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Strong representation results of the Kaplan-Meier estimator for censored negatively associated data

Qunying Wu1* and Pingyan Chen2

Author Affiliations

1 College of Science, Guilin University of Technology, Guilin, 541004, P.R. China

2 Department of Mathematics, Ji’nan University, Guangzhou, 510630, P.R. China

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Journal of Inequalities and Applications 2013, 2013:340  doi:10.1186/1029-242X-2013-340

The electronic version of this article is the complete one and can be found online at: http://www.journalofinequalitiesandapplications.com/content/2013/1/340


Received:4 February 2013
Accepted:10 July 2013
Published:25 July 2013

© 2013 Wu and Chen; licensee Springer

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

In this paper, we discuss the strong convergence rates and strong representation of the Kaplan-Meier estimator and the hazard estimator based on censored data when the survival and the censoring times form negatively associated (NA) sequences. Under certain regularity conditions, strong convergence rates are established for the Kaplan-Meier estimator and the hazard estimator, and the Kaplan-Meier estimator and the hazard estimator can be expressed as the mean of random variables, with the remainder of order <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M1">View MathML</a> a.s.

MSC: 60F15, 60F05.

Keywords:
NA sequence; random censorship model; Kaplan-Meier estimator; strong representation; strong convergence rate

1 Introduction and main results

Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M2">View MathML</a> be a sequence of true survival times. 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.) <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M3">View MathML</a> such that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M4">View MathML</a>. Let the r.v.s <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M5">View MathML</a> be censored on the right by the censoring r.v.s <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M6">View MathML</a>, so that one observes only <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M7">View MathML</a>, where

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M8">View MathML</a>

Here and in the sequel, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M9">View MathML</a> is the indicator random variable of the event A. In this random censorship model, the censoring times <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M6">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M11">View MathML</a>, are assumed to have the common distribution function <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M12">View MathML</a> such that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M13">View MathML</a>; they are also assumed to be independent of the r.v.s <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M5">View MathML</a>’s. The problem at hand is that of drawing nonparametric inference about F based on the censored observations <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M7">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M16">View MathML</a>. For this purpose, define two stochastic processes on <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M17">View MathML</a> as follows:

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M18">View MathML</a>

the number of uncensored observations less than or equal to t, and

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M19">View MathML</a>

the number of censored or uncensored observations greater than or equal to t. The following nonparametric estimation <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M20">View MathML</a> of F due to Kaplan and Meier [1] is widely used to estimate F on the basis of the data <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M7">View MathML</a>:

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M22">View MathML</a>

where <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M23">View MathML</a>.

Let L be the distribution of the <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M24">View MathML</a>’s, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M25">View MathML</a>. Since the sequences <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M26">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M27">View MathML</a> are independent, it follows that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M28">View MathML</a>. The empirical d.f. <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M29">View MathML</a> of L is defined by

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M30">View MathML</a>

where <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M31">View MathML</a>.

Define (possibly infinite) times <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M32">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M33">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M34">View MathML</a> by

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M35">View MathML</a>

Then <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M36">View MathML</a>. By setting

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M37">View MathML</a>

and the empirical d.f. of <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M38">View MathML</a> is defined by

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M39">View MathML</a>

We have then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M40">View MathML</a>

and

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M41">View MathML</a>

Another question of interest in survival analysis is the estimation of the hazard function h defined as follows when it is further assumed that F has a density f:

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M42">View MathML</a>

with <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M43">View MathML</a>. The quantity

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M44">View MathML</a>

(1.1)

is called the cumulative hazard function. The empirical cumulative hazard function <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M45">View MathML</a> is given by

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M46">View MathML</a>

(1.2)

where <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M47">View MathML</a>.

Since <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M48">View MathML</a> is a step function, and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M49">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M50">View MathML</a>, it can be easily seen that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M51">View MathML</a>

(1.3)

and

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M52">View MathML</a>

(1.4)

where <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M53">View MathML</a> denote the order statistics of <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M54">View MathML</a>, and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M55">View MathML</a> is the concomitant of <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M56">View MathML</a>.

There is extensive literature on the Kaplan-Meier and the hazard estimator <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M57">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M45">View MathML</a> for censored independent observations. We refer to papers by Breslow and Crowley [2], Foldes and Rejto [3] and Gu and Lai [4]. Martingale methods for analyzing properties of <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M57">View MathML</a> are described in the monograph by Gill [5]. 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] or Wei et al.[7]. In the context of censored time series analysis, Shumway et al.[8] considered (hourly or daily) measurements of the concentration of a given substance subject to some detection limits, thus being potentially censored from the right. Ying and Wei [9], Lecoutre and Ould-Saïd [10], Cai [11] and Liang and Uña-Álvarez [12] studied the convergence of <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M57">View MathML</a> for the stationary α-mixing data.

The main purpose of this paper is to study the strong convergence rates and strong representation of the Kaplan-Meier estimator and the hazard estimator based on censored data when the survival and the censoring times form the NA (see the following definition) sequences. Under certain regularity conditions, we find strong convergence rates of the Kaplan-Meier and hazard estimator, and the expression of the Kaplan-Meier estimator and the hazard estimator as the mean of random variables, with the remainder of order <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M1">View MathML</a> a.s.

Definition Random variables <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M62">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M63">View MathML</a> are said to be negatively associated (NA) if for every pair of disjoint subsets <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M64">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M65">View MathML</a> of <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M66">View MathML</a>,

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M67">View MathML</a>

where <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M68">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M69">View MathML</a> are increasing for every variable (or decreasing for every variable) so that this covariance exists. A sequence of random variables <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M70">View MathML</a> is said to be NA if every finite subfamily is NA.

Obviously, if <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M70">View MathML</a> is a sequence of NA random variables, and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M72">View MathML</a> is a sequence of nondecreasing (or non-increasing) functions, then <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M73">View MathML</a> is also a sequence of NA random variables.

This definition was introduced by Joag-Dev and Proschan [13]. A 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 in multivariate statistical analysis and reliability theory. Because of the wide applications of NA sampling, the limit behaviors of NA random variables have received more and more attention recently. One can refer to Joag-Dev and Proschan [13] for fundamental properties, Matula [14] for the three series theorem, and Wu and Jiang [15,16] for the strong convergence.

We give two lemmas, which are helpful in proving our theorems.

Lemma 1.1 (Yang [17], Lemma 1)

Let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M70">View MathML</a>be a sequence of negatively associated random variables with zero means and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M75">View MathML</a>, a.s. (<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M76">View MathML</a>). Let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M77">View MathML</a>be such that<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M78">View MathML</a>. Then, for all<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M79">View MathML</a>,

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M80">View MathML</a>

Lemma 1.2Let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M70">View MathML</a>be a sequence of NA r.v.s with continuous d.f. F, and let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M82">View MathML</a>be the empirical d.f. based on the segments<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M83">View MathML</a>. Then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M84">View MathML</a>

Proof Similar to the proof of Lemma 4 in Yang [17], we can prove Lemma 1.2. □

Theorem 1.3Let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M26">View MathML</a>and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M86">View MathML</a>be two sequences of NA random variables. Suppose that the sequences<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M26">View MathML</a>and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M27">View MathML</a>are independent. Then, for any<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M89">View MathML</a>,

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M90">View MathML</a>

(1.5)

and

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M91">View MathML</a>

(1.6)

here and in the sequel, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M92">View MathML</a>.

For positive reals z and t, and δ taking value 0 or 1, let

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M93">View MathML</a>

(1.7)

where <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M94">View MathML</a>.

Theorem 1.4Assume that the conditions of Theorem 1.3 hold. Then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M95">View MathML</a>

(1.8)

and

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M96">View MathML</a>

(1.9)

where<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M97">View MathML</a>a.s. <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M98">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M89">View MathML</a>.

2 Proofs

Proof of Theorem 1.3 It is easy to see from Property P7 of Joag-Dev and Proschan [13] that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M100">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M101">View MathML</a> are also two sequences of NA r.v.s. Therefore

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M102">View MathML</a>

(2.1)

and

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M103">View MathML</a>

(2.2)

follow from Lemma 1.2 and the fact that both <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M104">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M105">View MathML</a> are empirical distribution functions of L and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M38">View MathML</a>.

Now, by (1.1) and (1.2), let us write

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M107">View MathML</a>

(2.3)

Therefore, by the combination of equations (2.1) and (2.2), and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M108">View MathML</a>, for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M89">View MathML</a>, we obtain

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M110">View MathML</a>

Thus, (1.5) holds.

Now we prove (1.6). By (1.3) and (1.4),

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M111">View MathML</a>

Therefore, by combining the inequality <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M112">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M113">View MathML</a>, and (2.1), for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M89">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M115">View MathML</a>, we get that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M116">View MathML</a>

(2.4)

By (1.1),(1.6) and (2.4), using the Taylor expansion, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M117">View MathML</a>, we obtain

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M118">View MathML</a>

(2.5)

Thence, the combination (1.5), (1.6) holds. This completes the proof of Theorem 1.3. □

Proof of Theorem 1.4 By (2.1),

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M119">View MathML</a>

Thus, by the combination of (2.3),

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M120">View MathML</a>

(2.6)

Noting that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M121">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M122">View MathML</a> is a step function, we get

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M123">View MathML</a>

(2.7)

Therefore, to prove (1.8), it suffices to prove that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M124">View MathML</a> for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M125">View MathML</a>. Let us divide the interval <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M126">View MathML</a> into subintervals <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M127">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M128">View MathML</a>, where <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M129">View MathML</a>, and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M130">View MathML</a> are such that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M131">View MathML</a>. For <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M115">View MathML</a>, it is easy to check that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M133">View MathML</a>

(2.8)

To estimate <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M134">View MathML</a>, we further subdivide each <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M127">View MathML</a> into subintervals <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M136">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M137">View MathML</a>, where <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M138">View MathML</a> such that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M139">View MathML</a> uniformly in i, j. Now, by (2.1) and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M140">View MathML</a>, for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M141">View MathML</a>, it follows that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M142">View MathML</a>

(2.9)

For <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M143">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M144">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M145">View MathML</a>, let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M146">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M147">View MathML</a>. Then <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M148">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M149">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M150">View MathML</a> are NA sequences with <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M151">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M152">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M153">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M154">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M155">View MathML</a>.

Taking <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M156">View MathML</a> in Lemma 1.1, yields the following probability bound:

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M157">View MathML</a>

Using the bound and the Borel-Cantelli lemma, we deduce that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M158">View MathML</a> a.s. The estimation of <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M159">View MathML</a> is similar noting that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/340/mathml/M160">View MathML</a> for all x and y. Therefore, by (2.6)-(2.9), (1.8) holds. (1.9) follows from (2.5) and (1.8). □

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

QW conceived of the study and drafted, complete the manuscript. PC participated in the discussion of the manuscript. QW and PC read and approved the final manuscript.

Authors’ information

Qunying Wu, Professor, Doctor, working in the field of probability and statistics.

Acknowledgements

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, 2013GXNSFDA019001).

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