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An exponential nonuniform convergence rate for a class of normalized L-statistics
Journal of Statistical Planning and Inference 171 (2016) 135–146
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Journal of Statistical Planning and Inference journal homepage: www.elsevier.com/locate/jspi
An exponential nonuniform convergence rate for a class of normalized L-statistics✩ Jiang Hui a , Wang Weigang b,∗ , Yu Lei a a
Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, PR China
b
School of Statistics and Mathematics, Zhejiang Gongshang University, Hangzhou 310018, PR China
article
info
Article history: Received 1 May 2014 Received in revised form 7 October 2015 Accepted 18 October 2015 Available online 27 October 2015 MSC: 62F12 62M05 60F05 60H10
abstract In this paper, using the change of measure method (Petrov, 1975; Wang and Jing, 1999) and Rényi’s representation (Rényi, 1953), we derive the exponential nonuniform convergence rate in the central limit theorem for a class of normalized L-statistics. Moreover, our results are applied to Gini, Fortiana–Grané and Jackson statistics (Gail and Gastwirth, 1978; Fortiana and Grané, 2002; Jackson, 1967). © 2015 Elsevier B.V. All rights reserved.
Keywords: Change of measure Nonuniform convergence rate Normalized L-statistics Rényi’s representation
1. Introduction Let {Xn , X : n ≥ 1} be a sequence of nonnegative independent and identically distributed (i.i.d.) random variables, while X(1) ≤ X(2) ≤ · · · ≤ X(n) are the order statistics of X1 , X2 , . . . , Xn . In order to test whether X has the exponential distribution, Shorack and Wellner (1986) have proposed a class of normalized L-statistics with the form n
Tn =
wk,n X(k)
k=1 n
,
(1.1)
Xk
k=1
where {wk,n : n ≥ 1, k = 1, 2, . . . , n} is an array of coefficients. Different approaches may lead to statistics belonging to this class. For instance, Gail and Gastwirth (1978) suggested a test based on a Gini statistics which may be represented in the form (1.1), and recent tests based on maximum correlations by
✩ Research supported by Postdoctoral Science Foundation of China (2013M531341), the Fundamental Research Funds for the Central Universities (NS2015074), and by the Natural Science Foundation of Zhejiang Province (LY13A010003). ∗ Corresponding author. E-mail addresses: [email protected] (H. Jiang), [email protected] (W. Wang).
http://dx.doi.org/10.1016/j.jspi.2015.10.009 0378-3758/© 2015 Elsevier B.V. All rights reserved.
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H. Jiang et al. / Journal of Statistical Planning and Inference 171 (2016) 135–146
Fortiana and Grané (2002), Grané and Fortiana (2011) and Grané and Tchirina (2013) also belonged to this class. Furthermore, the large deviations for Tn have been investigated by Tchirina (2007) and Grané and Tchirina (2013) under exponentiality, while Jiang (2010a) studied the moderate deviations. It should be noted that the celebrated Rényi’s representation (Rényi, 1953) for exponential order statistics plays nan important role in getting the deviation properties of Tn . For the L-statistics of the form Ln = 1n k=1 wk,n X(k) , the necessary and sufficient conditions for the central limit theorem were given by Mason and Shorack (1992). Moreover, for convergence rate in the central limit theorem, i.e. the Berry–Esseen bound, one can refer to Bjerve (1977), Gribkova (1994), Helmers (1977), Helmers (1981) and Helmers et al. (1990) and the references therein. Under the assumption that X obeys exponential distribution with parameter 1, Jiang (2010b) has stated the following uniform Berry–Esseen bound
sup P
Bn
1
W (u)du
Tn −
x∈R
≤ x − Φ (x) ≤ Cn−1/2 ,
0
(1.2)
where C > 0 is an absolute constant, Φ (x) denotes the standard normal distribution function, and Bn , W (u) will be defined in Section 2 followed. Although the bound given by (1.2) is valid for all x ∈ R,it isonly useful for values of x near the center of the distribution. The reason is that, for x sufficiently large, the difference P Bn Tn −
1 0
W (u)du ≤ x − Φ (x) becomes
so close to 0 that the bound (1.2) is simply too crude to be of any use. One way to refine the Berry–Esseen bound is to reflect its dependence on x as well as n. In this article, by Rényi’s representation (Rényi, 1953) for exponential order statistics and the change of measure method (Petrov, 1975; Wang and Jing, 1999), we can establish an exponential nonuniform convergence rate of central limit theorem for Tn under suitable selection of its coefficients. Moreover, our results can be applied to Gini, Fortiana–Grané and Jackson statistics (Gail and Gastwirth, 1978; Fortiana and Grané, 2002; Jackson, 1967). Throughout this paper, C , C0 , C1 denote positive constants whose values can differ at different places. The rest of the paper is organized as follows. In Section 2, we state our main results. Then, the proofs are presented in Section 3. Finally, in Section 4, our results are applied to Gini, Fortiana–Grané and Jackson statistics. 2. Main results Corresponding to the coefficients {wk,n } in (1.1), define the score function wn (·): k−1
wn (u) = wk,n ,
≤u≤
n
k n
,
k = 1, 2, . . . , n.
Moreover, for 0 ≤ u < 1, k = 1, 2, . . . , n, set Wk,n =
n
1
n − k + 1 j =k
w j ,n ,
W ( u) =
1
1 1−u
w(v)dv u
n
and W n = 1n k=1 Wk,n . Now, we introduce the following assumptions. (Assumption A). X obeys exponential distribution with parameter 1. (Assumption B). Suppose that as n → +∞, the score function wn (u) convergences to w(u) at least pointwise on the open interval (0, 1). (Assumption C). Suppose that W n = O(1) and max1≤k≤n Wk,n = o n1/3 . (Assumption D). Suppose that n 1
n k =1
Wk,n − W n
2
1
W (u)du − 2
= 0
1
2
W (u)du
+ O(n−1/2 ).
0
Moreover, 1
W (u)du ̸= 2
0
2
1
W (u)du
.
0
(Assumption E). Suppose that
n
k=1
|Wk,n |3 = O(n).
Remark 2.1. (Assumption A) can be relaxed to the exponential distribution with parameter β > 0. As we can see in Section 3, (Assumption B)–(Assumption E) hold for Gini, Fortiana–Grané and Jackson statistics (Gail and Gastwirth, 1978; Fortiana and Grané, 2002; Jackson, 1967).
H. Jiang et al. / Journal of Statistical Planning and Inference 171 (2016) 135–146
137
Now, we present our main results as follows. Theorem 2.1. Let
∆n (x) = P Bn Tn − W n ≤ x − Φ (x) , where
1/2
Bn = 1
n
0
W2
(u)du −
1 0
W (u)du
2
.
Under (Assumption A)–(Assumption E), for n large enough and |x| ≤ C0 n1/6 , we have
∆n (x) ≤ C (1 + x )n 2
−1/2
2 x exp − .
(2.1)
2
By the mean value theorem, we can obtain the corollary below immediately. Corollary 2.1. Letting
ϑn = Bn W n −
1
¯ ∆n = P Bn Tn −
W (u)du , 0
1 0
W (u)du ≤ x − Φ (x) ,
under the conditions of Theorem 2.1
2 x (x − ϑn )2 ¯ n (x) ≤ C 1 + (x − ϑn )2 n−1/2 exp − ∆ + C ϑn exp − . 2
2
Moreover, Theorem 2.1 and Corollary 2.1 yield the following Cramér type moderate deviation results. Corollary 2.2. Let the sequence xn ≥ 0 satisfy xn
xn → ∞,
→ 0,
n1/6
n → ∞.
(i) Under the conditions of Theorem 2.1, we have
P Bn Tn − W n ≥ xn
1 − Φ (xn )
P Bn Tn − W n ≤ −xn
→ 1,
Φ (−xn )
→ 1,
n → ∞.
(ii) Assume the conditions of Corollary 2.1 hold and xn
xn → ∞,
n1/6
→ 0,
ϑn xn → 0,
n → ∞.
Then as n → ∞,
P Bn Tn −
1 0
W (u)du ≥ xn
1 − Φ (xn )
→ 1,
P Bn Tn −
1 0
W (u)du ≤ −xn
Φ (−xn )
→ 1.
3. Proof of Theorem 2.1 and Corollary 2.2 Using Rényi’s representation (Rényi, 1953) for exponential order statistics, we have n
Tn =
wk,n X(k)
k =1 n k=1
L
n L
= Xk
Wk,n Xk
k=1 n
,
(3.1)
Xk
k=1
where = denotes equal in distribution. To begin with, under (Assumption A)–(Assumption E), we establish some lemmas which are crucial in our analysis.
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H. Jiang et al. / Journal of Statistical Planning and Inference 171 (2016) 135–146 1
Lemma 3.1. Under (Assumption A)–(Assumption E), for n large enough and 2 ≤ x ≤ Cn 6 , we have P Bn Tn − W n > x = (1 − Φ (x)) (1 + Rn1 (x)) ,
(3.2)
where |Rn1 (x)| ≤ Cx(1 + x2 )n−1/2 . Proof. Set ak,n (x) = Wk,n − W n −
x Bn
,
k = 1, 2, . . . , n.
(3.3)
By (3.1), we can write that
P Bn Tn − W n > x = P
n
ak,n (x)Xk > 0 .
k=1
n
Since E
ak,n (x)Xk = − Bn x, then
k=1
n
P Bn Tn − W n > x = P
n
ηk >
k=1
where ηk = ak,n (x)Xk +
x . Bn
Letting h =
Bn x
max |hak,n (x)| ≤
n
1≤k≤n
Bn x , n
nx
,
Bn
(3.4)
by (Assumption C),
x2
max Wk,n − W n +
n
1≤k≤n
= o xn−1/6 ,
(3.5)
1
which implies that E exp(hηk ) always exists for 0 ≤ x ≤ Cn 6 and n large enough. Then, using the change of measure method (Petrov, 1975; Wang and Jing, 1999), we start the estimation of the right-side term in (3.4). Let ξ1 , . . . , ξn be independent random variables and each ξk has distribution function Vk (u) defined by Vk (u) =
E (exp{hηk }I (ηk ≤ u)) E exp{hηk }
.
(3.6)
Under the probability P, assume that P and Q are the measures induced by {η1 , . . . , ηn } and {ξ1 , . . . , ξn } respectively. From (3.6), the Random–Nikodym derivative of P with respect to Q can be written as dP dQ
(x1 , . . . , xn ) = exp −h
n
xk
k=1
n
E exp{hηk }.
k=1
Therefore,
P
n
ηk >
k=1
nx
=P
Bn
n
xk >
k=1
=
n
nx
Bn
E exp{hηk } EQ exp −h
k=1
=
n
n
xk
k=1
E exp{hηk }
k=1
exp{−hu}dP nx Bn
n k =1
where EQ denotes the expectation with respect to the probability Q. Define
n
(ξk − E ξk )
k=1
Gn (u) = P
Mn (h)
and
Rn (h) =
nx Bn
−
n
E ξk
k=1
Mn (h)
.
≤ u ,
Mn2 (h) =
I n k=1
∞
n k=1
Var (ξk )
xk > Bnx
n
ξk ≤ u ,
H. Jiang et al. / Journal of Statistical Planning and Inference 171 (2016) 135–146
From
∞ 0
exp{−ux}dΦ (u) = exp{x2 /2}(1 − Φ (x)), it follows that
P
139
n
nx
ηk >
=
Bn
k=1
n
E exp{hηk }
∞
exp −
0
k=1
hnx Bn
− hMn (h)u dGn (u + Rn (h))
= I0 (h) exp{−x2 } exp{x2 /2}(1 − Φ (x)) + I1 (h) + I2 (h) + I3 (h) ,
where I0 (h) = I1 (h) =
(3.7)
E exp{hηk } and
n
k=1
∞
exp{−hMn (h)u}d (Gn (u + Rn (h)) − Φ (u + Rn (h))) , 0
I2 (h) = I3 (h) =
∞
exp{−hMn (h)u}d (Φ (u + Rn (h)) − Φ (u)) ,
0 ∞
exp{−hMn (h)u} − exp{−xu}dΦ (u).
0
To prove this lemma, we have to estimate terms Ik (h) for k = 0, 1, 2, 3 step by step.
1
Lemma 3.2. Under (Assumption A)–(Assumption E), for n large enough and 2 ≤ x ≤ Cn 6 , we have I0 (h) = exp
x2 2
+ Rn2 (x) ,
1
where Rn2 (x) = O x3 n− 2 . Proof. By (3.5) and simple calculation, I0 (h) =
n
E exp{hηk }
k=1
=
n
2
n
hx
du
Bn
∞
exp (hak,n (x) − 1)u du
0
k=1
= ex
exp (hak,n (x) − 1)u +
0
k=1
= ex
∞
n 2
1
1 − hak,n (x) k=1
.
Therefore, we can write that
I0 (h) = exp x − 2
n
log 1 − hak,n (x) .
(3.8)
k=1
According to (3.5) and the following fact,
x + x2 /2 + x3 /3 ≤ − log(1 − x) ≤ x + x2 /2 + x3 , x + x2 /2 + x3 ≤ − log(1 − x) ≤ x + x2 /2 + x3 /6,
to estimate − h
n
n
k=1
if 0 ≤ x ≤ 2/3; if − 1 ≤ x ≤ 0,
log 1 − hak,n (x) , we have to study
ak,n (x),
h2
k =1
n
a2k,n (x),
h3
k=1
n
a3k,n (x).
k=1
In fact, by (3.3), n k=1
ak , n ( x ) =
n k=1
Wk,n − W n −
x Bn
=−
nx Bn
.
(3.9)
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H. Jiang et al. / Journal of Statistical Planning and Inference 171 (2016) 135–146
Now, using (Assumption D), we write that n
a2k,n
(x) =
k=1
n
Wk,n − W n
2
+
k=1
=
n2 B2n
+
nx2 B2n
x2
B2n
+ O(n1/2 ).
(3.10)
Moreover, according to (Assumption C) and (Assumption E), n
|ak,n (x)|3 ≤ 4
n 3 Wk,n − W n 3 + x
k=1
B3n
k=1
n 3 Wk,n |3 + 4n|W n 3 + nx ≤4 4 3
Bn
k=1
= O(n).
(3.11)
Therefore, (3.9)–(3.11) imply immediately that h
n
1
ak,n (x) = −x2 ,
2
k=1
h2
n
a2k,n (x) =
k=1
x2 2
+ O x2 n−1/2
and
n n h3 a3k,n (x) ≤ h3 |ak,n (x)|3 = O(x3 n−1/2 ). k=1 k=1 Together with above estimations, the proof of this lemma is completed.
In order to estimate I1 (h), I2 (h) and I3 (h), we consider the moments of ξk . Lemma 3.3. Letting ξk be defined by (3.6), we have E ξk =
ak,n (x) 1 − hak,n (x)
x
+
Bn
a2k,n (x) Var (ξk ) = 2 1 − hak,n (x)
,
and
24a3 (x) 4x3 k,n , E |ξk | ≤ + (1 − hak,n (x))3 B3n 3
1 ≤ k ≤ n.
Proof. By simple calculation, E ξk =
E (ηk exp{hηk }) E exp{hηk }
E ηk2 exp{hηk }
,
Var (ξk ) =
− (E ξk )2 ,
E exp{hηk }
and E |ηk |3 exp{hηk }
3
E |ξk | =
E exp{hηk }
.
Now, E (ηk exp{hηk }) , E ηk2 exp{hηk } and E |ηk |3 exp{hηk } can be estimated as follows.
E (ηk exp{hηk }) ∞
= 0
=
ak,n (x)u +
x Bn
exp (hak,n (x) − 1)u +
ak,n (x) x + (1 − hak,n (x))2 (1 − hak,n (x))Bn
exp
x
hx
Bn
2
n
du
H. Jiang et al. / Journal of Statistical Planning and Inference 171 (2016) 135–146
141
and E ηk2 exp{hηk }
∞
=
ak,n (x)u +
0
=
2
x Bn
2a2k,n (x)
(1 − hak,n (x))3
+
exp (hak,n (x) − 1)u + 2xak,n (x)
(1 − hak,n (x))2 Bn
+
hx
Bn
du
x2
(1 − hak,n (x))B2n
exp
x2 n
.
Moreover, E ηk3 exp(hηk )
∞ x3 hx |ak,n (x)|3 u3 + 3 exp (hak,n (x) − 1)u + du ≤4 Bn Bn 0 2 24a3 (x) 4x3 x k ,n ≤ . exp + (1 − hak,n (x))4 (1 − hak,n (x))B3n n Finally, from the calculations in (3.8), E exp(hηk ) =
1 1 − hak,n (x)
exp
x2
n
.
Hence, we can achieve the proof of this lemma.
1
Lemma 3.4. Suppose (Assumption A)–(Assumption E) hold. For n large enough and 2 ≤ x ≤ Cn 6 , we have
|I1 (h)| ≤ Cn−1/2 ,
|I2 (h)| ≤ Cx2 n−1/2 ,
|I3 (h)| ≤ Cn−1/2 .
Proof. Applying Berry–Esseen bound (Petrov, 1975) to the sequence {ξk , 1 ≤ k ≤ n},
|I1 (h)| ≤ supGn (u) − Φ (u) ≤ u∈R
n
C0
Mn3 (h) k=1
E |ξk − E ξk |3 .
Using Lemma 3.3
|I2 (h)| ≤ supΦ (v + Rn (h)) − Φ (v) ≤ v∈R
n ak,n (x) . Mn (h) k=1 1 − hak,n (x) C1
Moreover, 1 Bn Mn (h)
n2 − 1 max 1, 2 2 n Bn Mn (h) 2 n2 2 Mn (h) − B2n ≤ x n M (h) + n n Bn Bn 2 n2 ( h ) − M 2 n 2 Bn ≤ . 2
|I3 (h)| ≤
x
x
n B2n
Applying (3.5) and the following fact,
1 + x + x2 ≤
1 1−x 1
1 + x + x2 /2 ≤ to estimate n k=1
1−x
ak,n (x) k=1 1−hak,n (x) ,
n
ak,n (x),
h
≤ 1 + x + 2x2 ,
n k =1
if 0 ≤ x ≤ 1/2; (3.12)
≤ 1 + x + x2 ,
if − 1 ≤ x ≤ 0,
we have to study a2k,n (x),
h2
n k=1
a3k,n (x).
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H. Jiang et al. / Journal of Statistical Planning and Inference 171 (2016) 135–146
In fact, according to (3.10) and (3.11), we can write that h
n
a2k,n (x) =
k=1
nx Bn
+ O x + x3 n−1/2
and h2
n
|ak,n (x)|3 = O(x2 ).
k=1
Together with (3.9), we achieve for n large enough
n ak,n (x) ≤ Cx2 . k=1 1 − hak,n (x)
(3.13)
Then, we estimate Mn2 (h) =
n
Var (ξk ) =
a2k,n (x)
n
k=1
k=1
2 .
1 − hak,n (x)
In fact, from (3.5), (3.10), (3.11) and (3.12), it follows that Mn2
(h) =
n
a2k,n
(x) + 2h
k=1
=
n2
n
a3k,n
(x) + o h
k =1
n
|ak,n (x)|
3
k =1
+ O xn1/2 .
B2n
(3.14)
Finally, by (3.5), (3.11), Lemma 3.3 and (3.12), we can also obtain that for n large enough n
E |ξk − E ξk |3 ≤ C1 n.
(3.15)
k=1
Therefore, we can conclude the proof of this lemma by (3.13)–(3.15).
Now, we proceed to the proof of Lemma 3.1. Proof of Lemma 3.1. Using (3.7), Lemmas 3.2 and 3.4, we can write that P Bn Tn − W n > x
= exp −
x2 2
2 3 x + Rn2 (x) exp (1 − Φ (x)) + Ij (h) , 2
= (1 − Φ (x)) exp {Rn2 (x)} + exp − 3
where |Rn3 (x)| = exp{Rn2 (x)}
j = 1 Ij
j =1
x
2
2
Rn3 (x),
(h) ≤ C0 (1 + x2 )n−1/2 exp{Rn2 (x)}.
Since for x ≥ 2
3
√
4 2π x
exp −
x2
2
≤ 1 − Φ (x) ≤ √
1
2π x
exp −
it follows that P Bn Tn − W n > x = (1 − Φ (x)) (1 + Rn1 (x)) ,
where Rn1 (x) = −1 + exp {Rn2 (x)} + C1 xRn3 (x). Noting that Rn2 (x) = O(1), we have
|Rn1 (x)| ≤ Cx(1 + x2 )n−1/2 , and this lemma is proved.
x2 2
,
(3.16)
H. Jiang et al. / Journal of Statistical Planning and Inference 171 (2016) 135–146
143
Then, we give the proof of our main results. Proof of Theorem 2.1. Without of loss of generality, we assume that x > 0. For the case of 0 < x ≤ 2, (2.1) has been shown by Corollary 2.2 in Jiang (2010b). Therefore, it suffices to show that (2.1) also holds for x satisfying 2 ≤ x ≤ Cn1/6 . In fact, from Lemma 3.1 and (3.16), it follows that
P Bn Tn − W n > x − 1 − Φ (x) 2 x = (1 − Φ (x))|Rn1 (x)| ≤ C (1 + x2 )n−1/2 exp − , 2
which achieves the proof of Theorem 2.1.
Proof of Corollary 2.2. We only prove (i) of this corollary, while (ii) could be shown in the similar way. Using (3.16) and Theorem 2.1, we can write that
P B T − W ≥ x n n n n − 1 1 − Φ ( xn ) 2 x ∼ Cxn exp n P Bn Tn − W n ≥ xn − (1 − Φ (xn )) 2 ≤ C xn + x3n n−1/2 → 0, n → ∞. Similarly, one can see that
P Bn Tn − W n ≤ −xn
Φ (−xn )
→ 1,
n → ∞.
4. Applications to some statistics Now we apply our results to the following three statistics of type (1.1) to get their exponential nonuniform Berry–Esseen bounds. 4.1. Gini statistics (Gail and Gastwirth, 1978)
n k,j=1
Gn =
|Xk − Xj |
2(n − 1)
n
, Xk
k=1
where {Xk , k ≥ 1} be i.i.d random variables and exponential with parameter 1. Gail and Gastwirth (1978) show that Gn − is a normalized L-statistics with coefficients k−1 3 wk,n = 2 − . n−1 2 Moreover, Tchirina (2007) states that wn (u) defined in conditions (H0) converges to w(u) = 2u − W (u) = u −
1 . 2
Wk,n = and
1 0
Then we have
n − k + 1 j =k
W (u)du = 0,
Wn =
n
1
n 1
n k=1
1 0
wj,n =
n+k−2
W 2 (u)du =
Wk,n = 0,
n−1 1 . 12
n 1
n k=1
−
we can obtain immediately that
Wk2,n =
n k=1
|Wk,n |3 →
1
|W (u)|3 du, 0
and max1≤k≤n Wk,n ≤
1 . 2
1
Moreover, by simple calculation n 1
3 2
n→∞
W 2 (u)du + O (1/n) . 0
3 2
1 2
uniformly on [0, 1] and
144
H. Jiang et al. / Journal of Statistical Planning and Inference 171 (2016) 135–146
Therefore, (Assumption A)–(Assumption E) hold. Moreover,
ϑn = Bn W n −
1
W (u)du
= 0,
0
and then according to Corollary 2.1, we have the following result. Proposition 4.1. For n large enough and |x| ≤ C0 n1/6
2 √ 1 ≤ C (1 + x2 )n−1/2 exp − x . P 12n G − ≤ x − Φ ( x ) n 2 2 4.2. Fortiana–Grané statistics (Fortiana and Grané, 2002) Fortiana and Grané (2002) have represented a characterization-based 1 − FGn in the form of normalized L-statistics with coefficients
wk,n = 1 − log n − (n − k) log(n − k) + (n − k + 1) log(n − k + 1), which implies Wk,n = 1 + log 1 − k−n 1 . Moreover, we can see that wn (u) defined in condition (H0) converges pointwise to w(u) = 2 + log(1 − u) in (0, 1) and the corresponding function W (u) = 1 + log(1 − u) with 1 1 W (u)du = 0, W 2 (u)du = 1. 0
0
By simple calculation, as n → ∞ n 1
n k =1
1
3
|W (u)|3 du
|Wk,n | → 0
and max1≤k≤n Wk,n ≤ log n. Moreover,
n 1
1
W (u)du =
0 ≤ Wn −
n k=1
0
=
=
≤
≤
Wn,n n
1 n 1 n 1 n
+
− n−1 n
n −1 k=1
+
n −1 k=1
+
1
k n
log
u − k−n 1
k−1 n
1−u k n
k−1 n
+
n−1 1
n k=1
1−u
Wk,n −
W (u)du
0
du
du
1
n−1 n
1−u
k n
W (u)du
0
1 − k−n 1
k−1 n
n −1 1
n k =1
W (u)du
1
Wk,n −
du = O
log n n
.
Similarly, we can obtain that 1
W 2 (u)du −
0≤ 0
n 1
n k=1
Wk2,n ≤ 3
log2 n n
.
Therefore, (Assumption A)–(Assumption E) hold. Moreover, 1
ϑn = Bn W n −
W (u)du 0
=O
log n n1/2
.
Applying Corollary 2.1, we obtain the following result. Proposition 4.2. For n large enough and |x| ≤ C0 n1/6
2 √ P ≤ C (log n + x2 )n−1/2 exp − x . n 1 − FG ≤ x − Φ ( x ) n 2
(4.1)
H. Jiang et al. / Journal of Statistical Planning and Inference 171 (2016) 135–146
145
Remark 4.1. There are some mistakes in Theorem 3.2 of Jiang (2010b). By Corollary 2.2 in Jiang (2010b) and (4.1), we can see that
√ sup P n 1 − FGn ≤ x − Φ (x) ≤ Cn−1/2 log n. x∈R 4.3. Jackson statistics (Jackson, 1967) Jackson statistics (Jackson, 1967) is a L-statistics as follows: Jn =
1
n
n
−1 Xk
n k
nX(k)
n−i+1 k=1 i=1
k=1
.
Then, we have by Tchirina (2007) as 1 ≤ k ≤ n Wk,n = 1 −
n 11
n 1
−
n i =1 i
i=n−k+2
i
and
w(u) = 2 + log(1 − u),
W (u) = 1 + log(1 − u).
Since n 1
i i=n−k+2
=
n 1
i i=n−k+2
≤
n+1
−
1
−
n−k+2
1
n−k+2
u
1
du
n +1
n−k+2
− log 1 −
n+1
1
+
u
du
k−1 n+1
,
by (4.1), one can see that n 1
n 1
n k=1 i=n−k+2 i
=1+O
log n
n
.
Therefore, Wn =
n 1
n k=1
Wk,n = O
log n
n
n 1
,
n k=1
Wk2,n
−1=O
log2 n n
.
Moreover, by simple calculation n 1
n k=1
|Wk,n |3 →
1
|W (u)|3 du,
n→∞
0
and max1≤k≤n Wk,n ≤ log n. Hence, (Assumption A)–(Assumption E) hold. Moreover, 1
ϑn = Bn W n −
W (u)du 0
=O
log n n1/2
.
Applying Corollary 2.1, we obtain the following result. Proposition 4.3. For n large enough and |x| ≤ C0 n1/6
2 √ P ≤ C (log n + x2 )n−1/2 exp − x . nJ ≤ x − Φ ( x ) n 2 Acknowledgment The authors are very grateful to the anonymous referee for constructive comments which led to an improved presentation of this paper, especially the details in the proofs of Lemmas 3.2 and 3.4.
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References Bjerve, S., 1977. Error bounds for linear combinations of ordered statistics. Ann. Probab. 5, 357–369. Fortiana, G., Grané, A., 2002. A scale-free goodness-of-fit statistic for the exponential distribution based on maximum correlations. J. Statist. Plann. Inference 108, 85–97. Gail, M.H., Gastwirth, J.L., 1978. A scale-free goodness-of-fit statistic for the exponential distribution based on the Gini statistic. J. R. Stat. Soc. Ser. B 40, 350–357. Grané, A., Fortiana, G., 2011. A directional test of exponentiality based on maximum correlations. Metrika 73, 255–274. Grané, A., Tchirina, A.V., 2013. Asymptotic properties of a goodness-of-fit test based on maximum correlations. Statistics 47, 202–215. Gribkova, N.V., 1994. On analogues of the Berry–Esseen inequality for L-stastics. J. Math. Sci. 72, 2872–2879. Helmers, R., 1977. The order of the normal approximation for linear combinations of order statistics with smooth weight function. Ann. Probab. 5, 940–953. Helmers, R., 1981. A Berry–Esseen theorem for linear combinations of order statistics. Ann. Probab. 9, 342–347. Helmers, R., Janssen, P., Serfling, R., 1990. Berry–Esseen and bootstrap results for generalized L-statistics. Scand. J. Stat. 17, 65–77. Jackson, O.A.Y., 1967. An analysis of depatures from the exponential distribution. J. R. Stat. Soc. Ser. B 29, 540–549. Jiang, H., 2010a. Moderate deviations for a class of L-statistics. Acta Appl. Math. 109, 1165–1178. Jiang, H., 2010b. Berry–Esseen bound for a class of normalized L-statistics. Comm. Statist. Theory Methods 39, 3095–3107. Mason, D.M., Shorack, G.R., 1992. Necessary and sufficient conditions for asymptotic normality of L-statistics. Ann. Probab. 20, 1779–1804. Petrov, V.V., 1975. Sums of Independent Random Variables. Springer, Berlin. Rényi, A., 1953. On the theory of order statistics. Acta Math. Acad. Sci. Hungar. 4, 191–232. Shorack, G.R., Wellner, J.A., 1986. Empirical Process with Applications to Statistics. Wiley, New York. Tchirina, A.V., 2007. Asymptotic properties of exponentiality tests based on L-statistics. Acta Appl. Math. 97, 297–309. Wang, Q.Y., Jing, B.Y., 1999. An exponential nonuniform Berry–Esseen bound for self-normalized sums. Ann. Probab. 27, 2068–2088.
Contents lists available at ScienceDirect
Journal of Statistical Planning and Inference journal homepage: www.elsevier.com/locate/jspi
An exponential nonuniform convergence rate for a class of normalized L-statistics✩ Jiang Hui a , Wang Weigang b,∗ , Yu Lei a a
Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, PR China
b
School of Statistics and Mathematics, Zhejiang Gongshang University, Hangzhou 310018, PR China
article
info
Article history: Received 1 May 2014 Received in revised form 7 October 2015 Accepted 18 October 2015 Available online 27 October 2015 MSC: 62F12 62M05 60F05 60H10
abstract In this paper, using the change of measure method (Petrov, 1975; Wang and Jing, 1999) and Rényi’s representation (Rényi, 1953), we derive the exponential nonuniform convergence rate in the central limit theorem for a class of normalized L-statistics. Moreover, our results are applied to Gini, Fortiana–Grané and Jackson statistics (Gail and Gastwirth, 1978; Fortiana and Grané, 2002; Jackson, 1967). © 2015 Elsevier B.V. All rights reserved.
Keywords: Change of measure Nonuniform convergence rate Normalized L-statistics Rényi’s representation
1. Introduction Let {Xn , X : n ≥ 1} be a sequence of nonnegative independent and identically distributed (i.i.d.) random variables, while X(1) ≤ X(2) ≤ · · · ≤ X(n) are the order statistics of X1 , X2 , . . . , Xn . In order to test whether X has the exponential distribution, Shorack and Wellner (1986) have proposed a class of normalized L-statistics with the form n
Tn =
wk,n X(k)
k=1 n
,
(1.1)
Xk
k=1
where {wk,n : n ≥ 1, k = 1, 2, . . . , n} is an array of coefficients. Different approaches may lead to statistics belonging to this class. For instance, Gail and Gastwirth (1978) suggested a test based on a Gini statistics which may be represented in the form (1.1), and recent tests based on maximum correlations by
✩ Research supported by Postdoctoral Science Foundation of China (2013M531341), the Fundamental Research Funds for the Central Universities (NS2015074), and by the Natural Science Foundation of Zhejiang Province (LY13A010003). ∗ Corresponding author. E-mail addresses: [email protected] (H. Jiang), [email protected] (W. Wang).
http://dx.doi.org/10.1016/j.jspi.2015.10.009 0378-3758/© 2015 Elsevier B.V. All rights reserved.
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H. Jiang et al. / Journal of Statistical Planning and Inference 171 (2016) 135–146
Fortiana and Grané (2002), Grané and Fortiana (2011) and Grané and Tchirina (2013) also belonged to this class. Furthermore, the large deviations for Tn have been investigated by Tchirina (2007) and Grané and Tchirina (2013) under exponentiality, while Jiang (2010a) studied the moderate deviations. It should be noted that the celebrated Rényi’s representation (Rényi, 1953) for exponential order statistics plays nan important role in getting the deviation properties of Tn . For the L-statistics of the form Ln = 1n k=1 wk,n X(k) , the necessary and sufficient conditions for the central limit theorem were given by Mason and Shorack (1992). Moreover, for convergence rate in the central limit theorem, i.e. the Berry–Esseen bound, one can refer to Bjerve (1977), Gribkova (1994), Helmers (1977), Helmers (1981) and Helmers et al. (1990) and the references therein. Under the assumption that X obeys exponential distribution with parameter 1, Jiang (2010b) has stated the following uniform Berry–Esseen bound
sup P
Bn
1
W (u)du
Tn −
x∈R
≤ x − Φ (x) ≤ Cn−1/2 ,
0
(1.2)
where C > 0 is an absolute constant, Φ (x) denotes the standard normal distribution function, and Bn , W (u) will be defined in Section 2 followed. Although the bound given by (1.2) is valid for all x ∈ R,it isonly useful for values of x near the center of the distribution. The reason is that, for x sufficiently large, the difference P Bn Tn −
1 0
W (u)du ≤ x − Φ (x) becomes
so close to 0 that the bound (1.2) is simply too crude to be of any use. One way to refine the Berry–Esseen bound is to reflect its dependence on x as well as n. In this article, by Rényi’s representation (Rényi, 1953) for exponential order statistics and the change of measure method (Petrov, 1975; Wang and Jing, 1999), we can establish an exponential nonuniform convergence rate of central limit theorem for Tn under suitable selection of its coefficients. Moreover, our results can be applied to Gini, Fortiana–Grané and Jackson statistics (Gail and Gastwirth, 1978; Fortiana and Grané, 2002; Jackson, 1967). Throughout this paper, C , C0 , C1 denote positive constants whose values can differ at different places. The rest of the paper is organized as follows. In Section 2, we state our main results. Then, the proofs are presented in Section 3. Finally, in Section 4, our results are applied to Gini, Fortiana–Grané and Jackson statistics. 2. Main results Corresponding to the coefficients {wk,n } in (1.1), define the score function wn (·): k−1
wn (u) = wk,n ,
≤u≤
n
k n
,
k = 1, 2, . . . , n.
Moreover, for 0 ≤ u < 1, k = 1, 2, . . . , n, set Wk,n =
n
1
n − k + 1 j =k
w j ,n ,
W ( u) =
1
1 1−u
w(v)dv u
n
and W n = 1n k=1 Wk,n . Now, we introduce the following assumptions. (Assumption A). X obeys exponential distribution with parameter 1. (Assumption B). Suppose that as n → +∞, the score function wn (u) convergences to w(u) at least pointwise on the open interval (0, 1). (Assumption C). Suppose that W n = O(1) and max1≤k≤n Wk,n = o n1/3 . (Assumption D). Suppose that n 1
n k =1
Wk,n − W n
2
1
W (u)du − 2
= 0
1
2
W (u)du
+ O(n−1/2 ).
0
Moreover, 1
W (u)du ̸= 2
0
2
1
W (u)du
.
0
(Assumption E). Suppose that
n
k=1
|Wk,n |3 = O(n).
Remark 2.1. (Assumption A) can be relaxed to the exponential distribution with parameter β > 0. As we can see in Section 3, (Assumption B)–(Assumption E) hold for Gini, Fortiana–Grané and Jackson statistics (Gail and Gastwirth, 1978; Fortiana and Grané, 2002; Jackson, 1967).
H. Jiang et al. / Journal of Statistical Planning and Inference 171 (2016) 135–146
137
Now, we present our main results as follows. Theorem 2.1. Let
∆n (x) = P Bn Tn − W n ≤ x − Φ (x) , where
1/2
Bn = 1
n
0
W2
(u)du −
1 0
W (u)du
2
.
Under (Assumption A)–(Assumption E), for n large enough and |x| ≤ C0 n1/6 , we have
∆n (x) ≤ C (1 + x )n 2
−1/2
2 x exp − .
(2.1)
2
By the mean value theorem, we can obtain the corollary below immediately. Corollary 2.1. Letting
ϑn = Bn W n −
1
¯ ∆n = P Bn Tn −
W (u)du , 0
1 0
W (u)du ≤ x − Φ (x) ,
under the conditions of Theorem 2.1
2 x (x − ϑn )2 ¯ n (x) ≤ C 1 + (x − ϑn )2 n−1/2 exp − ∆ + C ϑn exp − . 2
2
Moreover, Theorem 2.1 and Corollary 2.1 yield the following Cramér type moderate deviation results. Corollary 2.2. Let the sequence xn ≥ 0 satisfy xn
xn → ∞,
→ 0,
n1/6
n → ∞.
(i) Under the conditions of Theorem 2.1, we have
P Bn Tn − W n ≥ xn
1 − Φ (xn )
P Bn Tn − W n ≤ −xn
→ 1,
Φ (−xn )
→ 1,
n → ∞.
(ii) Assume the conditions of Corollary 2.1 hold and xn
xn → ∞,
n1/6
→ 0,
ϑn xn → 0,
n → ∞.
Then as n → ∞,
P Bn Tn −
1 0
W (u)du ≥ xn
1 − Φ (xn )
→ 1,
P Bn Tn −
1 0
W (u)du ≤ −xn
Φ (−xn )
→ 1.
3. Proof of Theorem 2.1 and Corollary 2.2 Using Rényi’s representation (Rényi, 1953) for exponential order statistics, we have n
Tn =
wk,n X(k)
k =1 n k=1
L
n L
= Xk
Wk,n Xk
k=1 n
,
(3.1)
Xk
k=1
where = denotes equal in distribution. To begin with, under (Assumption A)–(Assumption E), we establish some lemmas which are crucial in our analysis.
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H. Jiang et al. / Journal of Statistical Planning and Inference 171 (2016) 135–146 1
Lemma 3.1. Under (Assumption A)–(Assumption E), for n large enough and 2 ≤ x ≤ Cn 6 , we have P Bn Tn − W n > x = (1 − Φ (x)) (1 + Rn1 (x)) ,
(3.2)
where |Rn1 (x)| ≤ Cx(1 + x2 )n−1/2 . Proof. Set ak,n (x) = Wk,n − W n −
x Bn
,
k = 1, 2, . . . , n.
(3.3)
By (3.1), we can write that
P Bn Tn − W n > x = P
n
ak,n (x)Xk > 0 .
k=1
n
Since E
ak,n (x)Xk = − Bn x, then
k=1
n
P Bn Tn − W n > x = P
n
ηk >
k=1
where ηk = ak,n (x)Xk +
x . Bn
Letting h =
Bn x
max |hak,n (x)| ≤
n
1≤k≤n
Bn x , n
nx
,
Bn
(3.4)
by (Assumption C),
x2
max Wk,n − W n +
n
1≤k≤n
= o xn−1/6 ,
(3.5)
1
which implies that E exp(hηk ) always exists for 0 ≤ x ≤ Cn 6 and n large enough. Then, using the change of measure method (Petrov, 1975; Wang and Jing, 1999), we start the estimation of the right-side term in (3.4). Let ξ1 , . . . , ξn be independent random variables and each ξk has distribution function Vk (u) defined by Vk (u) =
E (exp{hηk }I (ηk ≤ u)) E exp{hηk }
.
(3.6)
Under the probability P, assume that P and Q are the measures induced by {η1 , . . . , ηn } and {ξ1 , . . . , ξn } respectively. From (3.6), the Random–Nikodym derivative of P with respect to Q can be written as dP dQ
(x1 , . . . , xn ) = exp −h
n
xk
k=1
n
E exp{hηk }.
k=1
Therefore,
P
n
ηk >
k=1
nx
=P
Bn
n
xk >
k=1
=
n
nx
Bn
E exp{hηk } EQ exp −h
k=1
=
n
n
xk
k=1
E exp{hηk }
k=1
exp{−hu}dP nx Bn
n k =1
where EQ denotes the expectation with respect to the probability Q. Define
n
(ξk − E ξk )
k=1
Gn (u) = P
Mn (h)
and
Rn (h) =
nx Bn
−
n
E ξk
k=1
Mn (h)
.
≤ u ,
Mn2 (h) =
I n k=1
∞
n k=1
Var (ξk )
xk > Bnx
n
ξk ≤ u ,
H. Jiang et al. / Journal of Statistical Planning and Inference 171 (2016) 135–146
From
∞ 0
exp{−ux}dΦ (u) = exp{x2 /2}(1 − Φ (x)), it follows that
P
139
n
nx
ηk >
=
Bn
k=1
n
E exp{hηk }
∞
exp −
0
k=1
hnx Bn
− hMn (h)u dGn (u + Rn (h))
= I0 (h) exp{−x2 } exp{x2 /2}(1 − Φ (x)) + I1 (h) + I2 (h) + I3 (h) ,
where I0 (h) = I1 (h) =
(3.7)
E exp{hηk } and
n
k=1
∞
exp{−hMn (h)u}d (Gn (u + Rn (h)) − Φ (u + Rn (h))) , 0
I2 (h) = I3 (h) =
∞
exp{−hMn (h)u}d (Φ (u + Rn (h)) − Φ (u)) ,
0 ∞
exp{−hMn (h)u} − exp{−xu}dΦ (u).
0
To prove this lemma, we have to estimate terms Ik (h) for k = 0, 1, 2, 3 step by step.
1
Lemma 3.2. Under (Assumption A)–(Assumption E), for n large enough and 2 ≤ x ≤ Cn 6 , we have I0 (h) = exp
x2 2
+ Rn2 (x) ,
1
where Rn2 (x) = O x3 n− 2 . Proof. By (3.5) and simple calculation, I0 (h) =
n
E exp{hηk }
k=1
=
n
2
n
hx
du
Bn
∞
exp (hak,n (x) − 1)u du
0
k=1
= ex
exp (hak,n (x) − 1)u +
0
k=1
= ex
∞
n 2
1
1 − hak,n (x) k=1
.
Therefore, we can write that
I0 (h) = exp x − 2
n
log 1 − hak,n (x) .
(3.8)
k=1
According to (3.5) and the following fact,
x + x2 /2 + x3 /3 ≤ − log(1 − x) ≤ x + x2 /2 + x3 , x + x2 /2 + x3 ≤ − log(1 − x) ≤ x + x2 /2 + x3 /6,
to estimate − h
n
n
k=1
if 0 ≤ x ≤ 2/3; if − 1 ≤ x ≤ 0,
log 1 − hak,n (x) , we have to study
ak,n (x),
h2
k =1
n
a2k,n (x),
h3
k=1
n
a3k,n (x).
k=1
In fact, by (3.3), n k=1
ak , n ( x ) =
n k=1
Wk,n − W n −
x Bn
=−
nx Bn
.
(3.9)
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H. Jiang et al. / Journal of Statistical Planning and Inference 171 (2016) 135–146
Now, using (Assumption D), we write that n
a2k,n
(x) =
k=1
n
Wk,n − W n
2
+
k=1
=
n2 B2n
+
nx2 B2n
x2
B2n
+ O(n1/2 ).
(3.10)
Moreover, according to (Assumption C) and (Assumption E), n
|ak,n (x)|3 ≤ 4
n 3 Wk,n − W n 3 + x
k=1
B3n
k=1
n 3 Wk,n |3 + 4n|W n 3 + nx ≤4 4 3
Bn
k=1
= O(n).
(3.11)
Therefore, (3.9)–(3.11) imply immediately that h
n
1
ak,n (x) = −x2 ,
2
k=1
h2
n
a2k,n (x) =
k=1
x2 2
+ O x2 n−1/2
and
n n h3 a3k,n (x) ≤ h3 |ak,n (x)|3 = O(x3 n−1/2 ). k=1 k=1 Together with above estimations, the proof of this lemma is completed.
In order to estimate I1 (h), I2 (h) and I3 (h), we consider the moments of ξk . Lemma 3.3. Letting ξk be defined by (3.6), we have E ξk =
ak,n (x) 1 − hak,n (x)
x
+
Bn
a2k,n (x) Var (ξk ) = 2 1 − hak,n (x)
,
and
24a3 (x) 4x3 k,n , E |ξk | ≤ + (1 − hak,n (x))3 B3n 3
1 ≤ k ≤ n.
Proof. By simple calculation, E ξk =
E (ηk exp{hηk }) E exp{hηk }
E ηk2 exp{hηk }
,
Var (ξk ) =
− (E ξk )2 ,
E exp{hηk }
and E |ηk |3 exp{hηk }
3
E |ξk | =
E exp{hηk }
.
Now, E (ηk exp{hηk }) , E ηk2 exp{hηk } and E |ηk |3 exp{hηk } can be estimated as follows.
E (ηk exp{hηk }) ∞
= 0
=
ak,n (x)u +
x Bn
exp (hak,n (x) − 1)u +
ak,n (x) x + (1 − hak,n (x))2 (1 − hak,n (x))Bn
exp
x
hx
Bn
2
n
du
H. Jiang et al. / Journal of Statistical Planning and Inference 171 (2016) 135–146
141
and E ηk2 exp{hηk }
∞
=
ak,n (x)u +
0
=
2
x Bn
2a2k,n (x)
(1 − hak,n (x))3
+
exp (hak,n (x) − 1)u + 2xak,n (x)
(1 − hak,n (x))2 Bn
+
hx
Bn
du
x2
(1 − hak,n (x))B2n
exp
x2 n
.
Moreover, E ηk3 exp(hηk )
∞ x3 hx |ak,n (x)|3 u3 + 3 exp (hak,n (x) − 1)u + du ≤4 Bn Bn 0 2 24a3 (x) 4x3 x k ,n ≤ . exp + (1 − hak,n (x))4 (1 − hak,n (x))B3n n Finally, from the calculations in (3.8), E exp(hηk ) =
1 1 − hak,n (x)
exp
x2
n
.
Hence, we can achieve the proof of this lemma.
1
Lemma 3.4. Suppose (Assumption A)–(Assumption E) hold. For n large enough and 2 ≤ x ≤ Cn 6 , we have
|I1 (h)| ≤ Cn−1/2 ,
|I2 (h)| ≤ Cx2 n−1/2 ,
|I3 (h)| ≤ Cn−1/2 .
Proof. Applying Berry–Esseen bound (Petrov, 1975) to the sequence {ξk , 1 ≤ k ≤ n},
|I1 (h)| ≤ supGn (u) − Φ (u) ≤ u∈R
n
C0
Mn3 (h) k=1
E |ξk − E ξk |3 .
Using Lemma 3.3
|I2 (h)| ≤ supΦ (v + Rn (h)) − Φ (v) ≤ v∈R
n ak,n (x) . Mn (h) k=1 1 − hak,n (x) C1
Moreover, 1 Bn Mn (h)
n2 − 1 max 1, 2 2 n Bn Mn (h) 2 n2 2 Mn (h) − B2n ≤ x n M (h) + n n Bn Bn 2 n2 ( h ) − M 2 n 2 Bn ≤ . 2
|I3 (h)| ≤
x
x
n B2n
Applying (3.5) and the following fact,
1 + x + x2 ≤
1 1−x 1
1 + x + x2 /2 ≤ to estimate n k=1
1−x
ak,n (x) k=1 1−hak,n (x) ,
n
ak,n (x),
h
≤ 1 + x + 2x2 ,
n k =1
if 0 ≤ x ≤ 1/2; (3.12)
≤ 1 + x + x2 ,
if − 1 ≤ x ≤ 0,
we have to study a2k,n (x),
h2
n k=1
a3k,n (x).
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H. Jiang et al. / Journal of Statistical Planning and Inference 171 (2016) 135–146
In fact, according to (3.10) and (3.11), we can write that h
n
a2k,n (x) =
k=1
nx Bn
+ O x + x3 n−1/2
and h2
n
|ak,n (x)|3 = O(x2 ).
k=1
Together with (3.9), we achieve for n large enough
n ak,n (x) ≤ Cx2 . k=1 1 − hak,n (x)
(3.13)
Then, we estimate Mn2 (h) =
n
Var (ξk ) =
a2k,n (x)
n
k=1
k=1
2 .
1 − hak,n (x)
In fact, from (3.5), (3.10), (3.11) and (3.12), it follows that Mn2
(h) =
n
a2k,n
(x) + 2h
k=1
=
n2
n
a3k,n
(x) + o h
k =1
n
|ak,n (x)|
3
k =1
+ O xn1/2 .
B2n
(3.14)
Finally, by (3.5), (3.11), Lemma 3.3 and (3.12), we can also obtain that for n large enough n
E |ξk − E ξk |3 ≤ C1 n.
(3.15)
k=1
Therefore, we can conclude the proof of this lemma by (3.13)–(3.15).
Now, we proceed to the proof of Lemma 3.1. Proof of Lemma 3.1. Using (3.7), Lemmas 3.2 and 3.4, we can write that P Bn Tn − W n > x
= exp −
x2 2
2 3 x + Rn2 (x) exp (1 − Φ (x)) + Ij (h) , 2
= (1 − Φ (x)) exp {Rn2 (x)} + exp − 3
where |Rn3 (x)| = exp{Rn2 (x)}
j = 1 Ij
j =1
x
2
2
Rn3 (x),
(h) ≤ C0 (1 + x2 )n−1/2 exp{Rn2 (x)}.
Since for x ≥ 2
3
√
4 2π x
exp −
x2
2
≤ 1 − Φ (x) ≤ √
1
2π x
exp −
it follows that P Bn Tn − W n > x = (1 − Φ (x)) (1 + Rn1 (x)) ,
where Rn1 (x) = −1 + exp {Rn2 (x)} + C1 xRn3 (x). Noting that Rn2 (x) = O(1), we have
|Rn1 (x)| ≤ Cx(1 + x2 )n−1/2 , and this lemma is proved.
x2 2
,
(3.16)
H. Jiang et al. / Journal of Statistical Planning and Inference 171 (2016) 135–146
143
Then, we give the proof of our main results. Proof of Theorem 2.1. Without of loss of generality, we assume that x > 0. For the case of 0 < x ≤ 2, (2.1) has been shown by Corollary 2.2 in Jiang (2010b). Therefore, it suffices to show that (2.1) also holds for x satisfying 2 ≤ x ≤ Cn1/6 . In fact, from Lemma 3.1 and (3.16), it follows that
P Bn Tn − W n > x − 1 − Φ (x) 2 x = (1 − Φ (x))|Rn1 (x)| ≤ C (1 + x2 )n−1/2 exp − , 2
which achieves the proof of Theorem 2.1.
Proof of Corollary 2.2. We only prove (i) of this corollary, while (ii) could be shown in the similar way. Using (3.16) and Theorem 2.1, we can write that
P B T − W ≥ x n n n n − 1 1 − Φ ( xn ) 2 x ∼ Cxn exp n P Bn Tn − W n ≥ xn − (1 − Φ (xn )) 2 ≤ C xn + x3n n−1/2 → 0, n → ∞. Similarly, one can see that
P Bn Tn − W n ≤ −xn
Φ (−xn )
→ 1,
n → ∞.
4. Applications to some statistics Now we apply our results to the following three statistics of type (1.1) to get their exponential nonuniform Berry–Esseen bounds. 4.1. Gini statistics (Gail and Gastwirth, 1978)
n k,j=1
Gn =
|Xk − Xj |
2(n − 1)
n
, Xk
k=1
where {Xk , k ≥ 1} be i.i.d random variables and exponential with parameter 1. Gail and Gastwirth (1978) show that Gn − is a normalized L-statistics with coefficients k−1 3 wk,n = 2 − . n−1 2 Moreover, Tchirina (2007) states that wn (u) defined in conditions (H0) converges to w(u) = 2u − W (u) = u −
1 . 2
Wk,n = and
1 0
Then we have
n − k + 1 j =k
W (u)du = 0,
Wn =
n
1
n 1
n k=1
1 0
wj,n =
n+k−2
W 2 (u)du =
Wk,n = 0,
n−1 1 . 12
n 1
n k=1
−
we can obtain immediately that
Wk2,n =
n k=1
|Wk,n |3 →
1
|W (u)|3 du, 0
and max1≤k≤n Wk,n ≤
1 . 2
1
Moreover, by simple calculation n 1
3 2
n→∞
W 2 (u)du + O (1/n) . 0
3 2
1 2
uniformly on [0, 1] and
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H. Jiang et al. / Journal of Statistical Planning and Inference 171 (2016) 135–146
Therefore, (Assumption A)–(Assumption E) hold. Moreover,
ϑn = Bn W n −
1
W (u)du
= 0,
0
and then according to Corollary 2.1, we have the following result. Proposition 4.1. For n large enough and |x| ≤ C0 n1/6
2 √ 1 ≤ C (1 + x2 )n−1/2 exp − x . P 12n G − ≤ x − Φ ( x ) n 2 2 4.2. Fortiana–Grané statistics (Fortiana and Grané, 2002) Fortiana and Grané (2002) have represented a characterization-based 1 − FGn in the form of normalized L-statistics with coefficients
wk,n = 1 − log n − (n − k) log(n − k) + (n − k + 1) log(n − k + 1), which implies Wk,n = 1 + log 1 − k−n 1 . Moreover, we can see that wn (u) defined in condition (H0) converges pointwise to w(u) = 2 + log(1 − u) in (0, 1) and the corresponding function W (u) = 1 + log(1 − u) with 1 1 W (u)du = 0, W 2 (u)du = 1. 0
0
By simple calculation, as n → ∞ n 1
n k =1
1
3
|W (u)|3 du
|Wk,n | → 0
and max1≤k≤n Wk,n ≤ log n. Moreover,
n 1
1
W (u)du =
0 ≤ Wn −
n k=1
0
=
=
≤
≤
Wn,n n
1 n 1 n 1 n
+
− n−1 n
n −1 k=1
+
n −1 k=1
+
1
k n
log
u − k−n 1
k−1 n
1−u k n
k−1 n
+
n−1 1
n k=1
1−u
Wk,n −
W (u)du
0
du
du
1
n−1 n
1−u
k n
W (u)du
0
1 − k−n 1
k−1 n
n −1 1
n k =1
W (u)du
1
Wk,n −
du = O
log n n
.
Similarly, we can obtain that 1
W 2 (u)du −
0≤ 0
n 1
n k=1
Wk2,n ≤ 3
log2 n n
.
Therefore, (Assumption A)–(Assumption E) hold. Moreover, 1
ϑn = Bn W n −
W (u)du 0
=O
log n n1/2
.
Applying Corollary 2.1, we obtain the following result. Proposition 4.2. For n large enough and |x| ≤ C0 n1/6
2 √ P ≤ C (log n + x2 )n−1/2 exp − x . n 1 − FG ≤ x − Φ ( x ) n 2
(4.1)
H. Jiang et al. / Journal of Statistical Planning and Inference 171 (2016) 135–146
145
Remark 4.1. There are some mistakes in Theorem 3.2 of Jiang (2010b). By Corollary 2.2 in Jiang (2010b) and (4.1), we can see that
√ sup P n 1 − FGn ≤ x − Φ (x) ≤ Cn−1/2 log n. x∈R 4.3. Jackson statistics (Jackson, 1967) Jackson statistics (Jackson, 1967) is a L-statistics as follows: Jn =
1
n
n
−1 Xk
n k
nX(k)
n−i+1 k=1 i=1
k=1
.
Then, we have by Tchirina (2007) as 1 ≤ k ≤ n Wk,n = 1 −
n 11
n 1
−
n i =1 i
i=n−k+2
i
and
w(u) = 2 + log(1 − u),
W (u) = 1 + log(1 − u).
Since n 1
i i=n−k+2
=
n 1
i i=n−k+2
≤
n+1
−
1
−
n−k+2
1
n−k+2
u
1
du
n +1
n−k+2
− log 1 −
n+1
1
+
u
du
k−1 n+1
,
by (4.1), one can see that n 1
n 1
n k=1 i=n−k+2 i
=1+O
log n
n
.
Therefore, Wn =
n 1
n k=1
Wk,n = O
log n
n
n 1
,
n k=1
Wk2,n
−1=O
log2 n n
.
Moreover, by simple calculation n 1
n k=1
|Wk,n |3 →
1
|W (u)|3 du,
n→∞
0
and max1≤k≤n Wk,n ≤ log n. Hence, (Assumption A)–(Assumption E) hold. Moreover, 1
ϑn = Bn W n −
W (u)du 0
=O
log n n1/2
.
Applying Corollary 2.1, we obtain the following result. Proposition 4.3. For n large enough and |x| ≤ C0 n1/6
2 √ P ≤ C (log n + x2 )n−1/2 exp − x . nJ ≤ x − Φ ( x ) n 2 Acknowledgment The authors are very grateful to the anonymous referee for constructive comments which led to an improved presentation of this paper, especially the details in the proofs of Lemmas 3.2 and 3.4.
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