Empirical Likelihood For The Marginal Joint Probability Density Function Of A Negatively Associated Sample

In this paper, we study the construction of confidence intervals for the marginal joint proba-bility density function of a negatively associated (NA) sample by using the blockwise technique. It is shown that the blockwise empirical likelihood (EL) ratio statistic is asymptotically χ2-type distributed. The result is used to obtain EL-based confidence interval for the probability density function.The kernel estimation method of a probability density function (p.d.f.) was first proposed by Rosenblatt[Remarks on some nonparametric estimates of a density function[J], Ann. Math. Statist,1956,27:832-837]. Under i.i.d. samples, the properties of the kernel estimate of a p.d.f. have been studied extensively. There is also a vast literature on the estimation of a p.d.f. under dependent samples. Robinson [Nonparametric estimators for time series[J]. Time Ser. Anal,1983,4:185-197]studied the asymptotic normality of the kernel estimators of a p.d.f. under α-mixing samples. Lu [Asymptotic normality of kernel density estimators under dependence, Ann. Inst. Statist. Math,2001,53:447-468] investigated the asymptotic normality of the kernel estimators of a marginal joint p.d.f. under more general dependent samples. Lin [On the kernel estima-tion of a density for dependent sample[J], Science Bull,1983,28:709-713] studied the asymptotic normality of the kernel estimate of a p.d.f. under φ-mixing observations. Roussas [Asymptotic normality of the kernel estimate of a probability density function under association. Statist[J]. Probab. Lett,2000,50:1-12] studied the asymptotic normality of the kernel estimate of a uni-variate p.d.f. under positively associated (PA) samples and negatively associated (NA) samples. Masry [Multivariate probability density estimation for associated processes:strong consistency and rates[J], Statist. Probab. Lett,2002,58:205-219]investigated the moment consistency and strong consistency of the kernel estimators of the partial derivatives of the marginal joint p.d.f. of a PA sample. Chen [Empirical likelihood confidence intervals for nonparametric density estima-tion[J], Biometrika,1996,83:329-341] studied the asymptotic normality of the kernel estimators of a multiple p.d.f. under PA random fields. This paper focuses on constructing empirical likelihood (EL) confidence intervals for a multiple p.d.f. under NA samples. 1、Firstly, the results in this paper is a generalization of those in Qin.etc.[Empirical like-lihood for probability density functions under negatively associated samples[J]. Statist. Plann. Infer,2011,141:373-381] from one-dimensional marginal p.d.f. to multiple-dimensional marginal p.d.f, weakening applicable and more conditions.2、In studying the construction of confidence intervals(regions) for the multiple-dimensional marginal p.d.f, we use the blockwise EL method. It is shown that the blockwise empirical likeli-hood (EL) ratio statistic is asymptotically χ2-type distributed, Then we can construct confidence intervals(regions) for the multiple-dimensional marginal p.d.f. This would improve the accuracy of the EL confidence intervals.