Hypothesis Test And Estimation For Some Functional Models

With the advance of modern technology,more and more data are being recorded continuously during a time interval or collected with images.Such data that can be viewed as function is often referred to as functional data.The most significant features of these data are high-dimension and high correlation between adjacent data,which can not be treated by traditional multivariate statistics meth-ods.Whereas,functional data analysis has more advantage in dealing with these data,by viewing the sample data as an element of infinite dimension space.The field of functional data analysis has been rapid development over the last two decades.Its methods have been applied to a diverse range of subject areas in practice.These include economics,meteorology,psychology,neurophysiology,ar-chaeology,criminology,biomechanics,education,and so onThe aim of these thesis is to investigate hypothesis test and estimation for some functional data models,including functional linear quantile regression model.functional linear regression model with response missing at random,functional quadratic quantile regression model,partial functional linear quantile regression model with censored response.Specifically,the research contents of this disserta-tion have four parts as follows(1)For the functional linear quantile regression model,we focus on investigat-ing the hypothesis test of the model.That is,checking the assumption that the response is related to the functional predictor through a linear model for given quantile levels.We first propose a nonparametric U-process test statistic based on the functional principal component analysis.Under some regularity conditions,it is proved that the test statistic follows a normal distribution asymptotically under the null hypothesis and diverges to infinity for any misspecified models.Moreover,it is shown that the asymptotic properties of test statistic under local alternative hypothesis.Furthermore,we construct a Wild bootstrap procedure to determine the critical value.The finite sample properties of the test statistic are illustrated through extensive simulation studies.Finally,a real data set of ozone levels is analyzed by the proposed test(2)For functional quadratic quantile regression model,we consider the prob-lem of estimation of model parameter and checking of effects of quadratic term Firstly,the functional coefficients are estimated by functional principal compo-nents.The asymptotic behavior of the resulting estimators are established under some regularity conditions.Furthermore,we construct a rank score test procedure to test the significance of the nonlinear term in the model.The asymptotic dis-tribution of the proposed test statistic under the null hypothesis are established Finally,we examine the performance of the proposed method for finite sample sizes by Monte Carlo simulation studies and an illustrative example of Tecator data.(3)For functional linear regression model with response missing at random,we consider the problem of hypothesis test for model.Firstly,a U-process type test statistic is proposed by combining functional principle component analysis and nonparametric estimation techniques.The limiting behavior of the test statistic under the null and local alternative hypotheses are investigated.For determining the critical values easier in practice,a Wild bootstrap procedure is proposed Finally,the finite sample properties and usefulness of the test are illustrated in a simulation study and a real data analysis of Spanish weather(4)For partial functional linear quantile regression model with censored re-sponse,we focus on the problem of estimation of unknown coefficients of model We first use functional principle component basis to approximate the unknown slope functional coefficient.Then,the estimators of unknown coefficient are ob-tained by minimizing inverse probability weighted loss function.It is easy to implement via existing weighted quantile regression procedure.Under some regu-larity conditions,the asymptotic normality of the estimator of finite-dimensional parameter as well as the rate of convergence for the estimator of slope function are established.Finally,the finite sample performance of the estimators is illustrated via the extensive simulation studies and a real data analysis...