Contributions to the nonparametric methods for computer experiments
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Kriging, or Gaussian process modeling, is widely used in estimating unknown functions based on the (noisy) evaluations. The main idea of kriging is to assume the underlying function is a realization of a Gaussian random field. The accuracy of kriging, or more generally, nonparametric regression, depends very strongly on the manner in which data is collected and the properties of the underlying function, especially the smoothness of the underlying function. This dissertation addresses three important problems related to: (i) What type of data collection might be expected to enable one to build an accurate model; (ii) Based on a high-quality design, what is the accuracy of the model; and (iii) Can we construct estimators that achieve the optimal convergence rate without knowing the true smoothness in advance. In Chapter 1 we consider (i). We develop and justify a few guidelines for experimental design, which ensure accuracy of stochastic kriging emulators. In Chapter 2, we derive error bounds of the (simple) kriging predictor under a uniform metric, which provides some insights into (ii). We consider (iii) in Chapter 3, where we propose a method that can provide an estimator that achieves the optimal convergence rate without knowing the true smoothness in advance. Furthermore, the proposed method can provide an estimator of the smoothness.