Homogeneity Test Of Variance For Partial Functional Linear Regression Model

Functional data refers to the data that changes with a certain time,space or other continuous sets.They appear essentially as curves,surfaces or three-dimensional images,etc.,which are composed of functions.If we use traditional statistical methods to analyze such data,problems such as dimension curse and information loss may occur.Functional data analysis is a method of functional processing and analysis based on discretely observed statistical data.The basic idea is to process the observed data as elements in infinite dimensional function space,so above problems can be avoided.The functional linear model is an important basis for exploring and analyzing functional data,and the partial functional linear regression model discusses the relationship between a scalar response variable and mixed predictive variable.This thesis further extends the model by combining the heterogeneity of variance.Compared with models with homogeneous variances,heterogeneous models are more complex in structure and include the honogeneous model as a special case.Therefore,it is of great importance and practical significance to study whether the hetergeneous model can be simplified into a homogeneous one.In this thesis,the homogeneity test of variance for partial functional linear regression models is studied.Firstly,the error variance of the heterogeneous model is estimated according to the residual squares of the partial functional linear regression model whose variance is homogeneous.Then the parameters of the heterogeneous model are estimated by the generalized least squares estimation method.The asymptotic properties of the estimators are given,and the consistency of the estimators is verified.Secondly,according to the sums of the square residuals of the null model and the full model,the test statistics are constructed.The asymptotic properties of the test statistic under the null hypothesis and the alternative hypothesis are deduced and a bootstrap test algorithm is given.Finally,numerical simulation is used to evaluate the performance of the proposed test for finite sample sizes.