A Berry-Esseen Type Bound in Kernel Density Estimation for Negatively Associated Censored Data

Abstract

We discuss the kernel estimation of a density function based on censored data when the survival and the censoring times form the stationary negatively associated (NA) sequences. Under certain regularity conditions, the Berry-Esseen type bounds are derived for the kernel density estimator and the Kaplan-Meier kernel density estimator at a fixed point 𝑥.

1. Introduction

Let {𝑇_𝑖;𝑖≥1} be a sequence of the true survival times. The random variables (r.v.s.) are not assumed to be mutually independent; it is assumed, however, that they have a common unknown continuous marginal distribution function (d.f.) 𝐹(𝑥)=𝑃(𝑇_𝑖≤𝑥) and density function 𝑓(𝑥). Let the r.v.s. 𝑇_𝑖 be censored on the right by the censoring r.v.s. 𝑌_𝑖, so that one observes only (𝑍_𝑖,𝛿_𝑖), where 𝑍_𝑖=min𝑇_𝑖,𝑌_𝑖∶=𝑇_𝑖∧𝑌_𝑖,𝛿_𝑖=𝐼𝑇_𝑖≤𝑌_𝑖,𝑖=1,…,𝑛, (1) here and in the sequel, and 𝐼(𝐴) is the indicator random variable of the event 𝐴. In this random censorship model, the censoring times 𝑌_𝑖, 𝑖=1,…,𝑛, are assumed to have the common d.f. 𝐺(𝑦); they are also assumed to be independent of the r.v.s. 𝑇^′_𝑖s. Following the convention in the survival literature, we assume that both 𝑋_𝑖 and 𝑌_𝑖 are nonnegative random variables. In contrast to statistics for complete data, we observe only the pairs (𝑍_𝑖,𝛿_𝑖), 𝑖=1,…,𝑛, and the estimators are based on these pairs.

The following nonparametric estimation of the distribution functions 𝐹 and 𝐺 due to Kaplan and Meier [1] is widely used to estimate 𝐹 and 𝐺 on the basis of the data (𝑍_𝑖,𝛿_𝑖): 𝐹_𝑛(𝑥)=1−^𝑛_𝑘=1𝛿1−^(𝑘)𝑛−𝑘+1^{𝐼(𝑍(𝑘)≤𝑥)}=1−^𝑛_𝑘=1𝑛−𝑘𝑛−𝑘+1^{𝐼(𝛿(𝑘)=1,𝑍(𝑘)≤𝑥)},𝐺_𝑛(𝑥)=1−^𝑛_𝑘=1𝑛−𝑘𝑛−𝑘+1^{𝐼(𝛿(𝑘)=0,𝑍(𝑘)≤𝑥)}, (2) where 𝑍₍₁₎≤𝑍₍₂₎≤⋯≤𝑍_(𝑛) denote the order statistics of 𝑍₁,𝑍₂,…,𝑍_𝑛 and 𝛿_(𝑖) is the concomitant of 𝑍_(𝑖).

We introduce the kernel density estimator 𝑓_𝑛(𝑥)=1𝑛ℎ_𝑛^𝑛_𝑖=1𝐾𝑍_𝑖−𝑥ℎ_𝑛𝛿_𝑖1−𝐺𝑍_𝑖, (3) where 0<ℎ_𝑛→0 are bandwidths and 𝐾 is some kernel function. When 𝐺 is known, (3) can be used to estimate the common density of the lifetimes. However, in most practical cases 𝐺 is unknown and must be replaced by the Kaplan-Meier estimator 𝐺_𝑛, so the Kaplan-Meier kernel density estimator of the 𝑓 is defined by 𝑓_𝑛(𝑥)=1𝑛ℎ_𝑛^𝑛_𝑖=1𝐾𝑍_𝑖−𝑥ℎ_𝑛𝛿_𝑖1−𝐺_𝑛𝑍_𝑖. (4)

There is an extensive literature on the Kaplan-Meier estimator for censored independent observations. We refer to papers by Földes and Rejtő [2], Gu and Lai [3], Gill [4], and Sun and Zhu [5]. Sun and Zhu obtained the following Berry-Esseen bound for i.i.d. censored sequences.

Theorem A. Let 𝐾 be a bounded probability kernel function with compact support [−1,1] satisfying for integer 𝑟≥2, 1𝑢𝑗!^𝑗𝐾(𝑢)𝑐d𝑢=1,𝑗=0,0,𝑗=1,2,…,𝑟−1,_𝑟≠0,𝑗=𝑟. (5)

Let 𝑓 be 𝑟-order continuously differentiable and let 𝐺 be continuously differentiable in a neighborhood of 𝑥 with 𝑓(𝑥)>0 for 𝑥<𝜏_𝐿. Then sup_𝑦∈ℝ𝑃𝑛ℎ_𝑛^1/2𝑓_𝑛(𝑥)(𝑥(𝑥)𝑦𝑏−𝑓)≤𝑦𝜎−Φ=𝑂_𝑛, (6) where Φ(⋅) denotes the standard normal distribution function, 𝑏_𝑛=(𝑛ℎ_𝑛)^−1/2+𝑛^1/2ℎ_𝑛^𝑟+1/2+ℎ_𝑛^1/4 and 𝜎²(𝑥)=(𝑓(𝑥)/(1−𝐺(𝑥)))∫𝐾²(𝑡)d𝑡.

However, the censored dependent data appear in a number of applications. For example, repeated measurements in survival analysis follow this pattern; see Kang and Koehler [6]. In the context of censored time series analysis, Shumway et al. [7] considered (hourly or daily) measurements of the concentration of a given substance subject to some detection limits, thus being potentially censored from the right. Lecoutre and Ould-Said [8], Cai [9], and Liang and Uña-Álvarez [10] studied the convergence for the stationary 𝛼-mixing data. However, the convergence for the NA data has not been reported.

The main purpose of this paper is to study the kernel density estimator and the Kaplan-Meier kernel estimator of a density function based on censored data when the survival and the censoring times form the stationary NA (see the following definition) sequences. Under certain regularity conditions, the Berry-Esseen type bounds are derived for the kernel density estimator and the Kaplan-Meier kernel estimator at a fixed point 𝑥.

Definition 1. Random variables 𝑋₁,𝑋₂,…,𝑋_𝑛, 𝑛≥2 are said to be negatively associated (NA) if for every pair of disjoint subsets 𝐴₁ and 𝐴₂ of {1,2,…,𝑛}, cov𝑓₁𝑋_𝑖;𝑖∈𝐴₁,𝑓₂𝑋_𝑗;𝑗∈𝐴₂≤0, (7) where 𝑓₁ and 𝑓₂ are increasing for every variable (or decreasing for every variable) such that this covariance exists. A sequence of random variables {𝑋_𝑖;𝑖≥1} is said to be NA if every finite subfamily is NA.

Obviously, if {𝑋_𝑖;𝑖≥1} is a sequence of NA random variables and {𝑓_𝑖;𝑖≥1} is a sequence of nondecreasing (or nonincreasing) functions, then {𝑓_𝑖(𝑋_𝑖);𝑖≥1} is also a sequence of NA random variables.

This definition was introduced by Joag-Dev and Proschan [11]. Statistical test depends greatly on sampling. The random sampling without replacement from a finite population is NA but is not independent. NA sampling has wide applications such as those in multivariate statistical analysis and reliability theory. Because of the wide applications of NA sampling, the limit behavior of NA random variables has received more and more attention recently. One can refer to Joag-Dev and Proschan [11] for fundamental properties, Matuła [12] for the three-series theorem, and Wu and Jiang [13, 14] for the strong convergence.

2. Main Results

In what follows, let 𝐿 be the d.f. of the 𝑍_𝑖’s, 𝐿∶=1−𝐿. Since the sequences {𝑇_𝑛;𝑛≥1} and {𝑌_𝑛;𝑛≥1} are independent, it follows that 𝐿=1−𝐹𝐺∶=1−(1−𝐹)(1−𝐺).

Define (possibly infinite) times 𝜏_𝐹, 𝜏_𝐺, and 𝜏_𝐿 by 𝜏_𝐹=inf𝑦;𝐹𝑦=1,𝜏_𝐺𝑦,𝜏=inf𝑦;𝐺=1_𝐿=inf𝑦;𝐿𝑦=1. (8) Then, 𝜏_𝐿=𝜏_𝐹∧𝜏_𝐺.

We give the following four lemmas, which are helpful in proving our theorems.

Lemma 2 (Chang and Rao, [15]). Let 𝑋 and 𝑌 be random variables, then for any 𝑎>0sup_𝑦∈ℝ𝑃𝑋+𝑌≤𝑦−Φ𝑦≤sup_𝑦∈ℝ𝑃𝑦+𝑎𝑋≤𝑦−Φ√𝑌|),2𝜋+𝑃(|>𝑎 (9) here and in the sequel, where Φ(⋅) denotes the standard normal distribution function.

Lemma 3 (Su et al. [16, Theorem 1]). Let {𝑋_𝑖;𝑖≥1} be a sequence of NA r.v.s. with zero means and 𝔼|𝑋_𝑖|^𝑝<∞, 𝑖=1,2,… and 𝑝≥2. Then for 𝑆_𝑛=∑^𝑛_𝑖=1𝑋_𝑖, 𝔼𝑆_𝑛^𝑝≤𝑐_𝑝^𝑛_𝑖=1𝔼𝑋_𝑖^𝑝+^𝑛_𝑖=1𝔼𝑋²_𝑖^𝑝/2, (10) where 𝑐_𝑝>0 depends only on 𝑝.

Lemma 4. Let {𝑋_𝑖;𝑖≥1} be a sequence of NA r.v.s. with continuous d.f. 𝐹, and let 𝐹_𝑛(𝑥)∶=(1/𝑛)∑^𝑛_𝑖=1𝐼(𝑋_𝑖<𝑥) be the empirical d.f. based on the segments 𝑋₁,…,𝑋_𝑛. Then sup_𝑥∈ℝ𝐹_𝑛(𝑥)−𝐹(𝑥)=𝑂𝑛^−1/2ln^1/2𝑛𝑎.𝑠. (11)

Proof. Similar to the proof of Lemma 4 in Yang [17], we can prove Lemma 4.

Lemma 5 (Wu and Chen [18, Theorem 1.3]). Let {𝑇_𝑛;𝑛≥1} and {𝑌_𝑛;𝑛≥1} be two sequences of NA r.v.s. Suppose that the sequences {𝑇_𝑛;𝑛≥1} and {𝑌_𝑛;𝑛≥1} are independent. Then for any 0<𝜏<𝜏_𝐿, sup_{0≤𝑡≤𝜏}𝐹_𝑛(𝑡)−𝐹(𝑡)=𝑂𝑛^−1/2ln^1/2𝑛𝑎.𝑠. (12)

In order to formulate our main results, we now list some assumptions.(𝐴₁){𝑌_𝑖;𝑖≥1} and {𝑇_𝑖;𝑖≥1} are two sequences of stationary NA random variables, and {𝑌_𝑖} and {𝑇_𝑖} are independent.(𝐴₂) Suppose that 𝑥<𝜏_𝐿, 𝑓(𝑥)>0, and 𝑓 and 𝐺 have bounded derivative in a neighborhood of 𝑥.(𝐴₃) For all integers 𝑗≥1, the conditional distribution 𝑇_𝑗+1, given 𝑇₁=𝑥₁, has a density 𝑓_𝑗(⋅|𝑥₁), and for all 𝑥∈ℝ, 𝑓_𝑗(𝑥₂|𝑥₁)≤𝑀₀ for 𝑥₁,𝑥₂∈𝑈(𝑥) and some 𝑀₀>0, where 𝑈(𝑥) represents a neighborhood of 𝑥. (𝐴₄) The kernel 𝐾 is a bounded derivative function with 𝐾(𝑢)=0 for |𝑢|>1 and ∫¹₋₁𝐾(𝑢)𝑑𝑢=1.(𝐴₅) Let 𝑝_𝑛, 𝑞_𝑛, and 𝑘_𝑛=[𝑛/(𝑝_𝑛+𝑞_𝑛)] be positive integers with lim_𝑛→∞𝑘_𝑛=∞,lim_𝑛→∞𝑝_𝑛𝑘_𝑛𝑛=1,lim_𝑛→∞𝑝_𝑛ℎ_𝑛=0,lim_𝑛→∞𝑛ℎ_𝑛=∞,lim_𝑛→∞ℎ_𝑛⁻³𝑢𝑞_𝑛=0, (13) where 𝑢(𝑛)∶=∑^∞_𝑗=𝑛|Cov(𝑇₁,𝑇_𝑗+1)|.

Remark 6. (𝐴₅) Implies lim_𝑛→∞(𝑘_𝑛𝑞_𝑛/𝑛)=0 and lim_𝑛→∞(𝑝_𝑛/𝑛)=0.
Let 𝜎²_𝑛(𝑥)=𝑛ℎ_𝑛Var(𝑓_𝑛(𝑥)), 𝜎²(𝑥)=(𝑓(𝑥)/(1−𝐺(𝑥)))∫¹₋₁𝐾²(𝑡)d𝑡.

Theorem 7. Suppose that (𝐴₁)--(𝐴₅) are satisfied; then 𝜎²_𝑛(𝑥)−𝜎²(𝑥)=𝑂𝑎_𝑛, (14) where 𝑎_1𝑛=(𝑞_𝑛𝑘_𝑛/𝑛)+(𝑞_𝑛𝑘_𝑛𝑢(𝑝_𝑛)/𝑛ℎ³_𝑛), 𝑎_2𝑛=𝑝_𝑛/𝑛, 𝑎_3𝑛=𝑢(𝑞_𝑛)/ℎ³_𝑛, 𝑎_𝑛=𝑝_𝑛ℎ_𝑛+𝑎^1/2_1𝑛+𝑎^1/2_2𝑛+𝑎_3𝑛→0.
Consider the following: sup_𝑦∈ℝ𝑃𝑓_𝑛(𝑥)−𝔼𝑓_𝑛(𝑥)Var𝑓_𝑛(𝑥)≤𝑦−Φ𝑦=𝑂𝑏_𝑛, (15) where 𝑏_𝑛=1/(𝑛ℎ_𝑛)^1/2+𝑎^1/3_1𝑛+𝑎^1/3_2𝑛+𝑎^1/3_3𝑛→0.
Furthermore, if lim_𝑛→∞𝑛^1/2ℎ_𝑛^3/2=0, (16) then sup_𝑦∈ℝ𝑃𝑓_𝑛(𝑥)−𝑓(𝑥)Var𝑓_𝑛(𝑥)≤𝑦−Φ𝑦=𝑂𝑏_𝑛+𝑛^1/2ℎ_𝑛^3/2. (17)

Theorem 8. Assume that the conditions of Theorem 7 hold. Then sup_𝑦∈ℝ𝑃𝑓_𝑛(𝑥)−𝔼𝑓_𝑛(𝑥)Var𝑓_𝑛(𝑥)𝑦𝑏≤𝑦−Φ=𝑂_𝑛+ℎ_𝑛ln𝑛^1/4. (18) Furthermore, if (16) holds, then sup_𝑦∈ℝ𝑃𝑓_𝑛(𝑥)−𝑓(𝑥)Var𝑓_𝑛(𝑥)𝑦𝑏≤𝑦−Φ=𝑂_𝑛+ℎ_𝑛ln𝑛^1/4+𝑛^1/2ℎ_𝑛^3/2. (19)

3. Proofs

Proof of Theorem 7. We observe that, by (3), 𝑛ℎ_𝑛^1/2𝑓_𝑛(𝑥)=^𝑛_𝑖=11𝑛ℎ_𝑛^1/2𝐾𝑍_𝑖−𝑥ℎ_𝑛𝛿_𝑖1−𝐺𝑍_𝑖∶=^𝑛_𝑖=1𝑍_𝑛𝑖∶=𝑆_𝑛. (20)
Let 𝑘_𝑚=(𝑚−1)(𝑝_𝑛+𝑞_𝑛)+1, 𝑙_𝑚=(𝑚−1)(𝑝_𝑛+𝑞_𝑛)+𝑝_𝑛+1, 𝑚=1,2,…,𝑘_𝑛, where 𝑈_𝑛𝑚=^{𝑘𝑚+𝑝𝑛−1}_{𝑖=𝑘𝑚}𝑍_𝑛𝑖,𝑈^′_𝑛𝑚=^{𝑙𝑚+𝑞𝑛−1}_{𝑖=𝑙𝑚}𝑍_𝑛𝑖,𝑈^′_{𝑛,𝑘𝑛+1}=^𝑛_{𝑖=𝑘𝑛𝑝𝑛+𝑞𝑛+1}𝑍_𝑛𝑖,𝑆^′_𝑛=^𝑘𝑛_𝑚=1𝑈_𝑛𝑚,𝑆_𝑛^′′=^𝑘𝑛_𝑚=1𝑈^′_𝑛𝑚,𝑆_𝑛^′′′=𝑈^′_{𝑛,𝑘𝑛+1}, (21) and then 𝑆_𝑛=𝑆^′_𝑛+𝑆_𝑛^′′+𝑆_𝑛^′′′. (22) By (20), 𝜎²_𝑛(𝑥)=Var𝑆_𝑛=Var𝑆^′_𝑛+𝑆_𝑛^′′+𝑆_𝑛^′′′=Var𝑆^′_𝑛+Var𝑆_𝑛^′′+Var𝑆_𝑛^′′′+2Cov𝑆^′_𝑛,𝑆_𝑛^′′+2Cov𝑆^′_𝑛,𝑆_𝑛^′′′+2Cov𝑆_𝑛^′′,𝑆_𝑛^′′′. (23) We first estimate Var𝑆^′_𝑛, Var𝑆_𝑛^′′, and Var𝑆_𝑛^′′′. Obviously, (𝐴₁) implies that {𝑈_𝑛𝑚} and {𝑍_𝑛𝑖} are stationary; thus, Var𝑆^′_𝑛=Var^𝑘𝑛_𝑚=1𝑈_𝑛𝑚=^𝑘𝑛_𝑚=1Var𝑈_𝑛𝑚+2_{1≤𝑖<𝑗≤𝑘𝑛}Cov𝑈_𝑛𝑖,𝑈_𝑛𝑗=𝑘_𝑛Var𝑈_𝑛1+2_{1≤𝑖<𝑗≤𝑘𝑛}Cov𝑈_𝑛𝑖,𝑈_𝑛𝑗=𝑘_𝑛𝑝_𝑛Var𝑍_𝑛1+2𝑘_{𝑛1≤𝑖<𝑗≤𝑝𝑛}Cov𝑍_𝑛𝑖,𝑍_𝑛𝑗+2_{1≤𝑖<𝑗≤𝑘𝑛}Cov𝑈_𝑛𝑖,𝑈_𝑛𝑗∶=𝐼_𝑛1+𝐼_𝑛2+𝐼_𝑛3. (24)
From (𝐴₁), (𝐴₂), and (𝐴₄), we obtain Var𝑍_𝑛1=1𝑛ℎ_𝑛𝑍Var𝐾¹−𝑥ℎ_𝑛𝛿₁1−𝐺𝑍₁=1𝑛ℎ_𝑛𝔼𝐾²𝑍₁−𝑥ℎ_𝑛𝛿₁1−𝐺𝑍₁²−𝑍𝔼𝐾¹−𝑥ℎ_𝑛𝛿₁1−𝐺𝑍₁²=1𝑛ℎ_𝑛𝐾²min(𝑢,𝑣)−𝑥ℎ_𝑛×𝐼(𝑢<𝑣)(1−𝐺(min(𝑢,𝑣)))²(𝑢)(𝑣)−1d𝐹d𝐺𝑛ℎ_𝑛𝐾min(𝑢,𝑣)−𝑥ℎ_𝑛×𝐼(𝑢<𝑣)(((𝑢)(𝑣1−𝐺min𝑢,𝑣))d𝐹d𝐺)²=1𝑛ℎ_{𝑛𝑢<𝑣}𝐾²𝑢−𝑥ℎ_𝑛1(1−𝐺(𝑢))²(𝑢)(𝑣)−1d𝐹d𝐺𝑛ℎ_{𝑛𝑢<𝑣}𝐾𝑢−𝑥ℎ_𝑛1(𝑢)(𝑢)(𝑣1−𝐺d𝐹d𝐺)²=1𝑛ℎ_𝑛𝐾²𝑢−𝑥ℎ_𝑛𝑓(𝑢)(𝑢)−𝐾1−𝐺d𝑢𝑢−𝑥ℎ_𝑛𝑓(𝑢)d𝑢²=1𝑛¹₋₁𝐾²(𝑢)𝑓𝑥+𝑢ℎ_𝑛1−𝐺𝑥+𝑢ℎ_𝑛d𝑢−ℎ_𝑛¹₋₁𝐾(𝑢)𝑓𝑥+𝑢ℎ_𝑛d𝑢²1=𝑂𝑛. (25)(26) Hence, by (𝐴₅), 𝐼_𝑛1=𝑂(1).
For 𝑖<𝑗 and 𝑥<𝜏_𝐿, by (𝐴₁)−(𝐴₄), Cov𝑍_𝑛𝑖,𝑍_𝑛𝑗=Cov𝑍_𝑛1,𝑍_{𝑛,𝑗−𝑖+1}=1𝑛ℎ_𝑛𝐾𝑍Cov¹−𝑥ℎ_𝑛𝛿₁1−𝐺𝑍₁,𝐾𝑍_{𝑗−𝑖+1}−𝑥ℎ_𝑛𝛿_{𝑗−𝑖+1}1−𝐺𝑍_{𝑗−𝑖+1}≤𝑐𝑛ℎ_𝑛𝔼𝐾𝑇₁−𝑥ℎ_𝑛𝐾𝑇_{𝑗−𝑖+1}−𝑥ℎ_𝑛+𝑍𝔼𝐾¹−𝑥ℎ_𝑛²≤𝑐𝑛ℎ_𝑛𝐾𝑢−𝑥ℎ_𝑛𝐾𝑣−𝑥ℎ_𝑛×𝑓(𝑢)𝑓_𝑗−𝑖()+𝐾𝑣∣𝑢d𝑢d𝑣𝑢−𝑥ℎ_𝑛𝑓(𝑢)d𝑢²≤𝑐ℎ_𝑛𝑛¹₋₁|𝐾(𝑢)𝐾(𝑣+)|d𝑢d𝑣₁₋₁𝐾(𝑢)𝑓𝑥+ℎ_𝑛𝑢d𝑢²ℎ=𝑂^𝑛𝑛. (27)
Therefore, by (𝐴₅), 𝐼_𝑛2=2𝑘_{𝑛1≤𝑖<𝑗≤𝑝𝑛}Cov𝑍_𝑛𝑖,𝑍_𝑛𝑗𝑘=𝑂^𝑛𝑝²_𝑛ℎ_𝑛𝑛=𝑂𝑝_𝑛ℎ_𝑛⟶0. (28) By (𝐴₁), (𝐴₂), (𝐴₄), and Lemma 2.3 of Zhang [19], for 𝑙≥1, Cov𝑈_𝑛1,𝑈_𝑛,𝑙+1=^𝑝𝑛_𝑖=1^{𝑙(𝑝𝑛+𝑞𝑛)+𝑝𝑛}_{𝑗=𝑙(𝑝𝑛+𝑞𝑛)+1}Cov𝑍_𝑛𝑖,𝑍_𝑛𝑗=^𝑝𝑛_𝑖=1^𝑖−1_𝑗=1Cov𝑍_{𝑛,𝑖−𝑗+1},𝑍_{𝑛,𝑙(𝑝𝑛+𝑞𝑛)+1}+^𝑝𝑛_𝑗=𝑖Cov𝑍_𝑛1,𝑍_{𝑛,𝑙(𝑝𝑛+𝑞𝑛)+𝑗−𝑖+1}=^𝑝𝑛_𝑟=1𝑝_𝑛−𝑟+1Cov𝑍_𝑛1,𝑍_{𝑛,𝑙(𝑝𝑛+𝑞𝑛)+𝑟}+^𝑝𝑛−1_𝑟=1𝑝_𝑛−𝑟Cov𝑍_𝑛,𝑟+1,𝑍_{𝑛,𝑙(𝑝𝑛+𝑞𝑛)+1}≤𝑝_𝑛^{𝑙(𝑝𝑛+𝑞𝑛)+𝑝𝑛−1}_{𝑟=𝑙(𝑝𝑛+𝑞𝑛)}Cov𝑍_𝑛1,𝑍_𝑛,𝑟+1+𝑝_𝑛^{𝑙(𝑝𝑛+𝑞𝑛)−1}_{𝑟=𝑙(𝑝𝑛+𝑞𝑛)−(𝑝𝑛−1)}Cov𝑍_𝑛1,𝑍_𝑛,𝑟+1=𝑝_𝑛^{𝑙(𝑝𝑛+𝑞𝑛)+𝑝𝑛−1}_{𝑟=𝑙(𝑝𝑛+𝑞𝑛)−(𝑝𝑛−1)}Cov𝑍_𝑛1,𝑍_𝑛,𝑟+1=𝑝_𝑛𝑛ℎ_𝑛^{𝑙(𝑝𝑛+𝑞𝑛)+𝑝𝑛−1}_{𝑟=𝑙(𝑝𝑛+𝑞𝑛)−(𝑝𝑛−1)}𝐾𝑍Cov¹−𝑥ℎ_𝑛×𝛿₁1−𝐺𝑍₁,𝐾𝑍_𝑟+1−𝑥ℎ_𝑛×𝛿_𝑟+11−𝐺𝑍_𝑟+1𝑝≤𝑐^𝑛𝑛ℎ³_𝑛^{𝑙(𝑝𝑛+𝑞𝑛)+𝑝𝑛−1}_{𝑟=𝑙(𝑝𝑛+𝑞𝑛)−(𝑝𝑛−1)}Cov𝑇₁,𝑇_𝑟+1. (29)
Thus, by (𝐴₁) and (𝐴₅), 𝐼_𝑛3=2_{1≤𝑖<𝑗≤𝑘𝑛}Cov𝑈_𝑛𝑖,𝑈_𝑛𝑗=2^𝑘𝑛−1_𝑙=1𝑘_𝑛−𝑙Cov𝑈_𝑛1,𝑈_𝑛,𝑙+1≤𝑐𝑝_𝑛𝑘_𝑛𝑛ℎ³_𝑛^𝑘𝑛−1_𝑙=1^{𝑙(𝑝𝑛+𝑞𝑛)+𝑝𝑛−1}_{𝑟=𝑙(𝑝𝑛+𝑞𝑛)−(𝑝𝑛−1)}Cov𝑇₁,𝑇_𝑟+1≤𝑐𝑝_𝑛𝑘_𝑛𝑛1ℎ³_𝑛^∞_{𝑟=𝑞𝑛}Cov𝑇₁,𝑇_𝑟+1𝑢𝑞=𝑂^𝑛ℎ³_𝑛=𝑂𝑎_3𝑛⟶0. (30) Therefore, by the combination of (𝐴₅), (24), (26), (28), and (30), Var𝑆^′_𝑛=𝑂(1)+𝑂𝑝_𝑛ℎ_𝑛+𝑂𝑎_3𝑛=𝑂(1). (31)
Similarly, Var𝑆_𝑛^′′=𝑘_𝑛Var𝑈^′_𝑛1+2_{1≤𝑖<𝑗≤𝑘𝑛}Cov𝑈^′_𝑛𝑖,𝑈^′_𝑛𝑗=𝑘_𝑛𝑞_𝑛Var𝑍_𝑛1+2𝑘_{𝑛1≤𝑖<𝑗≤𝑞𝑛}Cov𝑍_𝑛𝑖,𝑍_𝑛𝑗+2_{1≤𝑖<𝑗≤𝑘𝑛}Cov𝑈^′_𝑛𝑖,𝑈^′_𝑛𝑗𝑞=𝑂^𝑛𝑘_𝑛𝑛+𝑞²_𝑛𝑘_𝑛ℎ_𝑛𝑛+𝑞_𝑛𝑘_𝑛𝑢𝑝_𝑛𝑛ℎ³_𝑛𝑞=𝑂^𝑛𝑘_𝑛𝑛+𝑞_𝑛𝑘_𝑛𝑢𝑝_𝑛𝑛ℎ³_𝑛=𝑂𝑎_1𝑛. (32)
By (26), (27), (𝐴₁), (𝐴₅), and 𝑛−𝑘_𝑛(𝑝_𝑛+𝑞_𝑛)≤𝑝_𝑛+𝑞_𝑛≤2𝑝_𝑛, Var𝑆_𝑛^′′′=Var^𝑛_{𝑖=𝑘𝑛(𝑝𝑛+𝑞𝑛)+1}𝑍_𝑛𝑖=𝑛−𝑘_𝑛𝑝_𝑛+𝑞_𝑛Var𝑍_𝑛1+2_{𝑘𝑛(𝑝𝑛+𝑞𝑛)+1≤𝑖<𝑗≤𝑛}Cov𝑍_𝑛𝑖,𝑍_𝑛𝑗𝑝=𝑂^𝑛𝑛+𝑝²_𝑛ℎ_𝑛𝑛𝑝=𝑂^𝑛𝑛=𝑂𝑎_2𝑛. (33)
By (25), (𝐴₂), and (𝐴₄), 𝑛Var𝑍_𝑛1−𝜎²(𝑥)=¹₋₁𝐾²(𝑢)𝑓𝑥+𝑢ℎ_𝑛1−𝐺𝑥+𝑢ℎ_𝑛−𝑓(𝑥)1−𝐺(𝑥)d𝑢−ℎ_𝑛¹₋₁𝐾(𝑢)𝑓𝑥+𝑢ℎ_𝑛d𝑢²≤¹₋₁𝐾²(𝑢)(1−𝐺(𝑥))𝑓𝑥+𝑢ℎ_𝑛−𝑓(𝑥)+𝑓(𝑥)𝐺𝑥+𝑢ℎ_𝑛−𝐺(𝑥)1−𝐺𝑥+𝑢ℎ_𝑛(1−𝐺(𝑥))d𝑢+ℎ_𝑛¹₋₁𝐾(𝑢)𝑓𝑥+𝑢ℎ_𝑛d𝑢²=𝑂ℎ_𝑛. (34)Note that |Cov(𝑋,𝑌)|≤(Var𝑋Var𝑌)^1/2 for any random variables 𝑋 and 𝑌; from (31)–(33), Cov𝑆^′_𝑛,𝑆_𝑛^′′=𝑂𝑎^1/2_1𝑛,Cov𝑆^′_𝑛,𝑆_𝑛^′′′=𝑂𝑎^1/2_2𝑛,Cov𝑆_𝑛^′′,𝑆_𝑛^′′′=𝑂𝑎^1/2_1𝑛𝑎^1/2_2𝑛. (35) Therefore, from the combination of (23) and (31)–(34), it follows that 𝜎²_𝑛(𝑥)−𝜎²(𝑥=)𝑛Var𝑍_𝑛1−𝜎²(𝑥)+𝑂𝑝_𝑛ℎ_𝑛+𝑎_3𝑛+𝑎^1/2_1𝑛+𝑎^1/2_2𝑛=𝑂𝑝_𝑛ℎ_𝑛+𝑎_3𝑛+𝑎^1/2_1𝑛+𝑎^1/2_2𝑛=𝑂𝑎_𝑛. (36) Thus, (14) holds.
Now, we prove (15). Let 𝑆_𝑛=(𝑆_𝑛−𝔼𝑆_𝑛)/𝜎_𝑛(𝑥), 𝑆^′_𝑛=(𝑆^′_𝑛−𝔼𝑆^′_𝑛)/𝜎_𝑛(𝑥), 𝑆_𝑛^′′=(𝑆_𝑛^′′−𝔼𝑆_𝑛^′′)/𝜎_𝑛(𝑥), 𝑆_𝑛^′′′=(𝑆_𝑛^′′′−𝔼𝑆_𝑛^′′′)/𝜎_𝑛(𝑥). Then, 𝑆_𝑛=𝑆^′_𝑛+𝑆_𝑛^′′+𝑆_𝑛^′′′. According to Lemma 2, (14), (20), (32), and (33), we have sup_𝑦∈ℝ𝑃𝑓_𝑛(𝑥)−𝔼𝑓_𝑛(𝑥)Var𝑓_𝑛(𝑥)≤𝑦−Φ𝑦=sup_𝑦∈ℝ𝑃𝑆^′_𝑛+𝑆_𝑛^′′+𝑆_𝑛^′′′≤𝑦−Φ𝑦≤sup_𝑦∈ℝ𝑃𝑆^′_𝑛𝑦+𝑎≤𝑦−Φ^1/31𝑛√𝑆2𝜋+𝑃_𝑛^′′>𝑎^1/3_1𝑛+𝑎^1/3_2𝑛√𝑆2𝜋+𝑃_𝑛^′′′>𝑎^1/3_2𝑛=sup_𝑦∈ℝ𝑃𝑆^′_𝑛≤𝑦−Φ𝑦+𝑂𝑎^1/3_1𝑛+𝑎^1/3_2𝑛. (37)
Let 𝜉_𝑛𝑚, 𝑚=1,2,…,𝑘_𝑛 be independent random variables with the same distribution as 𝑈_𝑛𝑚∶=(𝑈_𝑛𝑚−𝔼𝑈_𝑛𝑚)/𝜎_𝑛(𝑥) for 𝑚=1,2,…,𝑘_𝑛. Put 𝐻_𝑛=∑^𝑘𝑛_𝑚=1𝜉_𝑛𝑚, 𝐵²_𝑛=∑^𝑘𝑛_𝑚=1Var𝑈_𝑛𝑚=∑^𝑘𝑛_𝑚=1Var𝜉_𝑛𝑚=Var𝐻_𝑛. Obviously, sup_𝑦∈ℝ𝑃𝑆^′_𝑛≤𝑦−Φ𝑦≤sup_𝑦∈ℝ𝑃𝑆^′_𝑛≤𝑦−𝑃𝐻_𝑛≤𝑦+sup_𝑦∈ℝΦ𝑦𝐵_𝑛−Φ𝑦+sup_𝑦∈ℝ𝑃𝐻_𝑛𝑦≤𝑦−Φ𝐵_𝑛∶=𝐽_1𝑛+𝐽_2𝑛+𝐽_3𝑛. (38) Note that Var𝑆_𝑛=𝜎²_𝑛(𝑥) and 𝐵²_𝑛=(Var𝑆^′_𝑛−𝐼_𝑛3)/𝜎²_𝑛(𝑥) from (20) and (24). By (14), (30), (32), and (33), 𝐽_2𝑛=sup_𝑦∈ℝΦ𝑦𝐵_𝑛−Φ𝑦≤𝐵²_𝑛=−1Var𝑆^′_𝑛−𝐼_𝑛3−Var𝑆_𝑛𝜎²_𝑛(𝑥)≤𝑐Var𝑆^′_𝑛−Var𝑆_𝑛+𝑐𝐼_𝑛3≤𝑐Var𝑆_𝑛^′′+𝑆_𝑛^′′′+2Cov𝑆^′_𝑛,𝑆_𝑛^′′+𝑆_𝑛^′′′+𝑂𝑎_3𝑛=𝑂𝑎^1/2_1𝑛+𝑎^1/2_2𝑛+𝑎_3𝑛⟶0. (39) Note that 𝜉_𝑛𝑚, 𝑚=1,2,…,𝑘_𝑛, are independent random variables, and 𝐵²_𝑛=Var𝐻_𝑛. Therefore, by 𝐵_𝑛→1 (from (39)), (14), and Berry-Esseen inequality (cf. Petrov [20, page 154, Theorem 5.7]), there exists some constant 𝑐>0 such that 𝐽_3𝑛=sup_𝑦∈ℝ𝑃𝐻_𝑛𝐵_𝑛𝑦∑≤𝑦−Φ≤𝑐^{𝑘𝑛𝑚=1}𝔼𝜉_𝑛𝑚³𝐵³_𝑛≤𝑐^𝑘𝑛_𝑚=1𝔼𝑈_𝑛𝑚³=^𝑘𝑛_𝑚=1𝔼^{𝑘𝑚+𝑝𝑛−1}_{𝑖=𝑘𝑚}𝑍_𝑛𝑖³. (40)
Similar to (26), we can get 𝔼|𝑍_𝑛1|³=𝑂(1/𝑛^3/2ℎ_𝑛^1/2) and 𝔼𝑍²_𝑛1=𝑂(1/𝑛). It is easy to see from Property P7 of Joag-Dev and Proschan [11] that {𝑍_𝑛;𝑛≥1} is also sequence of NA r.v.s., so by using Lemma 3, we have 𝐽_3𝑛≤𝑐^𝑘𝑛_𝑚=1^{𝑘𝑚+𝑝𝑛−1}_{𝑖=𝑘𝑚}𝔼𝑍_𝑛𝑖³+^{𝑘𝑚+𝑝𝑛−1}_{𝑖=𝑘𝑚}𝔼𝑍²_𝑛𝑖^3/2≤𝑐𝑘_𝑛𝑝_𝑛1𝑛^3/2ℎ_𝑛^1/2+𝑝_𝑛𝑛^3/21=𝑂𝑛ℎ_𝑛^1/2+𝑎^1/2_2𝑛. (41)
Assume that 𝜑(𝑡) and 𝜓(𝑡) are the characteristic functions of 𝑆^′_𝑛 and 𝐻_𝑛, respectively. By Esseen inequality (cf. Petrov [20, page 146, Theorem 5.3]), for any 𝑇>0, there exists some constant 𝑐>0 such that 𝐽_1𝑛=sup_𝑦∈ℝ𝑃𝑆^′_𝑛≤𝑦−𝑃𝐻_𝑛≤≤𝑦_𝑇−𝑇𝜑(𝑡)−𝜓(𝑡)𝑡d𝑡+𝑇sup_{𝑦∈ℝ|𝑢|≤(𝑐/𝑇)}𝑃𝐻_𝑛≤𝑢+𝑦−𝑃𝐻_𝑛≤𝑦d𝑢∶=𝐽^′_1𝑛+𝐽^′′_1𝑛. (42)
By Theorem 10 in Newman [21], (14), and (30), 𝜑(𝑡)−𝜓(𝑡)=𝔼expi𝑡^𝑘𝑛_𝑚=1𝑈_𝑛𝑚−^𝑘𝑛_𝑚=1𝔼expi𝑡𝑈_𝑛𝑚≤2_{1≤𝑖<𝑗≤𝑘𝑛}𝑡²Cov𝑈_𝑛𝑖,𝑈_𝑛𝑗≤𝑐_{1≤𝑖<𝑗≤𝑘𝑛}𝑡²Cov𝑈_𝑛𝑖,𝑈_𝑛𝑗=𝑡²𝑂𝑎_3𝑛. (43) Therefore, 𝐽^′_1𝑛=𝑂𝑎_3𝑛𝑇². (44) On applying (39)–(41), we have sup_𝑦∈ℝ𝑃𝐻_𝑛≤𝑢+𝑦−𝑃𝐻_𝑛≤𝑦≤sup_𝑦∈ℝ𝑃𝐻_𝑛𝐵_𝑛≤𝑢+𝑦𝐵_𝑛−Φ𝑢+𝑦𝐵_𝑛+𝑃𝐻_𝑛𝐵_𝑛≤𝑦𝐵_𝑛𝑦−Φ𝐵_𝑛+Φ𝑢+𝑦𝐵_𝑛𝑦−Φ𝐵_𝑛1=𝑂𝑛ℎ_𝑛^1/2+𝑎^1/2_2𝑛+|𝑢|. (45) Thus, 𝐽^′′_1𝑛1=𝑂𝑛ℎ_𝑛^1/2+𝑎^1/2_2𝑛+1𝑇. (46) Choosing 𝑇=𝑎^−1/3_3𝑛, then by (42)–(46), 𝐽_1𝑛1=𝑂𝑛ℎ_𝑛^1/2+𝑎^1/2_2𝑛+𝑎^1/3_3𝑛. (47) Therefore, the combination of (37)–(39), (41), (47), and (15) holds.
Finally, we prove (17). By Lemma 2 and (15), for any 𝑎>0, sup_𝑦∈ℝ𝑃𝑓_𝑛(𝑥)−𝑓(𝑥)Var𝑓_𝑛(𝑥)≤𝑦−Φ𝑦=sup_𝑦∈ℝ𝑃𝑓_𝑛(𝑥)−𝔼𝑓_𝑛(𝑥)Var𝑓_𝑛(𝑥)+𝔼𝑓_𝑛(𝑥)−𝑓(𝑥)Var𝑓_𝑛(𝑥)≤𝑦−Φ𝑦≤sup_𝑦∈ℝ𝑃𝑓_𝑛(𝑥)−𝔼𝑓_𝑛(𝑥)Var𝑓_𝑛(𝑥)𝑦+𝑎≤𝑦−Φ√2𝜋+𝑃𝔼𝑓_𝑛(𝑥)−𝑓(𝑥)Var𝑓_𝑛(𝑥)𝑏>𝑎=𝑂_𝑛+𝑎√2𝜋+𝑃𝔼𝑓_𝑛(𝑥)−𝑓(𝑥)Var𝑓_𝑛(𝑥)>𝑎. (48)
Applying (14), (𝐴₃), (𝐴₄), and differential mean value theorem, there exists a constant 0<𝜃<1, such that 𝔼𝑓_𝑛(𝑥)−𝑓(𝑥)Var𝑓_𝑛(𝑥)≤𝑐𝑛ℎ_𝑛1ℎ_𝑛𝑍𝔼𝐾¹−𝑥ℎ_𝑛𝛿₁1−𝐺𝑍₁−𝑓(𝑥)=𝑐𝑛ℎ_𝑛¹₋₁𝐾(𝑢)𝑓𝑥+𝑢ℎ_𝑛−𝑓(𝑥)d𝑢=𝑐𝑛ℎ_𝑛¹₋₁𝐾(𝑢)𝑢ℎ_𝑛𝑓^′𝑥+𝜃𝑢ℎ_𝑛𝑛d𝑢=𝑂^1/2ℎ_𝑛^3/2. (49) Hence, there exists a constant 𝑀 sufficiently large such that |𝔼𝑓_𝑛(𝑥)−𝑓(𝑥)|/Var𝑓_𝑛(𝑥)<𝑀𝑛^1/2ℎ_𝑛^3/2. Let 𝑎=𝑀𝑛^1/2ℎ_𝑛^3/2 in (48); then 𝑃(|𝔼𝑓_𝑛(𝑥)−𝑓_𝑛(𝑥)|/Var𝑓_𝑛(𝑥)>𝑎)=𝑃(𝜙)=0. Therefore, by (48), (16) holds.

Proof of Theorem 8. Using (15) and Lemma 2, sup_𝑦∈ℝ𝑃𝑓_𝑛(𝑥)−𝔼𝑓_𝑛(𝑥)Var𝑓_𝑛(𝑥)≤𝑦−Φ𝑦=sup_𝑦∈ℝ𝑃𝑓_𝑛(𝑥)−𝔼𝑓_𝑛(𝑥)Var𝑓_𝑛(𝑥)+𝑓_𝑛(𝑥)−𝑓_𝑛(𝑥)−𝔼𝑓_𝑛(𝑥)−𝑓_𝑛(𝑥)Var𝑓_𝑛(𝑥)𝑦≤𝑦−Φ≤sup_𝑦∈ℝ𝑃𝑓_𝑛(𝑥)−𝔼𝑓_𝑛(𝑥)Var𝑓_𝑛(𝑥)𝑦+ℎ≤𝑦−Φ_𝑛ln𝑛^1/4√𝑓2𝜋+𝑃^𝑛(𝑥)−𝑓_𝑛(𝑥)−𝔼𝑓_𝑛(𝑥)−𝑓_𝑛(𝑥)Var𝑓_𝑛(𝑥)>ℎ_𝑛ln𝑛^1/4≤𝑂𝑏_𝑛+ℎ_𝑛ln𝑛^1/4𝔼+𝑐𝑛ℎ_𝑛^1/2𝑓_𝑛(𝑥)−𝑓_𝑛(𝑥)ℎ_𝑛ln𝑛^1/4. (50)
Let 𝐿_𝑛(𝑥)=𝑛⁻¹∑^𝑛_𝑖=1𝐼(𝑍_𝑖≤𝑥) be the empirical d.f. of 𝐿. Then, by (2), 𝐿_𝑛(𝑥)=1−1−𝐹_𝑛(𝑥)1−𝐺_𝑛(𝑥). (51)
Thus, by Lemmas 4 and 5, for 𝜏<𝜏_𝐿, sup_{0≤𝑥≤𝜏}𝐺_𝑛(𝑥)−𝐺(𝑥)≤sup_{0≤𝑥≤𝜏}11−𝐹_𝑛(𝑥)×𝐿_𝑛(𝑥)−𝐿(𝑥)+(1−𝐺(𝑥))𝐹_𝑛(𝑥)−𝐹(𝑥)=𝑂ln𝑛𝑛. (52)
Using (14), we get 𝔼𝑛ℎ_𝑛^1/2𝑓_𝑛(𝑥)−𝑓_𝑛(𝑥≤1)𝑛ℎ_𝑛^𝑛𝑖=1𝔼𝐾𝑍_𝑖−𝑥ℎ_𝑛×𝐺_𝑛𝑍_𝑖−𝐺𝑍_𝑖1−𝐺𝑍_𝑖1−𝐺_𝑛𝑍_𝑖𝛿_𝑖≤𝑐ln𝑛ℎ_𝑛𝑇𝔼𝐾¹−𝑥ℎ_𝑛=𝑐ln𝑛ℎ_𝑛ℎ_𝑛¹₋₁|𝐾(𝑡)|𝑓𝑥+ℎ_𝑛𝑡d𝑡=𝑂ℎ_𝑛ln𝑛. (53)
Therefore, (18) holds from (50) and (53).
Using (18), similar to the proof of (17), we can prove (19). This completes the proof of Theorem 8.

Acknowledgments

The authors are very grateful to the referees and the editors for their valuable comments and helpful suggestions that improved the clarity and readability of the paper. This paper is supported by the National Natural Science Foundation of china (11061012), project supported by Program to Sponsor Teams for Innovation in the Construction of Talent Highlands in Guangxi Institutions of Higher Learning ((2011) 47), and the Support Program of the Guangxi China Science Foundation (2012GXNSFAA053010 and 2013GXNSFDA019001).