Nonparametric Regression for Spatially Correlated Data
Open AccessThe main theme of this dissertation is to investigate different problems related to the modelling of the correlated data. First, we focus on bandwidth selection methods because the smoothing parameters (bandwidth) determine the smoothness of the curve estimate. In the presence of spatially correlated errors, the traditional data-driven bandwidth selection methods, such as cross-validation and generalized cross-validation, do not work well for providing efficient bandwidth values. Moreover, the existing methods are based on the estimation of correlation structure. As errors are observable, estimation of correlation is considered a big challenge. Thus, we have proposed different methods for selecting bandwidth in the case of correlated data.Second, because the kernel smoothing-based techniques suffer from the effect of boundary points, we get the asymptotic bias and variance for the local polynomial estimators at the boundary points in the presence of correlated errors. As the boundary point problem has not previously been studied under correlated errors, the major finding of this dissertation is to study both the boundary problem and the errors correlation problem simultaneously. Thus, we are able to derive the asymptotic properties of the estimator using weighted least squares matrix theory. This approach will help us to analyze the asymptotic bias and variance near the boundary of predictor support in the presence of correlated errors. However, they assume independence of errors while we assume that errors are correlated. Finally, some boundary correction methods are proposed to remove the boundary effects.Finally, we can consider posterior consistency as a theoretical justification for using the Bayesian approach. As we are interested in nonparametric Bayesian regression, a prior distribution on the mean function is assumed using Gaussian process in the presence of correlated errors. In this dissertation, the posterior consistency in nonparametric regression is studied. We establish the in-probability consistency under the case of correlated errors and verify its conditions when using Gaussian process priors for the mean function with correlated errors. A metric topology on the space of the regression function is defined, then the probability convergence is established. Also, both random and fixed designs are considered.
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