Hypothesis Test Of Functional Quadratic Regression

With the development of precision instruments and technology,functional data has been more and more widely used in various fields.Since the study of functional data,the focus of functional regression has been on linear models,which is the fundamental but most important model for describing the relationship between predictors and response variable.However,when the potential regression relationship is nonlinear,we may extend the functional linear model and explore more general models,such as functional quadratic regression,or more general polynomial regression.Compared with functional linear model,the quadratic model is more complex than the simple and popular linear model,[4]Horvath L,Kokoszka P.Inference for Functional Data with Applications[M].New York:but the flexibility and accuracy are higher.Therefore,whether the functional quadratic model can be reduced to a linear model is of interest and has practical significance.In this thesis,we mainly study the significance test of quadratic terms in functional quadratic regression.In Chapter 2,we first introduce the basic content of functional data analysis,mainly including the L2 space and functional principal component analysis,and then we briefly introduce the functional quadratic regression model.In Chapter 3,the functional quadratic regression model with scalar response variables is highlighted,and the test statistic is proposed for whether the quadratic regression parameter function is significant.We perform the dimensionality reduction of the stochastic process X(t)based on functional principal component analysis.The test statistic is constructed by the residuals sum of squares of the model under the null hypothesis and the alternative hypothesis.The asymptotic distribution of the test statistic under the null hypothesis is obtained.And the asymptotic behavior of the test statistics under the alternative hypothesis is also established,which indicates that the test method is consistent.In addition,we deduce the asymptotic distribution of test statistic under the local alternative hypothesis.In Chapter 4,the performance of the test under finite samples is shown by numerical simulations.Regardless of whether X(t)is a Gaussian process,the test method has good empirical sizes and high powers.Finally,we apply our test to near-infrared absorbance spectra data to verify the effectiveness of the proposed method.