Empirical Likelihood For Multi-dimensional Generalized Linear Model

Generalized Linear Models (GLMs) is an extension of normal linear model,which is suitable for both continuous data and discrete data. The common statis-tic inference methods for GLMs include likelihood estimating equation and quasi-likelihood estimating equation, or the combination of estimating equation with oth-er statistical methods. Empirical likelihood method is a popular nonparametric inference method, which has great important applications in constructing confi-dence region and hypothesis testing, and empirical likelihood has many advantages over normal-approximation with variance estimation method and bootstrap without Bartlett-correctable.This paper will study the empirical likelihood method for quasi-likelihood esti-mating equation with working correlation matrix for fixed design and adaptive design.Firstly, we will construct the log empirical likelihood ratio, and when the minimum eigenvalue of the Fisher information matrix converges to the infinity (which is the weakest assumption in the study of asymptotic theory for GLMs) and some other reg-ular conditions, we prove that the log-empirical likelihood ratio at the true parameter converges to the standard chi-square distribution. In the simulation, we set the sam-ple size 50,100,200,500 and confidence level 0.95 and select 4 different correlation matrices to compare: true correlation matrix, estimated correlation matrix ,identity correlation matrix , the correlation matrix for natural link function, which are in de-scending order according to the simulation results. In particular, with the sample size increasing, the constructed matrix is approaching to the true correlation matrix.