Variable Screening And Statistical Inference In Ultrahigh Dimensional Nonparametric Additive Models

With the rapid development of modern information technology,ultrahigh dimen-sional data are collected in many scientific fields where the predictor dimension is often much higher than the sample size.Moreover,nonlinear dependence between predictors and the response is often present in ultrahigh dimensional problems.Traditional statis-tic methods and linear model setting are unfeasible.And considering the "dimension curse" of full nonparametric model,we proposed a new variable screening method and overall significance test statistics for nonparametric additive model.To reduce the ultrahigh dimensionality effectively,many marginal screening ap-proaches and penalized regression methods are developed.However,there exist some drawbacks in these approaches.Thus,Wang(2009)proposed a novel Forward Re-gression(FR)approach for ultrahigh dimensional linear models.Motivated by the out-standing performance of FR,we further extended the FR to develop a Forward Additive Regression(FAR)method for selecting significant predictors in ultrahigh dimensional nonparametric additive models.We established the screening consistency for the FAR method and examined its finite-sample performance by Monte Carlo simulations.And we also apply the FAR method to a real data analysis in genetic studies.Our simula-tions and analysis indicate that,compared with marginal screenings,the FAR is shown to be much more effective to identify important predictors for additive models.When the predictors are highly correlated,the FAR even performs better than the iterative marginal screenings.To accommodate the latest need of modern biology industry,we proposed an over-all significance test for ultrahigh dimensional nonparametric additive model with two new test statistics Zn,p and Tn,p.We also applied refitted cross-validation to estimate the variance in the model.Actually,the newly proposed overall significance test is equal to the significance test of the regression coefficients in ultrahigh dimensional augmented linear models.And we also proposed a conditional U-type test(CUT)to enrich the ex-isting theories about the test of regression coefficients in ultra-high dimensional linear models.Moreover,the existing high-dimensional regression coefficients test and CUT test suffer from the decreasing power due to the noise from irrelevant predictors in the ultrahigh dimensional model.In order to improve the above problems effectively,we proposed the conditional U-type test with screening based on multiple splitting of the data set(CUTS).And we studied asymptotic normal null distribution in these newly proposed test statistics and examined their finite-sample performance by Monte Carlo simulations and case study.