Asymptotic Characteristic Of Nonparametric Estimation Under Negatively Associated Samples

Nonparametic satistics is one of the most important branch of statistics. For the fixed-design model.it has been studied by many schollars under independent case. For example,Priestly and Chao[l],Gass and MuIer[2],Ahmadalin[3]and so ori.But we have not found any result under Negatively Associated Samples. In this paper,we established the asymptotic nomality of the weighted function estimate of the fixed edsign regression under weak conditions. Using this result,we further disscuss the Gasser-Muller estimate.Priestley-Chao estimate and obtain the corresponding results. In the paper[5]. strong consistency of neighbor estimator has been proved under i.i.d. (independently identical distribution) samples.In this paper the same result is obtained under one dimension and NA(negatively associated) samples.At first,suppose function g and random errors n,in model(1.1.1) satisfying:(.4) (i) g :A R is a bounded function defined on the conpact subset .4 of Rd;(ii) For all. the joint distribution of is the same as that of ,where the later is strictly stationary negatively associated randon variables defined on the probability space (n.fi.P)(iii) (iv) p := p(n),q := q(n) integers with p + q < n,and u(q)Rn for all sufficiently large n.in this result.1=1is satisfied.(B1) The design points xni,i - 1, n arc chosen so thatn. for some positive constants d,62 > 0(D'2) The bounded pdf K(x) is countinuous almost everywhere in (-00,-foe) and has a majorantH(x);that is ,K(x) < J (x),where H is bounded,symmetric,nonincreasing in [0,+00).(Bi) The design points xn,are chosen scj that for some positive constsnts C ,C2 > 0(D,) Suppose there is a boundecl,symmetric function (i). nonincreasing in [0,+00) ,K(x) 1) andthen(1.1.5)is satisfied.Then (1.1.5) holds.-Theorem 1.3-1 Suppose AI AS At BI B2 are satisfied and for some e > 0 ,thenis satisfied.Theorm 1.3.2 suppose A A A+ are satisfied ,g satisfied a local Lipschitzis satisfied.Theorm 1.3.3is right.Theorm 1.3.4 Suppose the condition of lemmal.3.4 and A4 are satisfied,and are also satisfied ,thenis right.Theorru 2.1.1 Suppose are random variables .each having the same distribution as random variables ,Wni is nearest neightbor weight which is expressed by (2.1.4) ,and5' is a bounded function. Regression function m(-) which is defined on the support subset of distribution function F is continuous. (support subset is defined as follow:F(5ir) > o is right for any then we have the follow result.