Preservation of ILR and IFR aging classes in sums of dependent random variables

Abstract

Closure of aging classes with respect to sums is a relevant topic in different areas of applied probability. In the case of independent random variables (i.e., for convolutions) this preservation property has been proved in the literature for a number of classes such as the increasing in likelihood ratio (ILR) and the increasing failure rate (IFR) classes. These results were applied, for example, to sums of life lengths in reliability theory and to sums of incomes/returns in economic/risk studies. However, in many practical situations the independence assumption is unrealistic. In the present paper, we provide conditions such that these closure properties are satisfied by the ILR and IFR classes when the assumption of independence is dropped. The classical copula approach is used to model the dependence structure, but other dependence models, such as relevation transforms and load sharing models, are considered as well. Several illustrative examples and counterexamples show how to use the presented theoretical results and which preservation results do not hold.

On relevation transform involved with statistical dependence between two component lifetimes

“…The authors would like to thank the two anonymous reviewers for their instructive comments, which helped improve the presentation of the earlier version of this manuscript. In particular, one reviewer directed us to Johnson and Kotz, 22,23 the pioneering work on dependent relevation transform, and Navarro and Pellerey, 11 one recent reference on classic relevation transform. Chen Li's research is supported by National Natural Science Foundation of China (72201191), Natural Science Foundation of Tianjin (20JCQNJC00620), and 100 Youth Elite Project of Tianjin University of Commerce.…”

“…For example, by virtue of the relevation transform, Pellerey et al 7 generalized the nonhomogeneous Poisson process to the relevation counting process and studied its inter‐failure times, and Kapodistria and Psarrakos 8 represented a process of the successive failures of one component replaced on failure by another component of equal age based on one sequence of stochastically increasing random variables. Afterward, Sordo and Psarrakos 9 conducted stochastic comparison on the failure times and inter‐failure times of two systems in the context of the relevation replacement policy of Kapodistria and Psarrakos, 8 Belzunce et al 10 conducted comparison on the policy based on relevation transform and that of replacement by a new unit in reliability context, and Navarro and Pellerey 11 studied the preservation of IFR under the relevation transform with minimal and imperfect repairs, respectively. Besides, Psarrakos and Di Cresccenzo 12 investigated an inaccuracy measure closely related to the relevation transform of two absolutely continuous lifetime distributions, and Sankaran and Kumar 13 discussed several reliability properties of the so‐called proportional hazards relevation transform.…”

Statistical Modeling of Some Cancerous Diseases Using the Laplace Transform Approach of Basic Life Testing Issues

“…The most well-known classifications of life distributions are IFR, IFRA, NBU, NBUC, NBUC mgf , NBUCL, NBRUL, NBRU mgf , UBA, UBAC, UBAC(2), and UBA mgf . Properties and applications of these aging notions can be found, Navarro and Pellerey [ 1 ], Bryson and Siddiqui [ 2 ], Barlow and Proschan [ 3 ], Abu-Youssef and Bakr [ 4 ], Mahmoud et al [ 5 ], EL-Sagheer et al [ 6 ], Esary et al [ 7 ], Navarro J. [ 8 ], Cao and Wang [ 9 ], Fernandez-Ponce et al [ 10 ], Ali [ 11 ], Hassan and Said [ 12 ], Ahmad [ 13 ], Qureshi et al [ 14 ], Yusuf and Qureshi [ 15 ], and Abu Youssef et al [ 16 , 17 ] investigated the moment generating function for UBA (UBA mgf ).…”

Modeling Various Survival Distributions using a Nonparametric Hypothesis Testing Based on Laplace Transform Approach with Some Real Applications

“…We have some genuine data in this research investigation, and we are trying to figure out whether H 0 : data is exponential or H 1 : data is NBU (2).…”

“…Various life distribution classes have been proposed during the last century to model various elements of aging. See, for example, the studies of Bakr et al [ 1 ], Navarro and Pellerey [ 2 ], Bakr et al [ 3 ], Navarro [ 4 ], Qureshi and Kumar [ 5 ], and Qureshi and Yusuf [ 6 ], for definitions and preservation of several classes of life distributions, such as IFR, NBU, UBA, UBAmgf, NBUE, and their duals.…”