Empirical Lq Likelihood

Likelihood method which is very important and quite a wide range of application is one of the parametric statistical methods. Empirical likelihood is the promotion from the parametric method to non-parametric. The confidence interval constructed by the method of empirical likelihood has a lot of advantages,such as transformation invariance and the domain to maintain. Therefore, this method is widely applied to various fields. Lq likelihood method is used to a more general function instead of log(u) to get a wider class of likelihood. In fact, it is a promotion of the parameters likelihood and is higher accuracy than the likelihood method under certain conditions. After Da-vide and Yang (2010),some scholars get the attention of it, such as Qin and Priebe (2011). The existing literature is focused on the parameters of the situation, there is no paper involved the non-parametric.In real life we often meet the problem of the non-parameter. So the paper will extend Lq likelihood to the non-parametric, and discusses its nature.With the Lq likelihood estimation thinking of Davide and Yang (2010) and empirical like-lihood idea of Qin and Lawless(1994),this paper constructs empirical Lq likelihood function and its corresponding parameters estimated by empirical Lq likelihood. On this basis, we discuss the estimate of the nature of the large sample asymptotic distribution, the construction of confidence intervals. We discuss the existence of the estimates and asymptotic normality.We use asymptotic properties of the test to construct confidence intervals and so on. Finally, this paper simulates the parameter estimation methods discussed earlier to compare. Simulation results show that:When q is taken a fixed value, the coverage of the confidence interval increases as the sample size increasing. When the value of q is nearby1, the effect is the best,and is closer and closer to nominal coverage level of95%.The estimates obtained is unbiased estimator, similar to the empirical likelihood.The major achievements and innovations of this paper can be summarized as follows:l.This paper is that empirical likelihood and parameter Lq likelihood promoted to empirical Lq likelihood,and getting to difference functions, discussing the function and the existence of the estimates and asymptotic normality.The predecessors don’t discuss this part before. 2.Under taking some of the value of q, the estimated confidence region coverage derived from the conclusions is still relatively high.3.The conclusions of this paper can be used to enrich and improve the likelihood of non-parametric theory, providing a broader perspective, a more simple and feasible tool for the practical application of workers.