Consistency Of Density Kernel Estimation Under NA Or PA Sample

In recent years various methods of estimation about probability density function of random variable sequence that is independent and identically distributed (abbreviated as iid.) and large sample quality have been more discussed in research documentation. Most of those are kernel estimation of density function,such as Rosenblent,Parzen,PrakasaRao and Silverman and so on.However,random variables(abbreviated as r.v.)often aren't iid but definite consistency in the reliability theory,permeation theory and some analysis questions in several elements.Negative association(abbreviated as NA)and position association(abbreviated as PA) r.v. are two species commonly seen.So the research problems about density function estimation of NA and PA r.v. sequence and large sample quality are posed.In our country or allien there are many experts who discusses some quanlity of NA variable in great.For example,in documentation 6,there is a discussion about the consistency of density kernel estimation and many results have been gained,but the density estimation f(x) is limited to the arrange of x [a,6]which limits the arrange of application.Thus in this passage we call off this limitation and further discuss the density kernel estimation under NA sample.Consequently the counterpart results have been gained and the condition of result has been weaked.In the same time we discuss the density kernel estimation under PA sample and the counterpart result has been gained.Our main results are as following :Theorem2.1 Let be sequence of id.NA r.v,and Let K(u) be a bounded vaeiationprobability density function,If lim,then for any continuityx of f(x), ve getTheorem2.2 Let A,X-2, ,Xn be sequence of id.PA r.v,and let K(u) be a bounded variation densisty function. If lim = 0,then we getTheorems. 1 Let be sequence of id.PA r.v,and let K(u) be a bounded densistyfunction whose derivative is exist and bounded,we note such that Let Theorems. 2 Let be sequence of id.PA r.v,and let AT(u) be a bounded variation function which has decomposition,where are bounded monotone calculusable function whose derivative is bounded.and limn,such that and r + logn > C > 0. then Theorems. 3 LetXi,A'2, ,Arn be sequence of id.PA r.v,and let K(u) be a bounded variation function ,Theorems. 4 Let be sequence of id.PA r.v,and let . be a bounded variation density function which has decomposition ,where Ki(u),K2(u) are bounded monotone calculusable function whose derivative is bounded,If then...