Testing Independence And Condi-tional Expectation Independence Of Functional Data

With the advancement of measuring instruments and storage technology,the ob-served data are becoming more and more dense and often have characteristics such as trajectories or images,which can be viewed as functional data.These data can be encountered in many fields,such as medicine,economics,and meteorology.Functional data differ from vector data in the sense that in addition to the overall statistical charac-teristics exhibited by the repeated observation samples,there is also a representation of trajectories.Ignoring the orbital properties may make the whole study inefficient.It is worthwhile to study the hypothesis testing problems,such as independence,goodness-of-fit,and conditional expectation independence tests,by using these characteristics of functional data.This dissertation focuses on the independence test and conditional expectation independence test for functional data.The details are as follows:(1)Give a general method for testing the independence of functional data.For vector data,there are many excellent methods for independence testing,but these methods are often based on the framework of Euclidean space.Since functional data are often of infinite-dimension,these independence tests for vector data may not be directly applicable to functional data.For this reason,the dissertation makes use of the basis expansion method to approximate the functional data by the vector data,so that the independence tests applicable to the vector data can be applied.In addition,the procedure of applying the projection covariance and the corresponding theoretical results are given in the dissertation.A simulation study shows that the independence test methods applicable to vector type data can be easily applied to functional data by using the proposed method.(2)Although in(1),the test methods for vector data are applied to functional data,many of the independence tests for vector data are based on Euclidean space and do not take into account the characteristics of functional data.Therefore,many tests are not very efficient.In this dissertation,we propose a metric for measuring correlation based on functional characteristics—angle covariance.The angle covariance has zero equivalence,i.e.,it is equal to zero if and only if the corresponding random variables are independent.Based on the angle covariance,the corresponding independence test is given,and the limiting distributions are given under the null and alternative hypotheses,which implies that the test based on angle covariance is consistent against any alternative.The conducted simulations show that the new test performs better than the other compared methods for functional data.(3)In functional linear models,the predictor variables are often assumed to be independent of the errors.We investigates the joint test of goodness-of-fit of functional data linear model and independence of predictor and error for functional data.Firstly,the linear model is assumed to hold,and the corresponding parameter estimates are obtained,which in turn lead to the error estimates(residuals).Secondly,the generalized distance covariance is applied to the predictor variables and the residuals,and the test statistic is constructed and the test procedure is given.Finally,the asymptotic behavior under the null and alternative hypotheses is given.The validity of the method for finite sample is presented by simulations,and several specific applications to real data are given.(4)The test of conditional expectation independence is studied.The conditional expectation independence means that E_Y|Xis independent of.A new metric is proposed based on the kernel embedding approach.This measure has zero equivalence,i.e.,it is equal to zero if and only if E_Y|X=Ealmost surely.Then a constructed U-estimator of the measure is given,and the corresponding test procedure is given based on this estimator.The corresponding theoretical results are proved,and simulation results for finite samples and practical applications of the proposed method are given.