Mixture failure rate modeling with applications
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Date
2019-03
Authors
Kotelo, Taoana Thomas
Journal Title
Journal ISSN
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Publisher
University of the Free State
Abstract
This thesis is mostly on mixture failure rate modeling with some applications. The topic is very important in the modern statistical analysis of real world populations, as mixtures is the tool for modeling heterogeneous populations. Neglecting heterogeneity can result in serious errors in analyzing the corresponding statistical data. Many populations are heterogeneous in nature and the homogeneous modeling can be considered as some approximation. It is well known that the failure (mortality) rate in heterogeneous populations tends (as time increases) to that of the strongest subpopulation. However, this basic result had to be considered in a much more generality dealing with the shape of the failure rate and the corresponding properties for other reliability indices as well. This is done in the dissertation, which is (we believe), its main theoretical contribution which can have practical implications as well. We focus on describing aging characteristics for heterogeneous populations. A meaningful case of a population which consists of two subpopulations, which we believe was not sufficiently studied in the literature, is considered. It is shown that the mixture failure rate can decrease or be a bathtub (BT) shaped: initially decreasing to some minimum point and eventually increasing as t or show the reversed pattern (UBT). Otherwise, the IFR property is preserved. The mean residual life’s (MRL) ‘shape properties’ are analyzed and some relations with the failure rate are highlighted. We show that this function for some specific cases with, e.g., IFR or UBT shaped failure rates is decreasing for certain values of parameters, whereas it is UBT for other values. Some findings on the bending properties of the mixture failure rates are presented. It follows from conditioning on survival in the past interval of time that the mixture failure rate is majorized by the unconditional one. These results are extended to other main reliability indices. The mixture failure rate before and after a shock for ordered heterogeneous populations are compared. It turns out that the failure rate after the shock is smaller than the one without a shock, which means that shocks under some assumptions can improve the probabilities of survival for items in a heterogeneous population. We show that the population failure/mortality rate decreases with age and, even tend to reach a plateau for some specific cases of mortality (hazard) rate process induced by the non-homogeneous Poisson process of shocks. Our model can be used to model and analyze the damage accumulated by organisms experiencing external shocks. In this case, the cumulated damage is reflected by jumps in the failure rate. The focus in the literature has been mostly on the study of expectations for mixtures, however, the obtained results show that the variability characteristics in heterogeneous populations may change dynamically.
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Keywords
Thesis (Ph.D. (Mathematical Statistics))--University of the Free State, 2019, Mixtures of distributions, Mixture failure rate, Stochastic ordering, Mortality (failure) rate process, Shocks, Increasing (decreasing) failure rate