Empirical Likelihood For Generalized Linear Models

The theory of the generalized linear models (GLMs), which was introduced by Nelder and Wedderburm (1972), is an important extension of classical theory of linear models and is widely used in statistical analysis.Empirical likelihood (EL), which was introduced by Owen (1988,1990,1991), is a nonparametric method of inference based on a data-driven likelihood ratio func-tion. Similar to the bootstrap and jackknife.empirical likelihood inference need not to specify a family of distributions for data.Empirical likelihood has a lot of obvious advantages over classical statistical methods. For example, EL makes an automatic determination of the shape of confidence regions, straightforwardly incorporates side information expressed through constraints or prior distributions, can be extended to biased sampling and censored data, and has very favorable asymptotic power proper-ties.Yan and Chen (2013) extend the empirical likelihood inference to the generalized linear models with the known construction of first moment of response variable. Based on the estimation equation with quasi-natural link function, two kinds of empirical log-likelihood ratio functions for unknown parameter are constructed in generalized linear models with quasi-natural link function for fixed designs and adaptive designs respectively. Furthermore, asymptotic chi-square distribution are discussed for the above empirical likelihood ratios and the corresponding confidence regions for the unknown parameter are also constructed.In some practical problem, the construction of mean and variance is usually known, at this case, the loss of information can be caused if the mean construction is only used in establishing model. In this paper,with the known construction of mean and variance, the empirical likelihood methods are considered for the generalized lin-ear models with fixed designs and adaptive designs. Based on the estimation equation with non-natural link function, this paper discuss empirical likelihood ratio statistics for two designs, and under certain conditions, the properties of asymptotic chi-square distribution are obtained for two empirical likelihood ratio statistics, and then based on above results, the confidential interval for unknown parameter are constructed. Monte Carlo simulations show that, under the condition of known mean and variance con-struction, the methods in this paper are better than those in Yan and Chen (2003), in which only the mean construction is used in analysis.