Empirical Likelihood Methods For High Dimensional Linear Regression Models

High dimensional data is becoming more and more prevalent in many areas. The study of high dimensional data is now faced with many difficulties and challenges, but it is also the area where we can make progress. Multivariate statistical analysis methods in solving high dimensional data may come across such situations, for ex-ample, the data itself is not from normal distribution or it does not own enough prior information. Thus, nonparametric approaches are considered in general. Besides, the method based on traditional empirical likelihood (EL) has been shown as a power-ful one, because it owns both the effectiveness of parametric likelihood methods and the reliability of nonparametric approaches. It is also more precise than normal ap-proximation under many cases, especially when the data does not come from normal distribution or the estimate of variance is not stable; it also owns sampling properties like the bootstrap method, and its calculation method is simple. What’s more, the nonparametric version of Wilks’theorem holds true under some regularity conditions.In this paper, our focus is to test whether the regression coefficients in a possi-bly high dimensional linear model are equivalent to given values. By transforming the high dimensional estimating equation used in traditional EL approach into low dimen-sional case tactfully, we propose our novel but simple method which not only retains the optimal properties in traditional EL method but also owns other exciting results. We need to compute the maximum value limited by estimating equations when solv-ing profile EL function, which is the key step of EL method. To ensure type I errors to be closer to the given nominal levels, we add a pseudo-observation in our method further to get relative test statistics. Simulations have also been conducted to assess the performance of the proposed methods in several different models. Moreover, be-cause of the particularity of the regression coefficients, we come up with another EL method which is suitable for this situation. Simulation results are also presented.Our main innovation points are as follows:(1)We try to transform the high dimensional estimating equation used in tradi-tional EL approach into low dimensional case tactfully so that new estimating equa-tion which is not related to p can be constructed, and then relative test problems could be solved by this EL method.(2)Pseudo-observations have been added into both our method and the method proposed by predecessors so that a novel adjustment is gained. The adjusted method reserves all the properties. Moreover, the coverage ratios are closer to confidence levels, and its calculation procedure is not tedious.(3)For the general regression coefficients, we weight the components of the orig-inal estimating equation, and then add them. Thus, hypothesis testing under this case is solved.(4)Different numbers of estimating equations are used in different dimensions, which makes us satisfied in terms of both type Ⅰ errors and type Ⅰ errors, and the computational expense is saved to a large extent at the same time.