Author: Liu, Changyu
Title: High-dimensional tests based on random projection approach
Advisors: Zhao, Xingqiu (AMA)
Huang, Jian (AMA)
Degree: Ph.D.
Year: 2021
Subject: Statistical hypothesis testing
Mathematical statistics
Hong Kong Polytechnic University -- Dissertations
Department: Department of Applied Mathematics
Pages: xv, 139 pages : color illustrations
Language: English
Abstract: In this thesis, we consider hypothesis testing in high-dimensional models, where the dimension of covariates p is greater than the sample size n, which is common in data analysis currently. Novel test statistics are proposed in both linear model and single-index model for high-dimensional settings. First, we focus on the simple linear model and propose a novel test statistic for the problem of testing global regression coefficients. The proposed test is constructed based on the technique of random projection. Concretely, we first randomly project high-dimensional data into a lower-dimensional space and then apply the projected data to the classical F-test. The proposed test has a simple form and intuitive interpretation. The advantages of this random-projection-based approach are demonstrated both theoretically and numerically. Under mild conditions, we derive the asymptotic normality and the asymptotic local power functions of the proposed test. By comparison with some recent developed methods, our proposed test shows higher asymptotic relative efficiency in a sufficient condition. The proposed method is further extended to the problems of testing partial regression coefficients and we derive its asymptotic properties. Through simulation studies, we evaluate the finite-sample performances of the proposed tests and demonstrate its superior performance than the competing tests. Applications to real high-dimensional gene expression data are also provided for illustration.
Next, we investigate the single-index model, which includes many commonly used models. First, we study the feasibility of applying the classical F-test to a single-index model where p/n → ζ ϵ (0, 1). We derive its asymptotic null distribution and asymptotic local power function. For the ultrahigh-dimensional single-index model where p >> n, we construct F-statistics based on lower-dimensional random projections of the data. For the hypothesis testing of global and partial parameters in the p > n settings, the asymptotic null distribution and the asymptotic local power function of the proposed test statistics are analyzed. The newly proposed test possesses the advantages of intuitive interpretation and simplified computation. We compare the proposed test with other high-dimensional tests and show our test is more efficient in a sufficient condition. We conduct simulation studies to evaluate the finite-sample performances of the proposed tests and demonstrate that it has higher power than some existing methods in the models we consider. The application of real high-dimensional gene expression data is also provided to illustrate the effectiveness of the method. Overall, we propose new tests applicable to general models in high-dimensional settings. The proposed tests are easy to implement and would present a reasonable performance under mild conditions. For the testing power, it is shown to hold for a wide range of alternatives and possess certain advantages in sparse cases. As a result, our proposed method provides a practicable choice for hypothesis testing in modern data analysis.
Rights: All rights reserved
Access: open access

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