On nonparametric methods for strongly and weakly dependent lattice data
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This dissertation develops nonparametric inference procedures for data exhibiting dependencies of varied form and structure. There are four research papers, each concerning observations located (sampled) on some lattice (e.g. integers Zd ):  On optimal spatial subsample size for variance estimation;  On the approximation of differenced lattice point counts with application to statistical bias expansions;  Frequency domain empirical likelihood for short- and long-range dependent processes;  Empirical likelihood confidence intervals for the mean of a long-range dependent process.;Let Rn be a sampling region in Rd , t [is in] Rd , and suppose data are those t + Zd lattice points within Rn. Paper  considers the problem of optimally implementing a spatial subsampling method for nonparametrically estimating the variance of statistics on Rn. The results are applicable to a wide variety of stationary random fields exhibiting weak dependence and a broad range of sampling regions.;Paper  frames and addresses a mathematical problem in lattice point theory. Bias expansions for statistics of spatial lattice data often require subtracted lattice point counts for sets like Rn and Rn ∩ (t + R n). This paper provides new lattice point count approximations that facilitate asymptotic bias expansions with non-rectangular sampling regions. The count estimation tools are valid for numerous R2-R3 sampling regions, including all convex sets.;Papers  and  consider inference on dependent time series data, allowing for many possible process dependence strengths. Let X 1, X2,... be a sequence of stationary random variables with autocovariance function r( k) = Cov(X1, X 1+k) where r(k ) ~ k-alpha L(k) for alpha [is in] (0,1) and a slowly varying function L. Since k=1infinity |r(k)| = infinity , the process Xt exhibits strong or long-range dependence (LRD). Inference methods proposed for iid or weakly dependent data often do not work for strongly dependent data, or at least require suitable modifications. Paper  develops spectral empirical likelihood which allows nonparametric, likelihood-based confidence region estimation and hypothesis testing for processes exhibiting either LRD or weak dependence. Inference is possible for spectral parameters like autocorrelations and Whittle parameters and spectral goodness-of-fit tests can also be made.;Paper  develops new empirical likelihood confidence intervals for the process mean E(X1) = mu under LRD, using (time-domain) "blocking" techniques. The paper shows that empirical likelihood provides valid nonparametric estimation of mu, even in those instances of LRD where the block bootstrap is known to fail.