Ordering properties of the smallest and largest claim amounts in a general scale model

Suppose is a set of non-negative random variables with having the distribution function , for and are independent Bernoulli random variables, independent of the ’s, with , . Let , for . It is of interest to note that in actuarial science, corresponds to the claim amount in a portfolio of risks. In this paper, under certain conditions, by using the concept of vector majorization and related orders, we discuss stochastic comparison between the smallest claim amount in the sense of the usual stochastic and hazard rate orders. We also obtain the usual stochastic order between the largest claim amounts when the matrix of parameters changes to another matrix in a mathematical sense. We then apply the results for three special cases of the scale model: generalized gamma, Marshall–Olkin extended exponential and exponentiated Weibull distributions with possibly different scale parameters to illustrate the established results.

[1]  N. Balakrishnan,et al.  Stochastic comparisons of series and parallel systems with generalized linear failure rate components , 2017 .

[2]  Stochastic comparison of aggregate claim amounts between two heterogeneous portfolios and its applications , 2015 .

[3]  Narayanaswamy Balakrishnan,et al.  Stochastic Comparisons of Series and Parallel Systems With Generalized Exponential Components , 2015, IEEE Transactions on Reliability.

[4]  S. Nadarajah,et al.  The exponentiated Weibull distribution: a survey , 2013 .

[5]  Baha-Eldin Khaledi,et al.  Stochastic comparisons of order statistics in the scale model , 2011 .

[6]  Stochastic Orders , 2008 .

[7]  Baha-Eldin Khaledi,et al.  On stochastic comparison between aggregate claim amounts , 2008 .

[8]  N. Balakrishnan,et al.  Generalized gamma frailty model , 2006, Statistics in medicine.

[9]  Heterogeneity and the need for capital in the individual model , 2006 .

[10]  S. Loukas,et al.  On an extension of the exponential-geometric distribution , 2005 .

[11]  A note on multivariate stochastic comparisons of Bernoulli random variables , 2004 .

[12]  Mark A. McComb.  Comparison Methods for Stochastic Models and Risks , 2003, Technometrics.

[13]  A. Müller,et al.  Comparison Methods for Stochastic Models and Risks , 2002 .

[14]  A comparison between homogeneous and heterogeneous portfolios , 2001 .

[15]  Chunsheng Ma,et al.  Convex orders for linear combinations of random variables , 2000 .

[16]  S. Loukas,et al.  A lifetime distribution with decreasing failure rate , 1998 .

[17]  I. Olkin,et al.  A new method for adding a parameter to a family of distributions with application to the exponential and Weibull families , 1997 .

[18]  Alan D. Hutson,et al.  The exponentiated weibull family: some properties and a flood data application , 1996 .

[19]  N. L. Johnson,et al.  Continuous Univariate Distributions. , 1995 .

[20]  G. S. Mudholkar,et al.  Exponentiated Weibull family for analyzing bathtub failure-rate data , 1993 .

[21]  I. Olkin,et al.  Inequalities: Theory of Majorization and Its Applications , 1980 .

[22]  Samuel Karlin,et al.  Generalized convex inequalities , 1963 .

[23]  E. Stacy.  A Generalization of the Gamma Distribution , 1962 .