Stochastic Aspects of Proportional Vitalities Model

Abstract

In this paper, a family of models requiring proportional mean life vitalities is considered. The problem of estimation of the parameter(s) of the model is studied in two cases of known and unknown baselines along with some simulation studies to detect the adequacy of fitting. Closure and preservation properties of some ageing classes and stochastic orders are derived.

Further Results on the Proportional Vitalities Model

“…To present the state of scientific development in the context of recent survival models, we consider the works accomplished in the context of the PMRL model and the AMRL model that are closely related to the PVIT model. In fact, the PVIT model is a special case of an additive-multiplicative mrl model as stated in the proof of Theorem 2 in Shrahili et al [ 19 ]. Recently, Nair et al [ 20 ] carried out a reliability study of the PMRL model in the frame of quantile functions.…”

“…Clearly, the random variable is non-negative. It has been taken for granted that there is a realization of that equals Given that the conditional vitality function of the population is obtained as for all Shrahili et al [ 19 ] obtained the conditional cdf with corresponding conditional vitality function as and To integrate the effect of random variable we denote by U the unconditional random variable, which has sf . By replacing from (4), The pdf of U is also , which is obtained by substituting from (5) with The connection of three random variables , and U is as follows.…”

“…That is, it is a further conclusion of Theorem 1(i) that for all , , , and a conclusion of Theorem 1(ii) that for all non-negative , and . From relation (6) in Shrahili et al [ 19 ], for all . Thus, by the iterated expectation rule, .…”

“…Estimation procedures for coefficients of the regression PMRL model have been conducted by Maguluri and Zhang [ 17 ] and Chen et al [ 18 ]. Recently, Shrahili et al [ 19 ] proposed the proportional vitalities (PVIT) model as an alternative to the PMRL model and also the CPH model. Unlike the prevalent CPH and PMRL models, in the PVIT model, the sf and the pdf of the dependent random variable do not have an explicit closed form.…”

“…According to Shrahili et al [ 19 ] the PVIT model is identified by where is the vitality growth parameter with a time-dependent domain so that , for all , the function is the vitality function of the population or the dependent variable and is the baseline vitality function. The model (3) is a partial model but it stands valid and qualified under more general circumstances.…”