Hypothesis Testing In Functional Data

In functional data analysis,testing problems are also very important.With the process of technology,collecting and storing large amounts of data become possible,such as gene-expression data,stock transaction data and image data.It makes sense for further model establishment and analysis if we can tell the features from different sample data are the same and establish a reasonable relationship between the variables.For example,one sample and two sample hypothesis test problems,testing the dependence between the response and the functional predictor in functional linear models.Horvath and Kokoszka(2012)conducted a series of research in view of the testing problems,which project the variables from infinite space to finite space that expanded by the functional principal components.Hence,the test statistics are constructed based on the transformed variables.The estimation and test methods can be divided into parametric and non-parametric tests,which performed good large sample property in case of good estimation of the functional principal component.As the increasing data complexity,the finite dimensional principal component(usually p less than 5)cannot fully describe the statistical characteristics of data.However,the traditional test statistics will fail in the case of high dimension,especially with small sample size(Bai and Saranadasa(1996)).in the design of data pretreatment,it usually has two forms: One is to discrete continuous variables into higher-dimensional multivariate variables,such as 1000 grid points at equal intervals,and view the number of points as the dimension of the data;the other one is to transform the continuous variables to finite variables based on Karhunen-Loeve expansion,and view the number of bases as the dimension of the data.In this paper,we use the second method.In the case of “large p,small n”,based on the U statistic,we proposed the test statistics for the problem of sample mean test and the regression operator test of functional linear models.Due to the new test statistics do not require dimensionality reduction through the functional principal components,there is no need to estimate the covariance operator directly.We prove the asymptotic normality of the new test statistics under both the null hypothesis and the local alternative hypothesis.Further,simulation studies the efficacy of test statistics in different dimensions and samples,which can show that our test statistics perform better than the current benchmark methods under high dimension situations,especially in the small or moderate sample case.There are mainly four parts in this thesis.In Chapter 1,we introduce the properties of functional principal components and review the properties of U statistics in both classical and high dimension cases.In Chapter 2,we propose a new U statistic for testing the equality of mean functions,and prove the asymptotic normality of the new test statistics under both the null hypothesis and the local alternative hypothesis.We also give the power curve under the alternative hypothesis.In Chapter 3,we propose new U statistics for testing the dependence between the response and the functional predictor in functional linear models,the response can be real and functional variable,respectively.We prove the asymptotic normality of the FLUTE statistics under both the null hypothesis and the local alternative hypothesis,and give the power curve under the alternative hypothesis.In Chapter 4,we present the conclusions of the study,point out the insufficiencies in my study and give some propositions of the further study.