Empirical Likelihood For Probability Density Functions Under Ф-mixing Samples

The φ-mixing condition was introduced by Ibragimov [Some limit theorems for stochastic processes stationary in the strict sense, Dokl. Akad. Nauk SSSR.125(1959),711-714]. and was also studied by Cogburn [Asymptotic properties of stationary sequences, Univ. Calif. Publ. Statist.3(1960),99-146]. The concept of φ-mixing has been used extensively in the time series literature as measures of weak dependence. Bradley [Basic properties of strong mixing conditions:a survey and some open questions, Probab. Surveys.2(2005),107-144] made a good survey on the φ-mixing condition as well as other commonly used mixing conditions. The convergence of the sums of φ-mixing random variables, because of its wide applications, were studied extensively by among others.The EL method to construct confidence intervals, proposed by Owen [Empirical likelihood ratio confidence intervals for a single functional, Biometrika.75(1988),237-249; Empirical likelihood ratio confidence regions, Ann. Statist.18(1990),90-120], has many advantages over its counterparts like the normal-approximation-based method and the bootstrap method. Chen [Smoothed block empirical likelihood for quantiles of weakly dependent processes. Statistica Sinica.19(2009),71-81] developed the EL method in construction of p.d.f. It should be noted that the usual EL method can be only used properly for independent data, but not for dependent data.Kitamura [Empirical likelihood methods with weakly dependent processes, Ann. Statist.25(1997),2084-2102] first proposed blockwise EL method to construct confidence intervals for parameters under mixing samples. Chen and Wong [Smoothed block empirical likelihood for quantiles of weakly dependent processes. Statistica Sinica.19(2009),71-81] developed blockwise EL method to construct confidence intervals for quantiles under φ-mixing samples. The kernel estimation principle of probability density function was first proposed by Rosenblatt [Remarks on some nonparametric estimates of a density function. Ann. Math. Statist.27(1956),832-837]. This paper discussed the construction of confidence intervals for probability density function under φ-mixing sample by means of blockwise EL method. It is shown that the asymptotic distribution of blockwise empirical likelihood ratio statistics is chi-square distribution. The result is used to obtain EL-based confidence interval for the probability density function. Finally, the performances of the EL-based confidence regions and the normal-approximation-based confidence regions were compared through simulation studies.The new findings in this paper may be summarized as follows:(1) It is the first time used blockwise EL method to construct confidence intervals for proba-bility density function underφ-mixing sample, and extended Kitamura’s conclusions.(2) There is a lack of research on the construction of the EL-based confidence intervals for p.d.f. underφ-mixing observations.(3) This paper’s results will generate to broader mixing conditions, which is left for our future study.