Inferences on Modeling with Non-Monotone Failure Rate

Abstract

Non-monotonic pattern of data with respect to time frequently occur in life testing and biomedical studies. Such studies are directly related to human life and their welfare, so the analysis of such data need more care and exact inferences are needed. The non-monotone type of data is usually handled by the mixture of distribution or generalized family of distribution in case of parametric inferences. It is well known that the inferences based on such family of distributions are difficult to derive with standard statistical procedures. The problems related with inferences also become more cumbersome when observations are incomplete. This work is an attempt in this direction. The modest aim is to present a family of distribution that efficiently models the data which exhibits the non- monotone type of hazard shape along with other type of monotonic structures of the hazard. The inferences for such a family are discussed under the Bayesian and classical framework. When none of the parameters in the model is known, the three different estimators are proposed for their estimation under the Bayesian framework, These Bayes estimators are developed in such a way that their performance can be compared with the classical estimators. The probability density function of underlying model is suspected to have a J-shape for certain specific choice of the shape parameter and also the maximum likelihood estimator (MLE) does not exist in such cases. Therefore under such circumstances, the maximum product of spacing (MPS) method is proposed for the estimation of parameters. It is shown that although none of the proposed estimators exist in a closed form, still it is possible to obtain them through numerical techniques with some care during computational procedures. Performances of such methods as well as their suitability in the long run use are discussed on the basis of their risks which are computed under Linex and squared error loss functions. A Monte-Carlo simulation experiment is conducted to study the finite sample properties of the proposed estimators. These estimators are also exposed to a real life data set and the conclusions are presented.