Statistical inferences for functional data

TL;DR

This paper investigates the effects of local polynomial kernel smoothing on functional data analysis, demonstrating that substitution effects can be ignored asymptotically and developing estimators and tests for functional data with covariates.

Contribution

It introduces asymptotic theory for LPK-based estimators in functional data analysis, including mean, covariance, and covariate effects, with a GCV bandwidth selection rule.

Findings

Substitution effect can be ignored asymptotically under mild conditions.

LPK-based estimators for mean, covariance, and covariate effects are asymptotically normal.

A global hypothesis test for covariate effects is proposed and asymptotically justified.

Abstract

With modern technology development, functional data are being observed frequently in many scientific fields. A popular method for analyzing such functional data is ``smoothing first, then estimation.'' That is, statistical inference such as estimation and hypothesis testing about functional data is conducted based on the substitution of the underlying individual functions by their reconstructions obtained by one smoothing technique or another. However, little is known about this substitution effect on functional data analysis. In this paper this problem is investigated when the local polynomial kernel (LPK) smoothing technique is used for individual function reconstructions. We find that under some mild conditions, the substitution effect can be ignored asymptotically. Based on this, we construct LPK reconstruction-based estimators for the mean, covariance and noise variance functions of a functional data set and derive their asymptotics. We also propose a GCV rule for selecting good bandwidths for the LPK reconstructions. When the mean function also depends on some time-independent covariates, we consider a functional linear model where the mean function is linearly related to the covariates but the covariate effects are functions of time. The LPK reconstruction-based estimators for the covariate effects and the covariance function are also constructed and their asymptotics are derived. Moreover, we propose a $L^2$-norm-based global test statistic for a general hypothesis testing problem about the covariate effects and derive its asymptotic random expression. The effect of the bandwidths selected by the proposed GCV rule on the accuracy of the LPK reconstructions and the mean function estimator is investigated via a simulation study. The proposed methodologies are illustrated via an application to a real functional data set collected in climatology.