Hypothesis Testing For Several Functional Models

As an important research topic in statistics,the hypothesis testing of functional data analysis has been proposed in the literature.Many effective testing methods have been proposed.Although these methods have good performance when the number of principal components p is limited,but it will fail when p is infinite.Therefore,it is of great significance to comprehensively discuss the testing of complex data,both in theory and in practice.How to perform hypothesis testing when p is infinite is the main research content of this paper.Aiming at this situation,this paper studies the hypothesis testing of coefficient functions in one-sample mean function,functional linear model and functional partially linear model,and proves the asymptotic properties of the proposed tests.Finally,the proposed test is verified by numerical simulation and real data analysis.The research content mainly includes the following three parts:The first section studies the test of one sample mean function.For the function data with p infinite and covariance operator having spike eigenvalues(several eigenvalues of p eigenvalues are significantly greater than other eigenvalues),due to the shortage of sample size and the strong condition of covariance operator,the chi-square and mixed chi-square test statistics constructed by classical methods will have poor performance.Therefore,this part will conduct randomization test and prove asymptotic properties.Furthermore,the validity of the method is verified by the finite sample performance with the simulation studies,and this method is applied to Medflies data and Benchmark Phoneme data.The second section is to test the linearity between response variables and explanatory variables in functional linear models when the response variables are real data and functional data.The existing research constructed a fourth-order U-statistic with infinite p.However,this method takes a long calculation time,so this paper improves it and constructs a second-order U-statistic,which improves the running speed of the test.The asymptotic distribution of the test statistic will be proved theoretically.The validity of the method is verified by some numerical simulation studies.Finally,the method is applied to Canadian data and Diffusion Tensor Imaging data.The third part studies the test of coefficient function in partial linear model with finite p and infinite .For finite p,the test statistic is constructed based on the estimators of slope function and nonlinear part,and the asymptotic distribution of the test statistic is proved under some regularity conditions.For infinite p,the estimation of the unknown part may be infeasible due to more projection directions and less sample size.Therefore,similar as the test procedure constructed for testing the parametric coefficient vector in high-dimensional partial linear model,the second-order U-statistic is constructed based on the pseudo estimate of the nonlinear part without the estimators of covariance operator and coefficient function.The asymptotic distribution of the test statistic is proved under the null hypothesis and local alternative hypothesis.Finally,the proposed test methods in these two cases are assessed by simulation studies and real data analysis.