-~}(log
q 297
Volume
14, Number
4
STATISTICS
& PROBABILITY
17 July 1992
LETTERS
Acknowledgement The author
is grateful
to the referee
for a careful
reading
of the paper
and for helpful
comments.
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