Methods for Reliability Theory and Applications of Order-Restricted Inference

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The question is addressed of when the bathtub shape of a failure rate function is preserved by various functions used to describe optimal behavior. A second issue is when do mixtures have specified shapes such as a bathtub shape. A third topic introduces isotonic logistic discrimination, which is developed as a nonparametric generalization of linear logistic discrimination taking into account order restrictions. The fourth question addresses finding simultaneous confidence bands for isotonic response surfaces and the fifth topic develops an ordinal extension of the Pearson chi-squared for testing 2(k ordered tables. A new metric on permutations is proposed and developed as the sixth topic. Failure rates are a method used to describe the propensity of a system, either engineering or biological, to fail. They are a function of time and typically at an early age they are high (more likely to fail), at middle age they are low, and then at an old age they tend to increase (called wearout). This gives rise to the bathtub shaped failure rate. For mechanical or electrical systems, which have a bathtub failure rate, there are many early failures. Consequently, a product should not be released until this period of early failures is past. A method of insuring this is called burn-in. Typically this means testing such products for a length of time until this critical period is over. An important consideration is how to optimally determine such a period and this is one question studied by the researchers. Another research area deals with a common phenomenon in many scientific problems where the experimental responses tend to get larger as various underlying experimental parameters are increased. For example, an individual's likelihood of heart attacks within a five-year period will increases as baseline systolic blood pressure increases and as cholesterol levels increase. In order to properly analyze the data arising from such phenomena, new and more general statistical methods need to be developed. The researchers are considering several major new statistical methods to model and analyze such data.