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Some Research On Functional Regression And Functional Canonical Correlation Analysis

Posted on:2020-11-19Degree:DoctorType:Dissertation
Country:ChinaCandidate:H B ZhuFull Text:PDF
GTID:1367330620952035Subject:Statistics
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With the progress in modern technology of data collection and storage,functional data have been increasingly available in many scientific fields,such as meteorology,chem-istry,biomedicine and neuroimaging.The most striking feature of functional data is its inherent infinite dimensionality,which poses challenges both for theoretical analysis and statistical computation,and makes the traditional multivariate statistical analysis meth-ods no longer applicable.On the other hand,the infinite dimensional structure of the data is also a rich source of potential useful information,which brings many opportunities for theoretical research and data application.Hence,functional data analysis is getting more and more attention in the field of statistical research.In this thesis,we focus on functional regression and functional canonical correlation analysis.The main content is as follows.(1)We study estimation problem for extreme conditional quantiles of functional quantile regression models.Assuming that the functional covariate has a linear effect on the upper quantiles of the response,we develop a new estimation procedure for extreme conditional quantiles by first estimating intermediate conditional quantiles and then extrapolating these estimates to high tails.We obtain the estimators of the slope function and extreme conditional quantiles,and establish their asymptotic prop-erties by using functional principal component analysis and extreme value theory.We also demonstrate that the proposed method enjoys higher accuracy than the conventional functional linear quantile regression estimator by simulation studies.Finally,a real data example is analyzed for illustration(2)We study estimation and testing problems for partial functional linear models when the covariates in the non-functional linear component are measured with additive error.A corrected profile,least-squares based,estimation procedure is developed for the parametric component.Asymptotic properties of the proposed estimators are established under some regularity conditions.To test a hypothesis on the parametric component,a statistic based on the difference between the corrected residual sums of squares under the null and alternative hypotheses is proposed and its limiting null distribution is shown to be a weighted sum of independent standard ?12 variables Simulation studies are conducted to demonstrate the performance of the proposed procedure and a real data example is analyzed for illustration(3)We study nonlinear functional canonical correlation analysis based on distance covari-ance.We propose a nonlinear extension of functional canonical correlation analysis based on distance covariance.the estimators of nonlinear functional canonical cor-relation are established and consistency of these estimators is proved.Monte Carlo studies show that the proposed nonlinear functional canonical correlation analysis can uncover new association patterns between functional variables.
Keywords/Search Tags:Corrected profile least squares, Distance covariance, Extreme quantile, Extreme value theory, Functional principal component analysis, Functional quantile regression, Heavy-tailed distribution, Hypothesis test, Measurement error
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