Stochastic and Dependence Comparisons Between Extreme Order Statistics in the Case of Proportional Reversed Hazard Model

‎independent random‎ ‎variables $y_{1},ldots‎ ,‎y_{n}$ belongs to the‎ ‎proportional reversed hazard rate (prhr) model with‎ ‎proportionality parameters $lambda_1,...,lambda_n$‎, ‎if‎ ‎$y_{k}sim g^{lambda _{k}}(x)$‎, ‎for $k=1,...,n$‎, ‎where $g$ is an‎ ‎absolutely continuous distribution function‎. ‎in this paper we compare‎ ‎the smallest order‎ ‎statistics‎, ‎the sample ranges and the ratios of the smallest and‎ ‎largest order statistics of two sets of independent random‎ ‎variables belonging to prhr model‎, ‎in the sense of (reversed) hazard‎ ‎rate order‎, ‎likelihood ratio order and dispersive order‎, ‎when the‎ ‎variables in one set have proportionality parameters‎ ‎$lambda_1,...,lambda_n$ and the variables in the other set are‎ ‎independent and identically distributed with common parameter‎ ‎$overline{lambda}=sum_{k=1}^{n}lambda_k/n$‎. ‎we also compare‎ ‎the relative degree of dependence between the smallest and the largest‎ ‎order statistics of these samples whit respect to the monotone‎ ‎regression dependence order‎.