Maximum likelihood based estimation of hazard function under shape restrictions and related statistical inference
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The problem of estimation of a hazard function has received considerable attention in the statistical literature. In particular, assumptions of increasing, decreasing, concave and bathtub-shaped hazard function are common in literature, but practical solutions are not well developed. In this dissertation, we introduce a new nonparametric method for estimation of hazard function under shape restrictions to handle the above problem. This is an important topic of practical utility because often, in survival analysis and reliability applications, one has a prior notion about the physical shape of underlying hazard rate function. At the same time, it may not be appropriate to assume a totally parametric form for it. We adopt a nonparametric approach in assuming that the density and hazard rate have no specific parametric form with the assumption that the shape of the underlying hazard rate is known ( either decreasing, increasing, concave, convex or bathtub-shaped). We present an efficient algorithm for computing the shape restricted estimator. The theoretical justification for the algorithm is provided. We also show how the estimation procedures can be used when dealing with right censored data.We evaluate the performance of the estimator via simulation studies and illustrate it on some real data sets. We also consider testing the hypothesis that the lifetimes come from a population with a parametric hazard rate such asWeibull against a shape restricted alternative which comprises a broad range of hazard rate shapes. The alternative may be appropriate when the shape of the parametric hazard is not constant and monotone. We use appropriate resampling based computation to conduct our tests since the asymptotic distributions of the test statistics in these problems are mostly intractable.