Empirical Likelihood Asymptotic Theory For Multivariate Response Generalized Linear Models

Generalized linear models is suitable to analyze continuous or discrete response data.For the case of univariate response variables,the likelihood and quasi-likelihood methods can infer effectively when the mean and variance functions are specified correctly.But for multivariate response variables,besides marginal mean and variance functions,we also need to consider the correlation structure modeling of response variables.Empirical likelihood is an effective non-parametric statistical inference method.Compared with the classical or modern statistical methods,empirical likelihood has many advantages,such as invariance of the confidence interval,automatic determination of the shape of the confidence region by the data,Bartlett correctability and no need to construct pivot statistics.In this paper,under correlation matrix modeling,we will study asymptotic theories of empirical likelihood in generalized linear models with multivariate responses.In this paper,firstly,we construct the quasi-likelihood estimation equation by using the correlation matrix which estimated by data and combine it with empirical likelihood to establish statistical inference for the parameters in generalized linear models.Then,under certain conditions,we obtain the following theoretical results: the log empirical likelihood ratio under true parameters converges to chi-square distribution;the consistency and asymptotic normality of the empirical likelihood estimation for unknown parameters;asymptotic properties of the test statistic of the profile empirical likelihood ratio.Finally,we use numerical simulation to verify the validity of our theoretical results.In the simulation,we consider four cases of covariance matrix modeling methods,corresponding to three kinds of correlation matrix structure.From the simulation results,it can be seen that the effect of our method is very close to the true correlation matrix.In addition,we use empirical likelihood and the generalized estimation equation to obtain the estimation of parameters,and the result shows that our method also performs well.The innovation of this paper is that it does not assume the structure of correlation matrix at all,only estimates the correlation matrix of response variables by data,and establishes a series of asymptotic results of empirical likelihood in generalized linear models with multivariate responses.However,most of the existing literatures discuss asymptotic theories of empirical likelihood based on the specific correlation structure.Obviously,the method of specifying the correlation structure in advance has the risk of bad or even wrong inference results.The correlation matrix modeling method in this paper is completely determined by data,which can effectively avoid this situation.