usual stochastic order of α-mixtures with proportional odds model as a baseline

As a new flexible family of distributions, the α-mixture model includes many existing mixture models as special cases. this paper is an attempt to the usual stochastic order of this family when the underlying distribution follows from the proportional odds model. sufficient conditions are provided for comparing two finite α-mixtures of survival functions with the baseline survival functions following the po model in the sense of usual stochastic order when both the mixing proportions and the po parameters of the first α-mixture majorize the mixing proportions and the po parameters of the second one. also, an upper bound for the reliability function of the α-mixture of survival function. moreover, similar results are obtained for the α-mixture of cumulative distribution function. finally, our theoretical findings are evaluated by some numerical examples.