A Short and Elementary proof of the main Bahadur-Kiefer Theorem

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EINDHOVEN UNIVERSITY OF TECHNOLOGY Department of Mathematics and Computing Science

Memorandum

COSOR

94-26

A Short and Elementary proof

of the main Bahadur-Kiefer Theorem

John H.J. Einmahl

Eindhoven, August 1994 The Netherlands

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A

SHORT AND ELEMENTARY PROOF

OF THE MAIN BAHADUR-KIEFER THEOREM

John H.J. Einmahl

Eindhoven University of Technology

A short proof of the lower bound in the strong version of the famous Theorem 1A in Kiefer (1970) 011 the Bahadur-Kiefer process is presented. The proof is elementary and does in particular not use strong approximations.

AMS 1991 subject classifications. 62G30,60F15.

Key words and phrases. Bahadur-Kiefer process, empirical and quantile process, strong law.

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Let [11, U2 , ••• be a sequence of independent uniform-(O, 1) random variables and for each

n E N, let

n

Fn(t)

=

~

L:

l[o,tl(Ui) , 0 ::;

t ::;

1,

i=l

be the empirical distribution function at stage n. The uniform empirical process will be written as

1

On(t)

=

n

2(Fn(t) - t),

0 S; t ::; 1 ;

On(t)

=

0 for t

<

0 or t

>

1.

Also for each n E N,

Qn(t)

= inf{s :

Fn(s)

~

t},

0

<

t ::; 1;

Qn(O)

= 0, denotes the empirical quantile function and we write

J

!3n (t)

=

n'i (

Q

n (t) - t),

0 ::; t ::; 1,

for the corresponding uniform quantile process. The so-called Bahadur-Kiefer process is defined by

Rn(t)

=

on(t)

+

!3n(t) ,

0 ::;

t ::;

1.

This process is introduced in Bahadm (1966); in Kiefer (1970, Theorem 1A) the "in-probability-analogue" of the following statement is proved

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1

1· JIll n"4 1

II

Rn

II -_

1 1 a.s.,

n--+oo

(log

11.)2

lion 11'i

where

II 1 11=

SUPO<t<1

11(t)1

for any real-valued function

1

on [0,1]. In the latter paper a proof of (1) itselfls-claimed but not presented. However, it is proved in Shorack (1982) that, indeed, the expresssion on the left in (1) (with 'lim' replaced by 'limsup') is not larger than 1, almost surely, (note that

lion 11=11

!3n

II)

whereas in a. recent paper by Deheuvels and Mason (1990) it is established that the same expression is not smaller than 1, almost surely. The short and elegant proof in Shorack (1982) is based on the Kiefer process strong approximation of an, but in Shorack and Wellner (1986, pp. 590-591) a similar, direct proof of the "upper-bound-part" is given. The ingenious proof of the "lower-bound-part" (which finally led to a complete proof of (1)) in Deheuvels and Mason (1990) is extremely technical, moreover it is again based on the Kiefer process strong approximation of an.

It is the purpose of this note to give a new, short proof of the "lower-bound-part" of (1), i.e. we will prove t.hat

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(2) I liminf _1_1-4--;:-

II

Rn

II

>

1 1 -n ... oo (log

II

an

112

a.s.

Our proof is rather easy and not based on strong approximations. It uses as tools the following well-known facts on empirical and quantile processes, although most of them are not required at their full strength.

FACT 1 (Mogul'skii (1979»).

(3) liminf (loglog

n)~

II

an

11=

1r/8~

a.s. n--+oo

FACT 2 (easy).

(4) a.s.

FACT 3 (Kiefer (1970».

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Define the oscillation modulus of an by

let {an} ~=l be a sequence of positive numbers with an

1

0 and nan

i.

FACT 4 (Mason, Shorack and Wellner (1983». If log(l/an)floglog n ~ c E

[0,00),

then

(6) limsnp

n-+oo (an log log n)

1

(2(1

+

C»2

a.s.

FACT 5 (Stute (1982». Iflog(l/an )/ log log n ~

00

and rwn / log n ~

00,

then

(7) I11n .

wn(an)

I =

n-+oo (an 10g(1/ an

»2

a.s.

FACT 6 (Mallows (1968». If (NI , ... ,

Nk), kEN, has a multinomial distribution with

k

parameters m and ])1, . . . , ])k, where mEN and ])1, ... ,])k are non-negative with

L

Pi :::: 1, i=l

then for all )\}, ... , Ak

k

P(N

1 ~

AI, ... , Nk

~

Ak)

~

II

P(Ni

~

Ai) .

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FACT 7 (Kolmogorov (1929)). Let mEN and t E (0, ~). Then for every 8

>

0 there exist B:j,](2 E

(0,00)

such that for ](Id ::; ). ::; ](2mb

P(am(t)

> ).)

~ exp(

-(1

+

8)).2/(2t(1 -

t))).

FACT 8 (Dvoretzky, Kiefer and Wolfowitz (1956), Massart (1990)). Let n E N. Then for all ). ~

°

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P(II

an II~

).) ::;

2 exp( - 2).2).

PROOF OF (2). Let I denote the identity function on [0,1]. From (4), (3) and Qn

=

1+

(3n/nt

we see that it suffices to show that

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1 I

n4

II

an(I

+

(3n/n"2) -

an

II

liminf - - - - : : - 1 I ~ 1

n-= (log

nF

II

an

11"2

a .. s.

Using

(5), (7)

and

(3), (9)

can in turn be repla.ced by

(10) 1 I 1/."4

II

an

(I -

a n

/n"2) -

an

II

liminf ---,--1 1 ~ 1 (log

nF

II

an

Wi

a.s. Set, for 0 ::; t ::; 1, { a

(t)'

if lan(t)1

>

1/

log n ,

on(t)

=

₁/11_1"_og_n, _if

1 ()I

_an_t _::;₁

/

_log_{11. •}

Define the following grid on [0,1]: ti,n = i/[log n],i O,I, ... ,[log

nJ,

where

[xJ

denotes the integer pa.rt of :1: E R. Now using (6) and (7), and again (3), it follows that instead of proving (10), it suffices to prove that

(11) liminf a.s.

11·--+00

Ush]g t.he Borel-Cantelli lemma a proof ,of (11) is established if we show that for ill

=

£ E (0,1),

L

PAn

<

00,

where

11.=3

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::; ((1 -

E)

Vhite Cn

=

Cn( Cl,n, C2,n, . .. , C[log n]-l,n)

=

{O'n(ti,n)

=

Ci,n, 1 ::; i ::; [log

n] -

1}, Ci,n E

1 1

[- log 11., log 11.] and Ci,n such that 11. ti,n

+

11. 2" Ci,n E {O, 1, ... , n} and such that 11. ti,n

+

n2" Ci,n is non-decreasing in i. Observe that PCn

>

o.

Set cn = ( . max

ICi,nl)

V (1/log n)

l~t~[log nl-l

and, on Cn, let tn be the smallest ti,n,O ::; i ::; [log n], such that lan(ti,n)1 = cn, write

dn

=

O'n(tn) and

d

n

=

an(tn)j set t~

=

tn

+

1/[log 11.] and d~

=

O'n(t~). Now we have (12)

1

Write 1n_n

=

n/[log 11.]

+

n2"(d~ - dn). Note that, on Cn, 1n_nis the number of observations

falling in the interval

(tn,

t:J

Now it is not hard to see that on Cn, the process am" defined by

is a uniform empirical process based on 1n_n observations. Hence the right hand side of

(12) can be written as (13)

P(n"4

1 sup ",,,[lot n] ::; S ::; 1 _ ",,,[lot n]

I(

!l!:.u.)~

{a

(s -

dn[log n]) -

a

(s)}

11, tnn n~ mn n2 n2

Now observe that

1

::; 2(1og n)2 11.-"4 -+ 0, as n -+ 00.

Therefore, for la.rge n, the expression in (13) is bounded from above by

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P(1/."4(7)2"

1 1 sup ",,,[log

nJ

< <

1 _ ",,,[log

nJ

1. _ 8 _ I 71.2 n2

I - (-

dn _[log_{n] ) _ -} _{( )}

I

O'm" S nt O'mn S

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It is easy to check that, for large n, Fact 7 applies to the probability in (15). This yields, with (j

=

[/4, the following upper bound for the expression in (15)

Now we are ready to complete the proof. Combining (12)-(16) we have P(AnICn) ~

l/n

2

(11.

large). Set Dn

=

{II

an

II>

log

n}

and note that (8) implies that PDn ~

l/n

2

(n

~ 4). Hence for large n

where sup* denotes the supremum over all Cn as defined before. Now, of course, L:~=3 PAn

<

00 because of (17). This proves (11) and hence (2). 0

References

Bahadur, R.R. (1966). A note Oil quantiles in large samples. Ann. Math. Statist. 37

577-580.

Deheuvels, P. and Mason, D.M. (1990). Bahadur-Kiefer-type processes. Ann. Probab. 18 669-697.

Dvoretzky, A., Kiefer, J .C. and Wolfowitz, J. (1956). Asymptotic minimal char-ader of the sample distribution function and of the classical multinomial estimator.

Ann. A1alh. Statist. 27 642-669.

Kiefer, J .C. (1970). Devia.tions between the sample quantile process and the sample df. In Non-]Ja7'((.mct.l'ic Techniques in Statistical Inference (M. Puri, ed.) 299-319. Cambridge Univ. Press, London.

Kolmogorov, A.N. (1929). tTber das Gesetz des iterierten Logarithmus. Math. Ann. 101 126-135.

Mallows, C.L. (1968). An inequality involving multinomial probabilities. Biometrika 55 422-424.

Mason, D.M., Shorack, G.R. and Wellner, J .A. (1983). Strong limit theorems for oscillation moduli of the uniform empirical process. Z. Wahrsch. Verw. Gebiete 65 83-97.

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Massart, P. (1990). The tight constant in de Dvoretzky-Kiefer-Wolfowitz inequality.

Ann. P1'Obab. 18 1269-1283.

Mogul'skii, A.A. (1979). On the law of the iterated logarithm in Chung's form for functional spaces. Theory P1'Obab. Appl. 24405-413.

Shorack, G.R. (1982). Kiefer's theorem via the Hungarian construction. Z. Wahrsch.

Ve7'w. Gebiete 61 369-373.

Shorack, G.R. and Wellner, J .A. (1986). Empirical Processes with Applications to

Statistics. Wiley, New York.

Stute, W. (1982). The oscillation behaviour of empirical processes. Ann. Probab. 10 86-107.

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