Modified proportional hazard rates and proportional reversed hazard rates models via Marshall–Olkin distribution and some stochastic comparisons

Adding parameters to a known distribution is a useful way of constructing flexible families of distributions. Marshall and Olkin (1997) introduced a general method of adding a shape parameter to a family of distributions. In this paper, based on the Marshall–Olkin extension of a specified distribution, we introduce two new models, referred to as modified proportional hazard rates (MPHR) and modified proportional reversed hazard rates (MPRHR) models, which include as special cases the well-known proportional hazard rates and proportional reversed hazard rates models, respectively. Next, when two sets of random variables follow either the MPHR or the MPRHR model, we establish some stochastic comparisons between the corresponding order statistics based on majorization theory. The results established here extend some well-known results in the literature.     

Sensitivity Analysis for Dependent Variables*

Decision analysts use sensitivity analysis to identify influential variables, to determine which input variables to model stochastically, and to characterize scenarios that could affect a change in the rank ordering of the alternatives. A frequently recommended sensitivity analysis technique is “one‐way” sensitivity analysis, which determines a variable's influence by the degree to which the objective function changes as that variable is varied while all other variables are held fixed. Disadvantages of one‐way analysis are that it measures the influence of only one variable at a time and it assumes independence among the input variables. Clearly, however, there are situations when dependencies exist among the input variables that could possibly affect the sensitivity analysis results. This research develops a strategy that incorporates dependence relations among the input variables into the sensitivity analysis using rank correlations. Only decision problems with a finite number of alternatives and continuous state variables are considered.