Relative Stochastic Orders Of Univariate Mixture Models

In recent years,stochastic orders theory has developed into a relative active branch in the field of probability statistics research,and as a practical tool and method,it has been used in epidemiology,demographics,biomedical and other related fields.Mixture models appear in various fields related to probability statistics such as survival analysis,reliability engineering,and risk management by different research backgrounds and forms.The models can describe and reveal heterogeneous populations well and play an important role in analyzing,fitting and dealing with mixed data.So it is of great theoretical significance and practical application value to study the relative stochastic orders problem in the univariate mixture models.In this paper,we focus on the stochastic comparison between the overall random variables in the univariate mixture models by using the theory of stochastic orders under the sense of relative hazard rate order and relative mean residual life order and study two special univariate mixture models.For the scale change mixture model,firstly,two population variables with different distribution of frailty variables and the same distribution of baseline variables were stochastic compared under the sense of two relative stochastic orders;and secondly,under what conditions,two population variables with different baseline variables present two relative stochastic orders were investigated respectively.For the weighted frailty mixture model,firstly,the stochastic comparisons of the baseline variable and the overall population variable in the sense of two relative stochastic orders were discussed,then under certain conditions,the stochastic comparison results of two mixture models with different distribution of frailty variables under the sense of two relative stochastic orders were demonstrated,and the two population variables with different distribution of baseline variables were compared in the sense of two relative stochastic orders.Finally,for the different situations discussed in the above models,the results obtained were illustrated by specific distribution examples.