Heterogeneity in Populations and the Paradoxes of Survival: A Tribute to Nozer Singpurwalla

Abstract

We consider several survival models in heterogeneous settings. Heterogeneity in the failure rates of subpopulations results (as a specific case) in the famous failure rate paradox when the failure rate of a mixture of items with constant failure rates is decreasing. Random failure rate that is due to a point process that increases it at random times on fixed values also results in the "bending down" of the population failure rate. Similar effect is observed while analyzing the extreme shock models with shock processes that possess memory. Finally, another paradox when, due to heterogeneity in a vital parameter of a model, a terminating point process with decreasing rate after "mixing" becomes a non-terminating one with increasing rate is described. Those are the impacts of heterogeneity that are discussed from the unified perspective that employs the "principle": the weaker subpopulations are dying out first.