Mathematical Sciences: Implementation of Accurate Methods for Practical Inference

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DMS 9625440 Casella Although much research effort has been expended on developing accurate approximation techniques such as saddlepoint and improved likelihood-based procedures, less effort has been devoted to assessing the types of inferences that can be achieved in practice by using these methods. When this assessment is made, it is seen that the available inferences are severely limited in scope by both statistical issues and computational complexity. These two sources of limitation are intertwined, as computational difficulties can arise from the inherent demands of valid frequentist procedures. Ensuring the correctness of frequentist inferences can be computationally intensive, requiring many cumbersome evaluations of the complex expressions that derive from higher-order asymptotic approximations. In this research, these difficulties are overcome by a synthesis of frequentist and Bayesian inference, as the latter approach is simpler in outlook and implementation. In particular, the computational problem is addressed by adapting sampling-based techniques, such as Markov Chain Monte Carlo, to attain the higher-order approximations. The result will be improved inferences in a wide variety of practical problems; examples include logistic regression, censored data models, and variance component estimation. %%% In more complicated statistical models, statisticians have typically relied on approximate methods of inference, primarily because exact methods can be both difficult to derive and complex to compute. However, the validity of these approximate methods rests on the sample size being large, which means that such methods may not be accurate in problems with small samples. Now that inexpensive computational power has become widely available, statisticians are attempting to use the more realistic and complex models. For example, models used in analyzing global environmental change, or DNA assessment, are quite complex. The focus of t his research is to develop statistical methods that offer both accuracy and computational tractability. This work necessarily blends high-performance computing with modern statistical methodology. ***