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Abstract: With modern technology development, functional data are being observedfrequently in many scientific fields. A popular method for analyzing suchfunctional data is ``smoothing first, then estimation.- That is, statisticalinference such as estimation and hypothesis testing about functional data isconducted based on the substitution of the underlying individual functions bytheir reconstructions obtained by one smoothing technique or another. However,little is known about this substitution effect on functional data analysis. Inthis paper this problem is investigated when the local polynomial kernel LPKsmoothing technique is used for individual function reconstructions. We findthat under some mild conditions, the substitution effect can be ignoredasymptotically. Based on this, we construct LPK reconstruction-based estimatorsfor the mean, covariance and noise variance functions of a functional data setand derive their asymptotics. We also propose a GCV rule for selecting goodbandwidths for the LPK reconstructions. When the mean function also depends onsome time-independent covariates, we consider a functional linear model wherethe mean function is linearly related to the covariates but the covariateeffects are functions of time. The LPK reconstruction-based estimators for thecovariate effects and the covariance function are also constructed and theirasymptotics are derived. Moreover, we propose a $L^2$-norm-based global teststatistic for a general hypothesis testing problem about the covariate effectsand derive its asymptotic random expression. The effect of the bandwidthsselected by the proposed GCV rule on the accuracy of the LPK reconstructionsand the mean function estimator is investigated via a simulation study. Theproposed methodologies are illustrated via an application to a real functionaldata set collected in climatology.

Autor: Jin-Ting Zhang, Jianwei Chen



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