Penalized High Dimensional Empirical Likelihood For Generalized Linear Models

The empirical likelihood method, introduced by Owen[23], is one of the most importantstatistical inference methods. And it has many advantages. This paper considers the problemsof variable selection and estimation in generalized linear models via the penalized empiricallikelihood (PEL) method. In the generalized linear models, we select an appropriate penaltyfunction combined with the empirical likelihood method for statistical inference on theregression coefficients. Furthermore, we show that PEL has the oracle property.The paper consists of the following four chapters:Chapter1is mainly focused on introducing the form of generalized linear model, thedefinition and two theorem of empirical likelihood method, the form of kinds of penaltyfunctions, for high dimensionality data, the empirical likelihood method is still applied, andintroduced the variable selection method.In chapter2, Give the empirical likelihood function of β:The penalized empirical likelihood for parameter estimation and variable selection forproblems with diverging numbers of parameters is proposed. By using an appropriate penaltyfunction, we show that PEL has the oracle property. Our results are demonstrated regressioncoefficients in generalized linear models. That is, with probability tending to one, penalizedempirical likelihood identifies the model and estimates the nonzero coefficients as efficientlyas if the sparsity of the true model were known in advance. The advantage of penalizedempirical likelihood is illustrated in testing hypothesis and constructing confidence sets.In chapter3, we give the computation of the SCAD as well as the choice of the tuningparameter. and numerical simulations confirm our theoretical findings.The proof of main results is given in chapter4. Firstly, to prove main conclusion, weestablish several lemmas which are of significance on their own right, The estimationtechniques Lagrange multiplier method, central-limit theorem play an important role inderiving our results, which show the important position and role in the statistical inference.