The Several Results Of The Negatively Associated Random Variables In Statistical Inference Research

Negatively associated random variables was introduced by Joag-Dev and Proschan in 1983. Negatively associated random variables include independent random variabls have comprehensive use in practice.Random variables X1,X2,..., Xnare said to be nagatively associated (NA)if for every pair of disjoint sudsets A1, A2of {1,2, ..., n},where f1 and f2 are increasing for every variable (or are decreasing for every variable),such that this covariance exists.Random variables sequence{X_i,i∈N} are said to be negatively associated if for any positive integer n(n≤2)the random variables X1, X2, ..., Xn are negatively associated.Joag-Dev and Proschan pointed out a number of well known negatively associated random variables ,such as permutation distribution, random sampling without replacement from a finite population, the roder statistics of independent random variables,negatively correlated normal distribution and so on.In the. recent years many scholars have researched the asymptotic normality of negatively associated random. Roussas obtained some comprehensive discuss of PA and NA randoms,for example "asymptotic normality of random fields of positive or negatively associated processes"," the Berry-Esseen bounds of distributed function smooth estimator under association","the asymptotic normality of kernel estimates of a probability density function" .The scholars in our country researched the estimates of density functions and their asymptotic properties with applications.So in this article we conclude the results and it contains five parts.The first part is about uniformly strong consistency of nonparametric distribution function estimators.The main results is as follows:Theorem 1.1 Let the distribution function F(x) is continuous and the positive sequence dn satisfies the requirements: dn→0,nd_n²/log n→∞,thenTheorem 1.2 Let K(u) be a bounded variation function in R¹ and f(x) is uniformly continued in R¹, and alsothenThe second part is about the consistencies for the kernel-type density estimation in the case of NA samples under some conditions.Firstly it is the point wise strong consistency theorem.Theorem 2.1 Let {Xn,n≤1} be a sequence of identically distributed and negatively associated (NA)variables ,and condition (C1) holds, then|f_n(x) - f(x)|→0,a.s.for any x.Theorem 2.2 If {X_n, n≥1}are a sequence of identically distributed and negatively associated (NA)variables ,and condition (C1) holds, and the density function f(x) satisfies Lipschitz condition,thenThe third part is on the asymptotic normality of the recursive kernel estimate of a probability density function under negatively associated.The recursive kernel density function estimator is defined by Wolverton and Wagner.Let {X_n,n≤1} be a sequence of identically distributed and negatively associated (NA)variables ,and f(x) is the density function of X1,then the recursive kernel density function estimator is It is very convenient estimator because we don't need calculate again when the samples add.So we have the results.It is as follows.Theorem 3.1 Let{X_n, n≥1} be a NA stationary sequence.Under the condition (A1),if there exists pn,qn,pn+qn < n, kn=[n/(pn + qn)],and it satisfiesthenThe forth part is about strong consistency of regression function estimator for negatively associated samples.Let the sequence Y1,Y2,... ,Yn be the observe values at the fixed point and satisfies the model as followswhere g(x)is a unknown function in [0,1] and the value of g(x) is zero out of [0,1].{εi} are random errors and we assume 1. Priestley and Chao point out the weighted estimator of unknown function g(x) as followswhere K(u) is measurable function and 0 < hn→0(当n→0). And the main result follows that Theorem 4.1 If the condition (a)-(b) hold and (i){εi} is NA ,and Eεi = 0, |εi|≤b,a.s.;(ii)δn/hn = o((log n)^-1)and we have d > 0 so thatthenThe fifth part is about a class of nonparametric estimator for mean residual life and vitality function under NA samples, because mean residual life function is very important in use of the projection and biology. On the basis of the n.r.v's. (X1,X2,...,Xn) the secure estimator for the mean residual life function is a basic problem and many scholars in the world researched the problem.Let X be a real-valued and not negative random variable having the distribution function F(x) and density function f(x).R(x)=1-F(x)is its residual function.For any x > 0,the mean residual life function e(x) and vitality function m(x) of X defined byThe main result is as follows.Theorem 5.1 Let R(x) is the residual function,and its estimator is Rn(x) in (5.2). Some conditions are listed below.(A1)Letr.v.'sX1,X2,...is strict stationary NA sequence with the distribution function F and bounded density function/ and the covariance structure satisfies u(1) <∞; (A2)(i)Let 0 <α=αn < n,0 <β<βn < n be real-valued sequence and both run to infinite as n→∞.(ii)Let whereμ= [n/(α+β)} thenμ(α+β)≤n,μ(α+β)/n→1,μβ/n→0,α²/n→0,thenwhere...