On magnitude orderings between smallest order statistics from heterogeneous beta distributions

In this work, we pay attention to smallest order statistics arising from heterogeneous random variables satisfying the proportional reversed hazard rate (PRHR) model. In particular, we show various results on this model related to comparisons based on magnitude stochastic orders. Since beta distribution is the simplest element of PRHR models, we first face the problem of some stochastic comparisons under some majorization conditions on the beta parameters. We then extend these results to the general case of PRHR models.