Tuning Parameter Selection Using ERIC Criterion In The Generalized Linear Model

With the continuous development of the era of big data,the application of linear model is more general.The generalized linear model shows great advantages and has great development prospect and research value.It is generalized of linear model.And it is a very important and widely used model in statistical analysis.The dependent variable's distribution in generalized linear model only needs to follow the exponential family distribution,which makes the generalized linear model fit and model for various types of data.It includes not only continuous variables but also discrete variables.Besides it also includes large skewed distribution variables and symmetrical variables.At the same time,the generalized linear model can convert the non-linear relationship between the dependent variables and independent variables into the linear relationship through the link function to deal conveniently with complicated nonlinear relationship.Therefore,this paper studies the tuning parameter selection of penalty likelihood function based on the generalized linear model.In recent years,it has paid considerable attention to use the penalized likelihood function to continuously select variables and estimate unknown parameters.Variable selection is the basic method to deal with high-dimensional statistical model.In variable selection of the regression model,SCAD is a good kind of penalty function.It not only is a good way to choose the true model,but also is used to estimate the parameters,at the same time,it also has oracle property.The researchers will face two major challenges to apply the penalty likelihood function in regression analysis.The first difficulty is to compute the nonconcave penalty likelihood estimate.This problem has been detailedly studied in the recent literature,such as the local quadratic approximation(LQA)algorithm and least angle regression(LARS)algorithm etc..In addition,LARS also can be adopted to solve the optimization problem of nonconcave penalized likelihood functions with the help of the local linear approximation(LLA)algorithm.However,the above calculative process depends heavily on the correct selection of the tuning parameters.Therefore,the choice of this tuning parameter will be the second biggest challenge for researchers and it is the main part of this paper.These good properties of the SCAD penalty are based on the selection of an appropriate tuning parameter.Only when the selection of the tuning parameter is appropriate,can SCAD penalty get a good estimated result.In this paper,we propose the extended regularized information criterion(ERIC)for choosing the tuning parameters based on SCAD penalty function in the generalized linear model.At the same time,we prove that the model selected by the criterion is consistent in certain conditions.The experimental simulation and empirical analysis show that ERIC criterion is more advantageous than the CV criterion and BIC information criteria in choosing the true model.And the results also support the theoretical property of the criterion.