Algorithms, Approximations, and Valid Statistical Inference

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9971586New solutions to difficult statistical problems have been found through the use of algorithms and accurate approximations. Sometimes these solutions can be extremely difficult to implement and, moreover, the statistical properties of the solution, and hence the resulting statistical inference, may be difficult to describe and, in the worst cases, even invalid. The general theme of the proposed research is to develop computing algorithms, accurate approximations, and other numerical techniques for a variety of problems, and do it in such a way that a valid statistical inference results. Some specific projects are the development of algorithms for hierarchical and missing data models, including extensions of the EM algorithm to estimating equations; using accurate approximations and a synthesis of Bayesian and frequentist approaches to provide valid inferences in nuisance parameter problems; and computation and inference in discrete data problems, with particular attention to numerical alternatives to saddlepoint approximations and novel uses of Monte Carlo methods.Much current research in statistics is influenced by the availability of exceptionally fast computing, which has led to a reexamination of the approaches to many problems. These new approaches have provided some solutions to previously unsolved problems; however, their implementation is often not straightforward so the solutions will not be available to more than a few expert users. Such a situation arose in an epidemiological study to estimate the effect of clinical mastitis, where implementation of the currently available algorithm was not clear. The type of solution proposed here is straightforward to implement in this case, as well as in other applications such as estimating the location of cancer clusters relative to toxic waste sites. This new solution will also result in statistically valid conclusions, a property not always enjoyed by computationally intensive procedures. In addition to these algorithms, research in new statistical simulation methods is proposed which has application in genetics and biotechnology studies (the linkage of locations on a chromosome to particular diseases).