Empirical Lq Likelihood Method In Single-Index Models For Longitudinal Data

Single-index models is an important generalized regression models in statistics. Longitudinal data combines characteristics of cross-section data and time-series data, so longitudinal data can analyze the change of individuals and variation among individuals effectively, it has become one of the hot topics in the field of statistics research recently. The empirical Lq likelihood is a result of Box-Cox transformation theory application into empirical likelihood. When q→1, empirical Lq likelihood approaches empirical likelihood, in this sense, the empirical Lq likelihood extends the classic method, namely, empirical likelihood method of Qin and Lawless. When the sample size is large, the statistical properties of empirical Lq likelihood are consistent with the empirical likelihood.In this paper, we mainly study the adjusted empirical Lq likelihood method in single-index models for longitudinal data, based on the research results of predecessors. With the Lq like-lihood estimation thinking of Ferrari and Yang(2010) and empirical likelihood idea of Qin and Lawless(1994), For research method of the single-index models for longitudinal data, this paper extends empirical likelihood estimate to the empirical Lq likelihood estimate. But considering the problem that the proposed empirical Lq likelihood ratio statistics is asymptotically a weighted sum of chi-square distribution with unknown weights, needed to estimate weights, the amount of computations and the estimation precision are affected. So we propose the adjusted empirical Lq likelihood method, and the adjusted empirical Lq likelihood ratio statistics is asymptotically standard chi-square without weights so that the computation amount is reduced and the accuracy of estimation has been improved.We firstly discuss the confidence regions using the empirical Lq likelihood and the adjusted empirical Lq likelihood based on the single-index models for longitudinal data. Secondly, we study the limits properties of the estimations. In addition, a simulation study is conducted to com-pare the adjusted empirical Lq likelihood method with the adjusted empirical likelihood method. In theory, we find that it has same properties between adjusted empirical Lq likelihood and ad-justed empirical likelihood. The simulation shows that: Compared with the adjusted empirical likelihood estimation, when q is taken a suitable value, the higher coverage probabilities of the new confidence intervals can be obtained in most cases especially for small sample size. From a practical standpoint, the adjusted empirical Lq likelihood method can reduce cost and be practical, these results have important significance for the application workers.The major achievements and innovations of this paper can be summarized as follows:l.For research method of the single index models for longitudinal data, this paper extends empirical likelihood to the empirical Lq likelihood and gets to difference functions, the empirical Lq likelihood ratio statistics and the adjusted empirical Lq likelihood ratio statistics are suggested and proved the limits properties. The predecessors don’t discuss this part before.2.0n the choice of q, when q is taken a suitable value, the higher coverage probabilities of the new confidence intervals can be obtained in this paper compared with previous.3.The conclusions of this paper can be used to enrich and improve the research method of the single-index models for longitudinal data theoretical approaches, providing a broader perspective, a more simple and feasible tool for the practical application of workers.