| Let Xn,n 1 be a sample sequence with identical distribution ,having unknown probability density function fn(x). based on the sample Xt, ,Xn,the recursive density estimator for f(x) thatwas introduced by Wolverton and Wagner in 1969,it can be calculated recursively,i.e.From the practical point of view,the recursive density estimator has a clear advantage over non-recursive estimators in that it can be updated with each additional observation,in contrast to the nonrecursive case ofwhere the estimator must be entirely recomputed. Since it is more convenient to use the recursive density estimator fn(x),there has been a significant number of research papers on this subject matter. The most relevant papers are these by Wolverton and Wagner(1969),and Weg-man and Davis (1979) for independent samples those are very ideal;Masry(1986,1987) discussed both quadratic-mean consistency and asymptotic normality for dependent sample and asymptotically uncorrelated;Professor Cai Zongwu(1990,1992) discussed both rate of strong convergence and asymptotic normality for (a, 3)-dependent and p-dependent;Professor Yang Shanchao (1997) further discussed rate of strong convergence for -dependent,mended the result of Masry's. In a snapshot,one may say that negative association and positive association occur often in cetain reliability theory problems,filter theory problems,and multianalysis theory problems.Many more applications are to be anticipated in a host of other areas and ,in particular,those areas where spatial statistics play an important role,such area are ,for example,analysis of agricultural field experiments,geostatistical annalysis,image annalysis,oceanographic applications,signal processing in radar and sonars,and stereology. so it is more important to study asymptotic propertis of density estimation for positive and negative association,arousing our considerable regard. For example,Professor Wei Laiseng (2001),Professor Yang Shanchao (2000) studied the density estimation n(z) on consistency and asymptotic normality for negative association. However,up tothe present,there is no paper discussed asymptotic propertis of the recursive kernel estimate of probability density function under negative association and positive association. In this paper,the author investigates some asymptotic propertis of fn(x) under negative association that includes the consistency in r-order mean,the strong consistency and the asymptotic normality under certain conditions for the kernel function K(x) ,the unknown density function f(x) and the bandwidth hn. As applications in reliability theory problems,the hazard rate and survial function,will be estimated byand present the strongly consistency of rn (x) . The following is our main assumption and results 1 The main assumption(Al) The r.v.s X ,X2, ,are nagetive association sequence with identical distribution,satisfing (i) If f(x,y,k) is the joint p.d.f. of the r.v.s X ,Xi+k,then(iii) K(x) have bounded derivative,and(iv) There exists positive,increase constant2 The main resultTheorem 2.1 Suppose the assumption of (Al) are fulfilled ,if x and y is arbitrary continuity of (x),thenThereinto H(x,y) =Theorems. 1 Lct{Xn,n > 2} be stationary NA sequence with identical distribution,satisfiug assumption of (A2)(i) ,further ifTheorem4.1 Let {Xn,n > 2} be NA sequence with identical distribution,satisfing the assumption of Al(ii) and A2,further,if K(u) isn't increase or decrease,and exist a > 0,satisfingat each point x of continuity f(x).Theorem 5.1 Let{Xn,n > 1} be stationary NA sequence with identical distribution,in the condition of(Al),and let pn,qn,pn + qn < n and kn = ] exist,fulfillingCorollary5.1 Suppose all the assumption of the theorems. 1 and (A2) are fulfilled ,and,in addition,the second-order derivative (x) exists and is bounded ,and let ThenTheorem 6.1 Let {Xn,n > 2} be stationary NA sequence with identical distribution,satisfing assumption of theorem 4.1,then for all x c( (z)),we have...
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