Stochastic Ordering Inequalities in Parallel and Series Systems with Gumble Distributed Components

Abstract

The stochastic comparisons of parallel and series systems are worthy of study. In this article, we present some stochastic comparisons of parallel and series systems having independent components from Gumble distribution which is also known as an extreme value type-I distribution with two parameters (one location and one shape). This study only deals with the location parameter. Here, we first put a condition for the likelihood ratio ordering of the parallel systems and second, we use the concept of vector majorization technique to compare the systems by the reversed failure rate ordering, the failure rate ordering, the dispersive ordering, and the less uncertainty ordering with respect to the location parameter.

AMS 2020 subject classifications: 62G30, 60E15, 60K10

Keywords

Gumble distribution entropy Schur convex vector majorization stochastic comparisons

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References

Gumbel

EJ.

Statistical theory of extreme values and some practical applications: A series of lectures . US Government Printing Office, 1948: 33.

Fisher

and Tippett

LHC.

Limiting forms of the frequency distribution of the largest or smallest member of a sample. Math Proc Camb Philos Soc 1928; 24: 180–190

Gupta

, Patra

and Kumar

Stochastic comparisons in systems with fr‘echet distributed components. Oper Res Lett 2015; 43: 612–615.

Torrado

and Kochar

. Stochastic order relations among parallel systems from weibull distributions. J Appl Probab 2015; 52, 102–116.

Gómez

, Bolfarine

and Gómez

. Gumbel distribution with heavy tails and applications to environmental data. Math Comput Simul 2019; 157: 115–129.

Balakrishnan

, Haidari

and Masoumifard

. Stochastic comparisons of series and parallel systems with generalized exponential components. IEEE Transactionson Reliability 2014; 64: 333–348.

Fang

and Xu

. Ordering results of the smallest and largest order statistics from independent heterogeneous exponentiated gamma random variables. Stat Neerl 2019; 73: 197–210.

Bashkar

, Torabi

and Roozegar

. Stochastic comparisons of extreme order statistics in the heterogeneous exponentiated scale model. J Stat Theo App 2017; 16: 219–238.

Fang

and Balakrishnan

. Ordering results for the smallest and largest order statistics from independent heterogeneous exponential–weibull random variables. Statistics 2016; 50: 1195–1205.

10.

Fang

and Zhang

. New results on stochastic comparison of order statistics from heterogeneous weibull populations. J Korean Stat Soc 2012; 41: 13–16.

11.

Misra

and Misra

. New results on stochastic comparisons of two-component series and parallel systems. Stat Probab Lett 2012; 82, 283–290.

12.

Shaked

and Shanthikumar

. Stochastic orders . Springer Science Business Media, 2007.

13.

Shannon

CE.

Mathematical theory of communication. Bell Syst Tech J 1948; 27, 379–423.

14.

Ebrahimi

and Pellerey

. New partial ordering of survival functions based on the notion of uncertainty. J Appl Probab 1995; 32, 202–211.

15.

Marshall

, Olkin

and Arnold

. Inequalities: Theory of majorization and its applications . Springer, 1979: 143.

16.

Dewan

and Khaledi

. On stochastic comparisons of residual lifetime at random time. Stat Probab Lett 2014; 88: 73–79.