Statistical Inference For Complex Time Series,Sampling And Functional Data

This dissertation concerns efficient estimation and statistical inference methods for the complicated functions in the context of time series, sample survey and functional data analysis.First, we propose kernel estimator for the distribution function of unobserved errors in autoregressive time series, based on residuals computed by estimating the autoregressive coefficients with Yule-Walker method. Under mild assumptions, we establish oracle efficiency of the proposed estimator, i.e., it is asymptotically as efficient as the kernel estimator of the distribution function based on the unobserved error sequence itself. Applying result of [92], the proposed estimator is also asymptotically indistinguishable from the empirical distribution function based on the unobserved errors. A smooth simultaneous confidence band (SCB) is then constructed based on the proposed smooth distribution estimator and Kolmogorov distribution. Simulation examples support the asymptotic theory.SCBs are proposed for the distribution functions of a finite population and the latent superpopulation via the empirical distribution function (nonsmooth) and (ker-nel distribution estimator, KDE) based on a simple random sample (SRS), either with or without finite population correction. It is shown that both nonsmooth and smooth SCBs achieve asymptotically the nominal confidence level under standard assumptions. In particular, the uncorrected nonsmooth SCB for superpopulation is exactly the same as the Kolmogorov-Smirnov SCB based on iid sample as long as the SRS size is in- finitesimal relative to the finite population size. Extensive simulation studies confirm the asymptotic properties. The proposed SCBs are constructed for the population dis-tribution of the well-known Baseball data (Lohr 2009) from large number of SRSs, and the findings confirm the theoretical results.Next, KDE is also proposed for measurement error distribution function of dense functional data by using residuals from spline regression of individual trajectories. Under mild conditions, the proposed estimator is shown to be asymptotically as efficient as the infeasible KDE based on unobservable errors, thus it is oracally efficient and as efficient as the empirical cdf of unobservable errors. Consequently, Kolmogorov-Smirnov type SCB is constructed for the error distribution function based on the oracally efficient KDE. Simulation examples corroborate with the theoretical findings.In many applications, there are collections of trajectories densely sampled over time, space, and other continuum measures. This type of data is often considered as realizations of a deterministic mean function plus a stationary stochastic process. In this paper we propose nonparametric estimation and inference for covariance func-tions of such a stochastic process. A key strength of our approach is that it makes no parametric assumptions about the time or location effects. We propose a two-stage estimation procedure based on spline approximations, where the first stage involves estimation of the mean function; and the second stage involves estimation of the co-variance function through smoothing the empirical covariance function. The proposed covariance estimator is smooth, consistent and asymptotically positive definite. It is also shown to be as efficient as the oracle estimator when the mean function is known. Asymptotic SCB are developed for the true covariance function, and the coverage probabilities are shown to be asymptotically correct. Further, to ensure the positive definiteness, a constrained spline estimator is also proposed using a modified B-spline basis expansion. We conduct simulation experiments to compare the numerical perfor-mance of the proposed estimators and SCBs. The proposed method is also illustrated by a real data example.