Tilburg University
A short and elementary proof of the main Bahadur-Kiefer theorem
Einmahl, John
Published in:
Annals of Probability
Publication date:
1996
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Citation for published version (APA):
Einmahl, J. H. J. (1996). A short and elementary proof of the main Bahadur-Kiefer theorem. Annals of
Probability, 24(1), 526-531.
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The Annals of Probability
1996, Vol. 24, No. 1, 526 ] 531
A SHORT AND ELEMENTARY PROOF OF THE MAIN
BAHADUR–KIEFER THEOREM
BY 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 Ž1970. on the Bahadur]Kiefer process is presented.
The proof is elementary and, in particular, does not use strong approximations.
Let U1 , U2 , . . . be a sequence of independent uniform-Ž0,1. random variables
and, for each n g N, let
Fn Ž t . s
1
n
n
Ý 1w0 ,t x Ž Ui . ,
0 F t F 1,
is1
be the empirical distribution function at stage n. The uniform empirical
process will be written as
a n Ž t . s n1r2 Ž Fn Ž t . y t . ,
a n Ž t . s 0 for t - 0 or t ) 1.
0 F t F 1;
Also, for each n g N,
Q n Ž t . s inf s: Fn Ž s . G t 4 ,
0 - t F 1,
Q n Ž 0 . s 0,
denotes the empirical quantile function, and we write
bn Ž t . s n1r2 Ž Q n Ž t . y t . ,
0 F t F 1,
for the corresponding uniform quantile process. The so-called Bahadur]Kiefer
process is defined by
R n Ž t . s a n Ž t . q bn Ž t . ,
0 F t F 1.
This process is introduced in Bahadur Ž1966.; in Kiefer wŽ 1970., Theorem 1Ax
the ‘‘in-probability-analogue’’ of the following statement is proved:
Ž 1.
lim
nª`
n1r4
Ž log n .
5 Rn 5
1r2
5 a n 5 1r2
s 1 a.s.,
where 5 f 5 s sup 0 F t F 1 < f Ž t .< for any real-valued function f on w 0,1x . In the
latter paper a proof of Ž1. itself is claimed but not presented. However, it is
proved in Shorack Ž1982. that, indeed, the expression on the left in Ž1. Žwith
lim replaced by lim sup. is not larger than 1, almost surely Žnote that
5 a n 5 s 5 bn 5., whereas in a recent paper by Deheuvels and Mason Ž1990. it is
established that the same expression is not smaller than 1, almost surely.
Received July 1994; revised January 1995.
AMS 1991 subject classifications. 62G30, 60F15.
Key words and phrases. Bahadur]Kiefer process, empirical and quantile process, strong law.
526
527
PROOF OF BAHADUR]KIEFER THEOREM
The short and elegant proof in Shorack Ž1982. is based on the Kiefer process
strong approximation of a n , but in Shorack and Wellner wŽ 1986., pages
590]591x a similar, direct proof of the ‘‘upper-bound part’’ is given. The
ingenious and generally applicable proof of the ‘‘lower-bound part’’ w which
finally led to a complete proof of Ž1.x in Deheuvels and Mason Ž1990. is very
technical; moreover, it is again based on a strong approximation of a n .
It is the purpose of this note to give a new, short proof of the lower-bound
part of Ž1.. That is, we will prove that
Ž 2.
lim inf
nª`
n1r4
Ž log n .
5 Rn 5
1r2
5 a n 5 1r2
G 1 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 w Mogul’skii Ž1979.x . We have
Ž 3.
lim inf Ž log log n .
1r2
nª`
5 an 5 s
p
8
a.s.
1r2
FACT 2 ŽEasy.. We have
5 bn q a n ( Q n 5 s ny1 r2
Ž 4.
FACT 3 w Kiefer Ž1970.x .
Ž 5.
a.s.
We have
lim sup n1r4 Ž log n .
y1 r2
Ž log log n .
y1r4
nª`
5 R n 5 s 2y1r4
a.s.
Define the oscillation modulus of a n by
vn Ž a. s
sup
tysFa
0FsFtF1
< an Ž t . y an Ž s . <,
0 - a F 1;
let a n 4`ns1 be a sequence of positive numbers with a n x0 and na n ­.
FACT 4 w Mason, Shorack and Wellner Ž1983.x . If log Ž1ra n .rlog log n ª
c g w 0, `., then
Ž 6.
lim sup
nª`
vn Ž an .
Ž a n log log n .
1r2
s Ž 2Ž 1 q c . .
1r2
a.s.
FACT 5 w Stute Ž1982.x . If logŽ1ra n .rlog log n ª ` and na nrlog n ª `,
then
Ž 7.
lim
nª`
vn Ž an .
Ž a n log Ž 1ran . .
1r2
s 2 1r2
a.s.
528
J. H. J. EINMAHL
FACT 6 w Mallows Ž1968.x . If Ž N1 , . . . , Nk ., k g N, has a multinomial distribution with parameters m and p 1 , . . . , p k , where m g N and p 1 , . . . , p k are
nonnegative with Ý kis1 pi s 1, then, for all l1 , . . . , l k ,
k
P Ž N1 F l1 , . . . , Nk F l k . F
Ł P Ž Ni F l i . .
is1
FACT 7 w Kolmogorov Ž1929.x . Let m g N and t g Ž0, 12 .. Then for every
d ) 0 there exist K 1 , K 2 g Ž0, `. such that, for K 1 t 1r2 F l F K 2 m1r2 t,
ž
P Ž a m Ž t . ) l . G exp y
/
Ž 1 q d . l2
.
2 t Ž1 y t .
FACT 8 w Dvoretzky, Kiefer and Wolfowitz Ž1956. and Massart Ž1990.x . Let
n g N. Then, for all l G 0,
P Ž 5 a n 5 G l . F 2 exp Ž y2 l2 . .
Ž 8.
PROOF OF Ž2.. Let I denote the identity function on w 0, 1x . First we will
show that, as n ª `,
Ž 9.
ž
bn q a n ( I y
an
n1r2
/
s O Ž Ž log n .
3r4
Ž log log n .
1r8
ny3r8 .
a.s.
To prove Ž9., first observe that by Ž4. and Q n s I q bnrn1r2 we have that
ž
bn q a n ( I q
bn
n1r2
/
s ny1r2
a.s.
Now using Ž5. and Ž7. yields Ž9.. Observe that it immediately follows from Ž9.
and Ž3. that for a proof of Ž2. it is sufficient to show that
Ž 10.
lim inf
nª`
5 a n ( Ž I y a nrn1r2 . y a n 5
n1r4
Ž log n .
1r2
5 a n 5 1r2
G 1 a.s.
Set, for 0 F t F 1,
¡a Ž t . ,
an
Ž t . s~
n
1
¢log n ,
if < a n Ž t . < )
if < a n Ž t . < F
1
log n
1
log n
,
.
Define the following grid on w 0, 1x : t i, n s irw log n x , i s 0, 1, . . . , w log n x , where
w x x denotes the integer part of x g R. From Ž3. and Ž6. we have that
lim
nª`
max 0 F i F wlog nx < a n Ž t i , n . <
5 an 5
s 1 a.s.
529
PROOF OF BAHADUR]KIEFER THEOREM
Moreover, from Ž6., Ž7. and Ž3., it follows that
lim
nª`
n1r4
Ž log n .
1r2
max
sup
0FiF w log ny1 x t i , n FtFt i , nq1
ž
an t y
a n Ž ti , n .
n1r2
ž
ya n ty
anŽ t .
/
/
n1r2
5 a n 5y1r2s 0 a.s.
Hence, instead of proving Ž10., it suffices to prove that
lim inf
nª`
Ž 11.
n1r4
Ž log n .
=
1r2
max 0 F i F wlog nxy1 sup t i , n F t F t iq 1 , n a n Ž t y a n Ž t i , n . rn1r2 . y a n Ž t .
max 0 F i F wlog nx < a n Ž t i , n . < 1r2
G1 a.s.
Using the Borel]Cantelli lemma, a proof of Ž11. is established if we show
that, for all « g Ž0, 1., Ý`ns 3 PA n - `, where
½
A n s n1r4
max
sup
0FiF w log n x y1 t i , n FtFt iq1 , n
ž
F Ž1 y « .
ž
an t y
a n Ž ti , n .
n1r2
max
0FiF w log n x
/
y anŽ t .
< a n Ž t i , n . < log n
/
1r2
5
.
Write
Cn s Cn Ž c1 , n , c 2 , n , . . . , cwlog nxy1 , n . s a n Ž t i , n . s c i , n , 1 F i F w log n x y 1 4 ,
c i, n g w ylog n, log n x and c i, n is such that nt i, n q n1r2 c i, n g 0, 1, . . . , n4 and
such that nt i, n q n1r2 c i, n is nondecreasing in i Žobserve that PCn ) 0.. Set
cn s
ž
max
1FiF w log n x y1
/
< ci , n < k
ž /
1
log n
,
and, on Cn , let t n be the smallest t i, n , 0 F i F w log n x , such that < a nŽ t i, n .< s c n ;
write d n s a nŽ t n . and d n s a nŽ t n .; set tXn s t n q 1rw log n x and dXn s a nŽ tXn ..
Now we have
P Ž A n < Cn . F P n1r4
Ž 12.
sup
vyusc nrn 1r2
X
t nFuFvFt n
< an Ž v . y an Ž u. <
F Ž Ž 1 y « . c n log n .
1r2
0
Cn .
Write m n s nrw log n x q n1r2 Ž dXn y d n . and note that, on Cn , m n s nŽ FnŽ tXn . y
FnŽ t n ..; obviously < m n w log n x rn y 1 < F 2Žlog n. 2 ny1r2 ª 0 as n ª `. Now it is
530
J. H. J. EINMAHL
not hard to see that, on Cn , the process a
˜m n defined by
a
˜m nŽ s . s
1r2
ž / ½ ž
n
s
a n tn q
mn
w log n x
/
5
y Ž d n Ž 1 y s . q dXn s . ,
0 F s F 1,
is a uniform empirical process based on m n observations. Hence the righthand side of Ž12. can be written as
ž
P n1r4
mn
sup
syrsc nw log n x n
0FrFsF1
y1r2
Ž 13.
ž /
1r2
n
a˜m Ž s . y a˜m Ž r . 4
n
n
qd n w log n x Ž d n y dXn . ny1r2
F Ž Ž 1 y « . c n log n .
1r2
/
.
Now observe that
< n1r4 d n w log n x Ž d n y dXn . ny1r2 <
Ž c n log n .
1r2
F 2 c 1r2
n Ž log n .
3r2
ny1r4
2
F 2 Ž log n . ny1r4 ª 0 as n ª `.
Therefore, for large n, the expression in Ž13. is bounded from above by
P n1r4
Ž 14.
mn
ž /
1r2
sup
n
syrsc nw log n x n y1r2
0FrFsF1
F
žž
<a
˜m nŽ s . y a˜m nŽ r . <
1y
1
2
/
« c n log n
/
1r2
0
,
which by Fact 6 is less than or equal to
½ž
P n1r4
Ž 15.
mn
ž /
F
1r2
n
žž
1y
ã m n
1
2
ž
c n w log n x
/
/
n1r2
/ /5
1r2
« c n log n
n 1r 2 r Žlog n . 2
.
It is easy to check that, for large n, Fact 7 applies to the probability in Ž15..
This yields, with d s «r4, the following upper bound for the expression in
Ž15.:
Ž 16.
Ž 1 y nyŽ1y « r4.r2 .
n 1r 2 r Žlog n . 2
F exp
ž
yn « r8
Ž log n .
2
/
F
1
n2
.
PROOF OF BAHADUR]KIEFER THEOREM
531
We are now ready to complete the proof. Combining Ž12. ] Ž16., we have
P Ž A n < Cn . F 1rn2 Ž n large.. Set Dn s 5 a n 5 ) log n4 and note that Ž8. implies
that PDn F 1rn2 Ž n G 4.. Hence, for large n,
PA n F P Ž A n l Dnc . q PDnF Ž sup* P Ž A n < Cn . . q PDn
Ž 17.
F
1
n
2
q
1
n
2
s
2
n2
,
where sup* denotes the supremum over all Cn as defined before. Now, of
course, Ý`ns 3 PA n - ` because of Ž17.. This proves Ž11. and hence Ž2.. I
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