Asymptotic Properties For A Class Of Functionals Statistic Under Φ-Mixing Saples

The empirical likelihood method as a nonparametric technique for construction confidence regions in the nonparametric setting was introduced by Owen.It has been studied extensively by Chen,Chen and Hall,Qin and Lawless,Kitmura,Yong Song Chen,Cun Suet al , and many attractive properties with the method have been found . Chen and Qinshowed the empirical likelihood method can be naturally applied to make more accurate statistical inference in finite population estimating problems by employing auxiliary information efficiently. Chen et alemployed the empirical likelihood to formulate a test statistic that measures the goodness of fit of a time series, and found that the empirical likelihood formulation of the test has two attractive features. one is its automatic consideration of the variation that is associated with the nonparametric fit due to empirical likelihood's ability to Studentize internally. The other is that the asymptotic distribution of the test statistic is free of unknown parameters, avoiding plug-in estimation. Therefore the empirical likelihood not only has being attached importance to theoretics , but also has being used widely in applied Stat..In this paper we will use the empirical likelihood to study three parts hereinafter under a strongly stationary and mixing samples, and establish some asymptotic results.Asymptotic properties of empirical likelihood estimators for quantiles under mixing samplesletbe a strong stationary and mixing sequence(refer to Linfor the definition) with <1.for j≥1,we use F()andto denote the distribution functions of and,respectively .Assume that the q-th quantile ,,is uniquely defined .for ,assume that the density functions of 和 exist and denoted as and .We wish to construct a confidence interval for using blockwise empirical likelihood method.At first, we consider the ordinary empirical likelihood ratio for .For some let be a Borel measurable function satisfying (1.1) let ,.put.with same reason as in Chen and Hall[9],the ordinary empirical likelihood ratio statistic for iswhere is determined by (1.2)We need some assumptions.Assumption A:(1)For some exist in a neighborhood of and continuous at ,where is the derivative of of order .for has join partial derivatives up to r in a neighborhood of and continuous at .(2) satisfies(1.1)and is bounded and compactly supported.,as ,(3)<1,is non-increasing,0,then , as As we do not know,the above result could not be used in practice .we will use the blockwise empirical likelihood to overcoming of the ordinary empirical likelihood.Let <α≤1/3,ɡ= Putɡ.We consider the following blockwise empirical likelihood ratio :.It is easy to obtain the (log) blockwise empirical likelihood ratio statistic : , (1.3)where is determined by (1.4)Theorem 1.2 Suppose that Assumption A is true and that >0,then, as .2. Asymptotic properties of empirical likelihood estimators for conditional quantiles with auxiliary information under mixing samplesLet be a strong stationary and mixing sequence from a population .Note whereand are the jth component of X and,respectively .For given Define the qth quantile of Y given as .It is assumed that some auxiliary information about the conditional distrbution function givenis available in sense that there exist known functions such that (2.0) where is a dimensional vector .This model is of interest in many circumstances where only some partial information about the conditional distribution of the population is known.We assume that is a strong stationary,mixing sequence with For ,assume that the density functions o...