On Parametric and Nonparametric Methods for Dependent Data

Abstract

In recent years, there has been a surge of research interest in the analysis of time series and spatial data. While on one hand more and more sophisticated models are being developed, on the other hand the resulting theory and estimation process has become more and more involved. This dissertation addresses the development of statistical inference procedures for data exhibiting dependencies of varied form and structure. In the first work, we consider estimation of the mean squared prediction error (MSPE) of the best linear predictor of (possibly) nonlinear functions of finitely many future observations in a stationary time series. We develop a resampling methodology for estimating the MSPE when the unknown parameters in the best linear predictor are estimated. Further, we propose a bias corrected MSPE estimator based on the bootstrap and establish its second order accuracy. Finite sample properties of the method are investigated through a simulation study. The next work considers nonparametric inference on spatial data. In this work the asymptotic distribution of the Discrete Fourier Transformation (DFT) of spatial data under pure and mixed increasing domain spatial asymptotic structures are studied under both deterministic and stochastic spatial sampling designs. The deterministic design is specified by a scaled version of the integer lattice in IRd while the data-sites under the stochastic spatial design are generated by a sequence of independent random vectors, with a possibly nonuniform density. A detailed account of the asymptotic joint distribution of the DFTs of the spatial data is given which, among other things, highlights the effects of the geometry of the sampling region and the spatial sampling density on the limit distribution. Further, it is shown that in both deterministic and stochastic design cases, for "asymptotically distant" frequencies, the DFTs are asymptotically independent, but this property may be destroyed if the frequencies are "asymptotically close". Some important implications of the main results are also given.