Penalized High Dimensional Empirical Likelihood For Single-index Models

Single index model is a special kind of generalized semi-parametric model and is an effec-tive tool for dealing with the problem of multivariate nonparametric regression, and is widely used.Nearly two decades, variable selection of high-dimensional data has become one of the hot areas of statistics and related research.In dealing with high-dimensional data, the droping Witt of single-index model not only effectively avoid the "curse of dimensionality" problems, but also to seize the properties of sparse high-dimensional data.The use of a single index model dis-cussed article about variable selection method after another, but most are for parameter p is the dimension of the fixed time.However, in many high-dimensional variable selection, the parameter dimension p generally increases as the sample size n, while increases.Therefore, in this paper, we propose a single index model robust variable selection methods:Based on the SCAD(Smoothly Clipped Absolute Deviation)penalty function and empirical likelihood of punishment empirical likelihood.Under certain regularity conditions, we found that the parameter p dimension increases with the sample size n increases punishing experience likelihood estimation Oracle still has a na-ture, that is, if the true sparse model, then the probability tends to1, the experience seems to punish non-zero parameters of the model to determine the estimated contingent with sparsity.Based on the results of previous research has been on the basis of a comprehensive analysis of the single index model, mainly on high-dimensional single-index model parameter estimation and testing problems.We mainly with Fan&Peng (2004) the likelihood of punishment ideas and Hjort (2009), Chen (2009) empirical likelihood thought, for a single high-dimensional index model proposed punishment empirical likelihood method.Theoretical proof and simulation results show that the variables in the selection of a single index model processing and inspection issues, em-pirical likelihood method of punishment than the traditional single empirical likelihood method is more simple and effective.From a practical standpoint, the use of punishment empirical likelihood method can be cost-effective, more practical, with high promotional value.This thesis is mainly reflected in the following characteristics:1.of regrouping existing methods, learn from each other and improve the estimation accuracy and broaden the range of applications;2.punitive empirical likelihood method avoids the use of separate punishment likelihood or empirical likelihood of some shortcomings. Punishment likelihood requires proper distribution is assumed, and the likelihood of punishment experienced only need to satisfy some moment con-ditions.As we all know, the distribution of moment conditions more robust than assumed; addi-tion, the experience seems to penalize natural methods of statistical inference problem not under estimate the variance parameters, so that the statistical inference therefore easier to punish the experience more meaningful likelihood of statistical inference.This paper conclusions can enrich and improve the likelihood of punishment empirical theory for the practical application of workers to provide simple and feasible tool.