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ACTA MATHEMATICA VIETNAMICA
Volume 25, Number 2, 2000, pp. 161–179
161
RATE OF CONVERGENCE IN BOOTSTRAP
APPROXIMATIONS WITH RANDOM SAMPLE SIZE
NGUYEN VAN TOAN
Abstract. We give the convergence rates of the random sample size
bootstrap approximations
£¡ N of¢2the
¤ distribution of the sample mean. Using
2
n
the quantity αn =E
, where Nn is the random sample size, we
n −1
1
1
¡
1
1
¢
obtain a convergence rate O(n− 2 (log log n) 2 ) if αn =O n− 2 (log log n) 2 .
1. Introduction
Efron [2] introduced a very general resampling procedure, called the
bootstrap, for estimating the distributions of statistics based on independent observations. It may be explained briefly as follows: Let X1 , . . . , Xn
be independent identically distributed (i.i.d.) observations with distribution F ; let θ = θ(F ) be a parameter of interest, and let θn = θn (X1 , . . . , Xn )
be an estimator of θ. The bootstrap principle is to estimate the unknown
∗
∗
, . . . , Xnn
) and
distribution of θn by θˆn where θˆn is distributed as θn (Xn1
∗
∗
Xn1 , . . . , Xnn are i.i.d. from the empirical distribution of (X1 , . . . , Xn ).
In sufficiently regular cases, the bootstrap approximation to an unknown distribution function has been established as an improvement over
the simpler normal approximation (see [1], [3-4]). In the case where the
bootstrap sample size N is in itself a random variable, Mammen [7] has
considered bootstrap with a Poisson random sample size which is independent of the sample. Our work on this problem is limited to [8-11] and
[13-14]. In these references we find sufficient conditions for random sample size that random sample size bootstrap distribution can be used to
approximate the sampling distribution. The purpose of this paper is to
study the convergence rates to zero of the discrepancy between the actual
Received January 26, 1999; in revised form November 2, 1999.
1991 Mathematics Subject Classification. Primary: 62E20; Secondary: 62G05, 62G15.
Key words and phrases. Bootstrap, Berry-Esseen bound, Liapounov’s inequality, random sample size.
This research is supported by the National Basic Research Program in Natural Sciences,
Vietnam.
162
NGUYEN VAN TOAN
distribution of the sample mean and the random sample size bootstrap
approximations of it.
We derive our results by studying the rates of convergence of the bootstrap approximations with a random sample size. Before giving more
detail, it is necessary to introduce some notations. Let X, X1 , X2 , . . . be
i.i.d. random variables with finite variance. For any fixed natural number n, denote by Fn the empirical distribution of (X1 , . . . , Xn ) and by
∗
∗
Xn1
, Xn2
, . . . i.i.d. random variables with the distribution Fn . Let Nn be
a positive integer-valued random variable independent of X1 , X2 , . . . . Set
µ = EX is the expectation of X,
0 < σ 2 = DX is the variance of X,
¯ n = n−1
X
n
X
Xi ,
i=1
s2n = n−1
n
X
¯ n )2 ,
(Xi − X
i=1
¯ n∗Nn = n−1
X
Nn
X
∗
,
Xni
i=1
∗
¯N
X
= Nn−1
n
Nn
X
∗
Xni
.
i=1
In the case where Nn →p ∞ as n → ∞, where →p denotes convergence in probability,√the random sample size bootstrap distribution of
√
∗
¯∗ − X
¯ n ) and Nn (X
¯N
¯ n ) can be used to approximate the
Nn (X
−X
Nn
n
sn
√
√ ¯
n ¯
sampling distributions of n(Xn − µ) and
(Xn − µ), respectively (see
σ
Nn
→p 1 as n → ∞, the random sample size bootstrap
[9], [13]). When
n
√
√ ¡ ¯ ∗Nn Nn ¯ ¢
n ¡ ¯ ∗Nn Nn ¯ ¢
distribution of n Xn −
Xn and
Xn −
Xn can be used
n
sn
n
as given in [8] and [14].
Singh [12] has studied the uniform convergence
to zero of the discrep√ ¯
ancy between the actual distribution of n(X
−
µ) and the bootstrap
n
approximation
of it. The same convergence problem for the distribution of
√
n ¯
(Xn −µ) was also studied by Singh in [12]. In this paper we extend the
σ
results of Singh [12] to the random sample size bootstrap approximations.
163
RATE OF CONVERGENCE
Namely, we study the uniform and non-uniform convergence
√ ¯ rates to zero
of the discrepancy between the actual distribution of n(X
n − µ) and its
random sample size bootstrap approximations. We
√ also investigate the
n ¯
same convergence problem for the distribution of
(Xn − µ).
σ
2. Results
For simplicity we let
n
∆∗N
1
¯ £√
h√ ³
i¯
¤
Nn ¯ ´
¯
¯
∗
∗Nn
¯
¯
n Xn −
=
sup
X n < x ¯,
¯P n(Xn − µ) < x − P
n
−∞<x<+∞
n
∆∗N
2
¯ h √n
i
h √n ³
´
i¯
¯
∗
¯ n − µ) < x − P
¯ ∗Nn − Nn X
¯ n < x ¯¯,
=
sup
(X
X
¯P
n
σ
sn
n
−∞<x<+∞
δ1∗Nn (x)
¯ x ¯3 ´¯ £√
³
h√ ³
i¯
¤
Nn ¯ ´
¯ ¯ ¯
¯
∗Nn
∗
¯
¯
= 1 + ¯ ¯ ¯P n(Xn − µ) < x − P
n Xn −
X n < x ¯,
σ
n
∗Nn
δ2 (x)
¯ x ¯3 ´¯ h √n
³
´
i¯
i
h √n ³
¯ ¯ ¯
∗
¯ n < x ¯¯,
¯
¯ n∗Nn − Nn X
= 1 + ¯ ¯ ¯P
(Xn − µ) < x − P
X
σ
σ
sn
n
where P ∗ denotes conditional probability P (. . . |X1 , . . . , Xn ), and let
h³ N
´2 i
1
n
2
−1
, βn = max(n− 2 , αn ),
αn = E
n
1
1
γn = max(n− 2 (log log n) 2 , αn ).
Our main result are presented in the next two theorems. In Theorem
n
n
2.1, we study the convergence rates to zero of ∆∗N
, δ1∗Nn (x), ∆∗N
and
1
2
∗Nn
δ2 (x).
Theorem 2.1. Suppose that the sequence Nn of positive integer-valued
random variables satisfies the condition
(a) αn → 0 as n → ∞.
A) If EX 4 < ∞, then
(2.1)
n
∆∗N
= O(γn ) a.s.,
1
(2.2)
δ1∗Nn (x) = O(γn ) a.s..
164
NGUYEN VAN TOAN
B) If E|X 3 | < ∞, then
(2.3)
n
= O(βn ) a.s.,
∆∗N
2
(2.4)
δ2∗Nn (x) = O(βn ) a.s..
In particular, we have the following theorem which is analogous to
Theorem 2.1 of [10].
Theorem 2.1’.
A) If EX 4 < ∞ and if the sequence Nn of positive integer-valued random
variables satisfies the condition
1
1
(b) αn = O(n− 2 (log log n) 2 ), then
1
1
1
1
(2.1’)
n
∆∗N
= O(n− 2 (log log n) 2 ) a.s.,
1
(2.2’)
δ1∗Nn (x) = O(n− 2 (log log n) 2 ) a.s..
B) If E|X 3 | < ∞ and if the sequence Nn of positive integer-valued random
variables satisfies the condition
1
(c) αn = O(n− 2 ), then
1
(2.3’)
n
= O(n− 2 ) a.s.,
∆∗N
2
(2.4’)
δ2∗Nn (x) = O(n− 2 ) a.s..
1
Let
¯ £√
hp
i¯
¤
¯
¯
∗
∗
¯
¯
¯
=
sup
Nn (XNn − Xn ) < x ¯,
¯P n(Xn − µ) < x − P
−∞<x<+∞
¯ h √n
i
h √N
i¯
¯
n ¯∗
∗2
∗
¯
¯ n ) < x ¯¯,
RNn =
sup
(Xn − µ) < x − P
(XNn − X
¯P
σ
sn
−∞<x<+∞
¯ £√
p
¡
¢
¤
£
¤¯¯
3 ¯
∗
∗
∗1
¯
¯
¯
Nn (XNn − Xn ) < x ¯,
rNn (x) = 1 + |x| ¯P n(Xn − µ) < x − P
√
√
i
h N ³
i¯
¡
¢¯ h n
Nn ¯ ´
¯
n
∗
∗
∗2
3 ¯
¯
¯
XNn −
X n < x ¯,
(Xn − µ) < x − P
rNn (x) = 1 + |x| ¯P
σ
sn
n
∗1
RN
n
and
−1
1
1
1
ηn = E[Nn 2 ], ζn = max(n− 2 , ηn ), θn = max(n− 2 (log log n) 2 , ηn ).
165
RATE OF CONVERGENCE
∗1
∗1
∗2
The result on the same convergence problem for RN
, rN
(x), RN
and
n
n
n
∗2
rNn (x) is the following theorem.
Theorem 2.2.
A) If EX 4 < ∞, then
(2.5)
∗1
RN
= O(θn ) a.s.,
n
(2.6)
∗1
rN
(x) = O(θn ) a.s..
n
B) If E|X 3 | < ∞, then
∗2
RN
= O(ζn ) a.s.,
n
(2.7)
∗2
(x) = O(ζn ) a.s.
rN
n
(2.8)
In particular, we have the following theorem which is analogous to
Theorem 2.2 of [10].
Theorem 2.2’. Suppose that E[Nn−1 ] = O(n−1 ).
A) If EX 4 < ∞, then
p
(2.5’)
1
2
− 12
lim sup n (log log n)
n→∞
(2.6’)
1
∗1
RN
≤
n
D[(X − µ)2 ]
√
,
2σ 2 πe
1
∗1
rN
(x) = O(n− 2 (log log n) 2 ) a.s..
n
B) If E|X 3 | < ∞, then
1
(2.7’)
∗2
RN
= O(n− 2 ) a.s.,
n
(2.8’)
∗2
rN
(x) = O(n− 2 ) a.s..
n
1
3. Proofs
In this section we give the proofs of the above theorems. Towards the
end some remarks concerning these conclusions will be given.
For the simplicity of the proofs we state some facts and easily derived
results will be given.
166
NGUYEN VAN TOAN
Lemma 3.1. For every c > 0 we have
(3.1)
1
|xφ(cx)| = √
,
−∞<x<+∞
c 2πe
(3.2)
√
1
2 2
√ ,
sup
|(1 + |x| )xφ(cx)| ≤ √
+
−∞<x<+∞
c 2πe e2 c4 π
(3.3)
|Φ(x) − Φ(cx)| ≤ min{1, |x|φ(min(1, c)x)|1 − c|},
sup
3
where Φ(x) and φ(x) are the standard normal distribution function and
density, respectively.
By the proof of Theorem 1 of [12] we have
Lemma 3.2. Let X, X1 , X2 , . . . be i.i.d. random variables. If EX 4 < ∞,
then
(3.4)
1
1
lim sup n 2 (log log n)− 2 |s2n − σ 2 | =
n→∞
p
2D[(X − µ)2 ] a.s.,
(3.5)
¯ ³x´
³x´ ³ 1
1 ´ ³ x ´¯¯
¯
sup
−Φ
−
−
xφ
¯Φ
¯ = O(n−1 log log n) a.s.
sn
σ
sn
σ
σ
−∞<x<+∞
We will also need the following results of Kruglov and Korolev [6].
Lemma 3.3 [6, Lemma 6.3.1]. Let X, X1 , X2 , . . . be i.i.d. random variables with EX = 0. If E|X 3 | < ∞, then
√
(1 + |x|3 )|P (Sn < xσ n) − Φ(x)| ≤
where Sn =
n
P
i=1
cρ
√ ,
n
σ3
Xi , ρ = E|X 3 | and c is absolute constant, c ≤ 30.5378.
Theorem 3.1 [6, Theorem 6.2.1]. Let N, X, X1 , X2 , . . . be independent
random variables, where N takes values among the natural numbers and
X, X1 , X2 , . . . are identically distributed with EX = 0. If E|X 3 | < ∞,
then for all a ∈ (0, 1)
RATE OF CONVERGENCE
167
¯
¯ N
¯
³ x ´¯
Kρ
¯
¯
¯
¯
sup
+ Q(a)E ¯
− 1¯ ,
¯P (SN < x) − Φ √
¯≤ √
EN
−∞<x<+∞
σ EN
σ 3 a3 EN
where K is the universal appearing in the Berry-Esseen bound,
n 1
N
P
1
1 o
√ .
,√
SN =
Xi and Q(a) = max
,
1−a
2πea 1 + a
i=1
Theorem 3.2 [6, Theorem 6.3.1]. With N, X, X1 , X2 , . . . as in Theorem
3.1, if E|X 3 | < ∞ and EN 2 < ∞, then for all a ∈ (0, 1), b ∈ (1, ∞)
¯
³ x ´¯
¯
¯
(1 + |x|3 )¯P (SN < x) − Φ √
¯≤
σ EN
¯ (DN ) 34 o
n ¯ N
ρ
¯
¯
− 1¯ ,
+ K2 (a, b, 3) max E ¯
K1 (a, 3) √
,
3
3
EN
σ EN
(EN ) 2
where
3
K1 (a, 3) = c + 0.7655a− 2 , c ≤ 30.5378,
n w(b, 3)
v(3) o
b2 u(3)
1
√ ,
K2 (a, b, 3) = max
+
+
,
(b − 1)2
1−a
a+ a 1−a
¯
³ x ´¯
¯
¯
w(b, 3) =
sup
¯(1 + |x|3 )xφ √ ¯,
−∞<x<+∞
b
¯
n φ(x) o¯
¯
¯
v(3) =
sup
¯ < 1.2936,
¯(1 + |x|3 ) min 1,
|x|
−∞<x<+∞
¯
n r 2 Γ(2) o¯
¯
¯
3
u(3) =
sup
< 2.5958.
¯(1 + |x| ) min 1,
3 ¯
π
|x|
−∞<x<+∞
Proof of Theorem 2.1. We first note that
(3.6)
ENn ∼ n as n → ∞,
by the inequality
(3.7)
p
|ENn − n| ≤ E|Nn − n| ≤ E[(Nn − n)2 ] (by Liapounov’s inequality),
and condition (a).
168
NGUYEN VAN TOAN
Part A. It is worth noting that
n
∆∗N
≤ A1 + A2 + A3 + A4 + A5 ,
1
where
¯ £√
³ x ´¯
¤
¯
¯
¯
P
n(
X
−
µ)
<
x
−
Φ
¯,
¯
n
σ
−∞<x<+∞
¯ ³ x´
³ x ´¯
¯
¯
=
sup
−Φ
¯Φ t n
¯,
σ
σ
−∞<x<+∞
¯ ³ x´
³ x´
³1
1 ´ ³ x ´¯¯
¯
=
sup
− Φ tn
− tn
−
xφ tn ¯,
¯Φ t n
sn
σ
sn
σ
σ
−∞<x<+∞
¯ ³1
´
³
´¯
1
x ¯
¯
=
sup
−
xφ tn ¯,
¯t n
sn
σ
σ
−∞<x<+∞
¯ h√ ³
´¯
´
i
³
¯ ∗
¯ n∗Nn − Nn X
¯ n < x − Φ tn x ¯¯
=
sup
n X
¯P
n
sn
−∞<x<+∞
A1 =
A2
A3
A4
A5
sup
and
r
tn =
n
.
ENn
Firstly, due to the Berry-Esseen bound, we have
(3.8)
A1 ≤
Kρ
√ ,
σ3 n
where K is defined as in Theorem 3.1. By Theorem 3.1,
¯ N
¯
K ρˆn
¯ n
¯
(3.9)
A5 ≤ 3 √ 3
+ Q(a)E ¯
− 1¯,
EN
sn a ENn
n
n
1X
¯ n |3 .
where ρˆn =
|Xi − X
n i=1
The condition EX 4 < ∞ implies s2n → σ 2 and ρˆn → ρ a.s., and hence
K ρˆ
− 12
√ n
=
O(n
) a.s.,
s3n a3 ENn
(3.10)
because of (3.6). Further, by (3.1), (3.6) and Lemma 3.2 we see that under
the condition EX 4 < ∞ we get
p
(3.11)
1
2
lim sup n (log log n)
n→∞
(3.12)
− 12
A4 =
D[(X − µ)2 ]
√
a.s.,
2σ 2 πe
A3 = O(n−1 log log n) a.s..
RATE OF CONVERGENCE
Now, by Lemma 3.1 we see that
A2 ≤ √
(3.13)
|tn − 1|
.
2πe min(1, tn )
But we have
|tn − 1| ≤ |t2n − 1| =
(3.14)
|n − ENn |
.
ENn
Therefore, (2.1) follows from (3.6)-(3.14) and the inequality
(3.15)
E|Nn − ENn | ≤ 2E|Nn − n|.
To show (2.2) we note that
δ1∗Nn (x) ≤ B1 (x) + B2 (x) + B3 (x) + B4 (x),
where
¯ x ¯3 ´¯ £√
³
³ x ´¯
¤
¯
¯ ¯ ¯
B1 (x) = 1 + ¯ ¯ ¯P n(Xn − µ) < x − Φ
¯,
σ
σ
¯ x ¯3 ´¯ ³ x ´
³ x ´¯
³
¯
¯ ¯ ¯
−Φ
B2 (x) = 1 + ¯ ¯ ¯Φ tn
¯,
σ
σ
σ
¯ x ¯3 ´¯ ³ x ´
³ x ´¯
³
¯ ¯ ¯
¯
−Φ
B3 (x) = 1 + ¯ ¯ ¯Φ tn
¯,
σ
sn
σ
¯ x ¯3 ´¯ h√ ³
i
³ x ´¯
³
Nn ¯ ´
¯ ¯ ¯ ∗
¯
∗Nn
¯
n Xn −
Xn < x − Φ tn
B4 (x) = 1 + ¯ ¯ ¯P
¯.
σ
n
sn
Since EX 4 < ∞, Lemma 3.3 implies
(3.16)
B1 (x) ≤
cρ
√ .
σ3 n
We now apply Lemma 3.1 to get
¯ x ¯3 ´
n ¯x¯ ³
o
x´
¯ ¯
¯ ¯
B2 (x) ≤ 1 + ¯ ¯ min 1, ¯ ¯φ min(1, tn ) |1 − tn |
σ
σ
σ
√
h
i
2 2
1
≤ √
+ √
|1 − tn |
2πe min(1, tn ) e2 π[min(1, tn )]4
³
(3.17)
169
170
NGUYEN VAN TOAN
and
¯ x ¯3 ´
³
n ¯ x¯ ³
¡ σ ¢ x ´¯¯
σ ¯¯o
¯ ¯
¯
¯
(3.18) B3 (x) ≤ 1 + ¯ ¯ min 1, ¯tn ¯φ min 1,
t n ¯1 − ¯
σ
σ
sn
σ
sn

¯ x ¯3 
√
¯
1+¯ ¯
1
σ ¯¯
2 2

¯
σ
≤
¯ x ¯3  √
¡ σ¢+ √ h
¡ σ ¢i4  ¯1 − sn ¯.
2 π min 1,
1 + ¯tn ¯
2πe min 1,
e
σ
sn
sn
By (3.7) and (3.14) we get
(3.19)
B2 (x) = O(αn ).
Since s2n → σ 2 a.s. and because of (3.6), a repeated application of Lemma
3.2 enables us to write
1
1
B3 (x) = O(n− 2 (log log n) 2 ) a.s..
(3.20)
Theorem 3.2 now shows that
(3.21)
¯ x ¯3
¯ ¯
1+¯ ¯ n
ρˆn
√
B4 (x) ≤
K
(a,
3)
¯σ
¯
1
¯ x ¯3
s3n ENn
1+¯ ¯
sn
¯ (DN ) 34 oo
n ¯ N
¯ n
¯
n
+ K2 (a, b, 3) max E ¯
− 1¯,
3
ENn
(EN ) 2
for all a ∈ (0, 1), b ∈ (1, ∞). From (3.7), (3.10), (3.15) and the inequality
(3.22)
£
¤
DNn ≤ E (Nn − n)2 ,
it follows that
(3.23)
B4 (x) = O(βn ) a.s..
Combining (3.16),(3.19),(3.20) and (3.23) we obtain (2.2).
Part B. In order to prove (2.3) we note that
n
∆∗N
≤ A1 + A2 + A5 ,
2
171
RATE OF CONVERGENCE
where A1 , A2 and A5 are defined as in Part A, and if EX 4 < ∞ is replaced
by E|X 3 | < ∞. Then (3.8)-(3.10), (3.13)-(3.15) also hold. Hence (2.3)
follows from (3.7).
To derive (2.4) note that
δ2∗Nn (x) ≤ C1 (x) + C2 (x) + C3 (x),
where
¯
¯ ³ √n(X
´
¯ n − µ)
¯
¯
< x − Φ(x)¯,
C1 (x) = (1 + |x| )¯P
σ
C2 (x) = (1 + |x|3 ) |Φ(tn x) − Φ(x)| ,
µ
¶
√
Nn ¯
∗Nn
¯
¯ h n Xn − n Xn
¯
i
¯
3 ¯ ∗
C3 (x) = (1 + |x| )¯P
< x − Φ (tn x) ¯.
sn
3
Since E|X 3 | < ∞, by Lemma 3.3 we immediately have
C1 (x) ≤
cρ
√ ,
n
σ3
and by applying Theorem 3.2,
¯ (DN ) 43 o
n ¯ N
ρˆn
¯ n
¯
n
√
C3 (x) ≤ K1 (a, 3) 3
+ K2 (a, b, 3) max E ¯
− 1¯,
3
ENn
sn ENn
(ENn ) 2
for all a ∈ (0, 1), b ∈ (1, ∞). From (3.7), (3.10), (3.15) and (3.22), we
deduce that
C3 (x) = O(βn ) a.s..
Finally, applying Lemma 3.1, C2 (x) is bounded above by the right-hand
side of (3.17), and hence, by (3.7), it follows that
C2 (x) = O(αn ).
This with the previous estimates prove (2.4). The proof of Theorem 2.1
is now complete.
Proof of Theorem 2.1’.
Part A. If condition (b) holds, then
1
1
γn = O(n− 2 (log log n) 2 ).
172
NGUYEN VAN TOAN
Therefore, (2.1’) and (2.2’) immediately follow from (2.1) and (2.2), respectively.
Part B. In the case where condition (c) holds, we have
1
βn = O(n− 2 ),
and hence, by (2.3) and (2.4), we get (2.3’) and (2.4’), respectively.
Proof of Theorem 2.2.
Part A. It is easy to check that
∗1
RN
≤ A1 + A3 + A4 + A05 ,
n
where A1 , A3 and A4 are defined as in the proof of Theorem 2.1, and
A05
¯ £p
³ x ´¯
¡ ∗
¢
¤
¯
¯ ∗
¯
¯
=
sup
Nn XNn − Xn < x − Φ
¯P
¯.
sn
−∞<x<+∞
Because of EX 4 < ∞ one gets (3.8), (3.11) and (3.12). By the law of total
probability and the fact that Nn and X1 , X2 , . . . are independent, we get
A05
≤
≤
∞
X
m=1
∞
X
P [Nn = m]
¯ £√ ¡
³ ´¯
¢
¤
¯ ∗
∗
¯ n < x − Φ x ¯¯
¯m
−X
m X
¯P
sn
−∞<x<+∞
P [Nn = m]
K ρˆn
√
s3n m
m=1
=
sup
(by the Berry-Esseen bound)
K ρˆn
−1
E[Nn 2 ].
3
sn
Note that if EX 4 < ∞, s2n → σ 2 and ρˆn → ρ a.s., and hence
(3.24)
A05 = O(ηn ) a.s..
Combining theses results we obtain (2.5).
Using the argument leading to (2.2) and (3.24) and noting that
¯ x ¯3
¯ ¯
³ ´
1+¯ ¯
∗1
σ ¯ B x + B 0 (x),
rN
(x)
≤
B
(x)
+
¯
1
3
4
n
tn
¯ x ¯3
1 + ¯t n ¯
σ
RATE OF CONVERGENCE
173
where B1 (x) and B3 (x) are defined as in the proof of Theorem 2.1, and
¯ x ¯3 ´¯ £p
³
³ x ´¯
¡ ∗
¢
¤
¯
¯ ¯ ¯ ∗
0
¯
¯
B4 (x) = 1 + ¯ ¯ ¯P
Nn XNn − Xn < x − Φ
¯,
σ
sn
we obtain (2.6).
Part B. If E|X 3 | < ∞, s2n → σ 2 and ρˆn → ρ a.s. Therefore, (2.7) follows
from the inequality
∗2
RN
≤ A1 + A05 ,
n
and (3.8) and (3.24).
To obtain (2.8) notice that
∗2
rN
(x) ≤ B1 (σx) +
n
1 + |x|3
0
¯ s ¯3 B4 (sn x),
¯ n ¯
1 + ¯ x¯
σ
and arguing as in the proof of result (2.6). This completes the proof of
Theorem 2.2.
Proof of Theorem 2.2’. If E[Nn−1 ] = O(n−1 ), then by Liapounov’ inequal−1
1
ity we have E[Nn 2 ] = O(n− 2 ). Using the proof of Theorem 2.2 we obtain
Theorem 2.2’.
Remark 3.1. Theorem 2.1 can be stated more precisely as follows: Suppose that the sequence Nn of positive integer-valued random variables
satisfies the condition:
(a) αn → 0 as n → ∞.
A) If EX 4 < ∞, then for all a ∈ (0, 1), b ∈ (1, ∞)
p
D[(X − µ)2 ]
1
n
√
≤
lim sup γn−1 ∆∗N
+√
+ 2Q(a) a.s.,
1
2
2σ πe
n→∞
8πe
e 2 + 4 ³p
≤ 2√ 2
lim sup
D[(X − µ)2 ]
2e πσ
n→∞
√
2 2´
+
σ + 2K2 (a, b, 3) a.s..
2
B) If E|X 3 | < ∞, then for all a ∈ (0, 1), b ∈ (1, ∞)
3
γn−1 δ1∗Nn (x)
3
(a 2 + 1)Kρ
1
+ 2Q(a) a.s.,
n→∞
8πe
a σ3
3
¡
¢ρ
e2 + 4
−1 ∗Nn
lim sup βn δ2 (x) ≤ c + K1 (a, 3) 3 + √ + 2K2 (a, b, 3) a.s..
σ
n→∞
e2 8π
lim sup
n
βn−1 ∆∗N
2
≤
3
2
+√
174
NGUYEN VAN TOAN
Remark 3.2. Theorem 2.2 can be stated more precisely as follows:
A) If EX 4 < ∞, then
p
D[(X − µ)2 ] Kρ
∗1
√
+ 3 a.s.,
≤
lim sup θn−1 RN
n
σ
2σ 2 πe
n→∞
p
3
(e 2 + 4) D[(X − µ)2 ]
cρ
∗1
√
lim sup θn−1 rN
(x)
≤
+
a.s..
n
σ3
2σ 2 e2 π
n→∞
B) If E|X 3 | < ∞, then
2Kρ
a.s.,
σ3
n→∞
2cρ
∗2
lim sup ζn−1 rN
(x) ≤ 3 a.s..
n
σ
n→∞
∗2
lim sup ζn−1 RN
≤
n
Remark 3.3. We can describe a general model that allows formulae such
as the expansion of Edgeworth type. Let X, X1 , X2 , . . . be independent
¯n =
and identically distributed random variables with mean µ, and put X
n
X
1
Xi . We assume that the parameter of interest has form θ0 = g(µ),
n i=1
where g : IR → IR is a smooth function. The parameter estimate is
2
¯ n ), which is assumed to have asymptotic variance σ = h(µ) ,
θˆ = g(X
n
n
where h is a known, smooth function. It is supposed that σ 2 6= 0. The
¯ n )2 .
sample estimate of σ 2 is σ
ˆ 2 = h(X
θˆ − θ0
θˆ − θ0
We wish to estimate the distribution of either
or
· Note
σ
σ
ˆ
¯ n ), where
that both these statistics may be written in the form A(X
(3.25)
A(x) =
g(x) − g(µ)
h(µ)
A(x) =
g(x) − g(µ)
h(x)
in the former case and
(3.26)
∗
in the later. To construct the bootstrap distribution estimates, let Xn1
,...,
∗
Xnm denote a sample drawn randomly, with replacement, from X1 , . . . ,
RATE OF CONVERGENCE
175
m
X
¯∗ = 1
Xni ∗ for the resample mean. Define
Xn , and write X
nm
m i=1
∗
∗
¯ nm
θˆnm
= g(X
),
∗
∗
¯ nm
σ
ˆnm
= h(X
).
Both
(3.27)
(3.28)
∗
− θˆ
θˆ∗ − θˆ
θˆnm
ˆX
¯ ∗ ), where
and nm ∗
may be expressed as A(
nm
σ
ˆ
σ
ˆ
¯n)
g(x) − g(X
ˆ
A(x)
=
,
¯n)
h(X
¯n)
g(x) − g(X
ˆ
A(x)
=
h(x)
in the respective cases. The bootstrap estimate of the distribution of
∗
¯ n ) is given by the distribution of A(
ˆX
¯ nm
A(X
), conditional on X1 , . . . , Xn .
Let Nn be a positive integer-valued random variable independent of
X1 , X2 , . . . . The random sample size bootstrap estimate of the distribu¯ n ) is given by the distribution of A(
ˆX
¯ ∗ ), conditional on
tion of A(X
nNn
X1 , . . . , Xn .
Theorem 2.2 of Hall [5] gave an Edgeworth expansion of the distribution
¯ n ), which may be stated as follows. Assume that for an integer
of A(X
ν ≥ 1, the function A has ν + 2 continuous derivatives in a neighbourhood
of µ, that A(µ) = 0, that E|X|ν+2 < ∞, that the characteristic function
ϕ of X satisfies
(3.29)
lim sup|ϕ(t)| < 1,
|t|→∞
and that the asymptotic variance of
√
¯ n ) equals 1. Then
nA(X
ν
X
√
j
ν
¯ n ) ≤ x} = Φ(x) +
P { nA(X
n− 2 πj (x)φ(x) + o(n− 2 )
j=1
uniformly in x, where πj is a polynomial of degree of 3j − 1, odd for even
j and even for odd j, with coefficients depending on moments of X up to
order j + 2. The form of πj depends very much on the form of A. We
denote πj by pj if A is given by (3.25), and by qj if A is given by (3.26).
176
NGUYEN VAN TOAN
The bootstrap version of this result is described in Theorem 5.1 of Hall
[5]. Let Aˆ be defined by either (27) or (28), put πj = pj in the former case
and πj = qj in the latter, and let π
ˆj be the polynomial obtained from πj
on replacing population moments by the corresponding sample moments.
Arguing as in the proof of Theorem 5.1 [5], we can show the following
result:
Let λ > 0 be given, and let l = l(λ) denote a sufficiently large positive
number (whose value we do not specify). Assume that g and h each have
ν + 3 bounded derivatives in a neighbourhood of µ, that E|X|l < ∞, and
that Cramer’s condition (3.29) holds. Then there exists a constant C > 0
such that
(3.30)
ν
h°
i
X
°
√
∗
∗
− ν+1
− 2j
ˆ
¯
°
°
2
P P { mA(Xnm ) ≤ x} − Φ(x) −
ˆj (x)φ(x) > Cm
m π
j=1
= O(n−λ ),
©
P max
sup
1≤j≤ν −∞<x<∞
ª
(1 + |x|)−(3j−1) |ˆ
πj (x)| > C = O(n−λ ).
If we take λ > 1 in this result and apply the Borel-Cantelli lemma, we
may deduce from (3.30) that
ν
°
°
X
ν+1
° ∗ √ ˆ ¯∗
°
− 2j
(3.31) °P { mA(Xnm ) ≤ x}−Φ(x)−
m π
ˆj (x)φ(x)° = O(m− 2 )
j=1
with probability one. And from (3.31) we deduce that
P
∗
©p
ν
³ £
´
X
ª
£ −j ¤
− ν+1 ¤
∗
ˆ
¯
Nn A(XnNn ) ≤ x = Φ(x)+
E Nn 2 π
ˆj (x)φ(x)+O E Nn 2
j=1
with probability one.
Remark 3.4. If Nn is a Poisson variable with ENn = n, then
r
¯ N
¯ r 2
n+1
2n
¯ n
¯
−n n
DNn = n, E|Nn − ENn | = 2e
∼
, E¯
− 1¯ ∼
,
n!
π
ENn
πn
¯ ³ x´
³ x ´¯
¯
¯
−Φ
¯Φ t n
¯ = 0.
σ
σ
Therefore, if EX 4 < ∞, applying results (3.8), (3.9), (3.11)-(3.14), we
obtain
177
RATE OF CONVERGENCE
− 12
1
2
lim sup n (log log n)
n→∞
n
∆∗N
1
p
D[(X − µ)2 ]
√
≤
a.s.,
2σ 2 πe
and from (3.16)-(3.18) and (3.21) we have
lim sup n
1
2
¡
n→∞
log log n
¢− 12
√ ip
D[(X − µ)2 ]
1
2
2
δ1∗Nn (x) ≤ √
+ 2√
a.s.
2σ 2
2πe e π
h
However, if E|X 3 | < ∞, by the proof of Part B of Theorem 2.1 we only
have
1
2
n
lim sup n ∆∗N
2
n→∞
1
2
lim sup n (1 + |x|
n→∞
3
r
Kρ ³
1 ´
2
+ Q(a)
a.s. ∀a ∈ (0, 1),
≤ 3 1+ √
σ
π
a3
)δ2∗Nn (x)
ρ
≤ 3 (c + K1 (a, 3)) + K2 (a, b, 3)
σ
r
2
a.s.
π
for all a ∈ (0, 1), b ∈ (1, ∞).
Remark 3.5. Also, in the case where Nn is a Poisson random variable
with ENn = n, we have
∞
X
nm
1 − e−n
=
,
(m + 1)!
n
m=1
r
q
1 − e−n
− 21
−1
E[Nn ] ≤ E[Nn ] =
·
n
E[Nn−1 ]
=
e−n
(where Nn−1 = 0 if Nn = 0). Hence, from the proof of Theorem 2.2 it
follows that
A) If EX14 < ∞, then
p
D[(X1 − µ)2 ]
√
a.s.,
2σ 2 πe
n→∞
√ ip
h 1
D[(X − µ)2 ]
1
1
2
2
∗1
√
√
lim sup n 2 (log log n)− 2 rN
(x)
≤
+
a.s..
n
2σ 2
n→∞
2πe e2 π
1
2
lim sup n (log log n)
− 12
∗1
RN
n
≤
178
NGUYEN VAN TOAN
B) If E|X 3 | < ∞, then
2Kρ
a.s.,
σ3
n→∞
1
2cρ
∗2
lim sup n 2 rN
(x) ≤ 3 a.s..
n
σ
n→∞
1
∗2
≤
lim sup n 2 RN
n
Remark 3.6. If the positive integer-valued random variable Nn has the
uniform distribution function on [1, n], then
n
1 X 1
ln n
∼
,
n m=1 m
n
q
1
1
−1
E[Nn 2 ] ≤ E[Nn−1 ] ∼ n− 2 (ln n) 2 .
E[Nn−1 ] =
Hence, from the proof of Theorem 2.2 we get
A) If EX 4 < ∞, then
Kρ
a.s.,
σ3
n→∞
1
1
cρ
∗1
(x) ≤ 3 a.s..
lim sup n 2 (ln n)− 2 rN
n
σ
n→∞
1
1
∗1
lim sup n 2 (ln n)− 2 RN
≤
n
B) If E|X 3 | < ∞, then
Kρ
a.s.,
σ3
n→∞
1
1
cρ
∗2
(x)x ≤ 3 a.s..
lim sup n 2 (ln n)− 2 rN
n
σ
n→∞
1
1
∗2
lim sup n 2 (ln n)− 2 RN
≤
n
Acknowledgements
The author owes thanks to Professor T. M. Tuan and Professor T. H.
Thao for their guidance and continuous support. The author is grateful
to the anonymous referee who gave a carefully and thoroughly compiled
list of places where presentation could be improved.
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RATE OF CONVERGENCE
2.
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Department of Mathematics, College of Science,
Hue University