Stochastic comparisons of parallel systems of heterogeneous exponential components

Dykstra, Richard L.; Kochar, Subhash C.; Rojo, Javier

doi:10.1016/s0378-3758(97)00058-x

Cited by 145 publications

(94 citation statements)

References 10 publications

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“…It is evident that the result in Theorem 1 generalizes and strengthens those in [7,10] from the exponential case to the gamma distribution. One may wonder here whether we could establish more general comparison results between two largest order statistics both of which arise from heterogeneous variables under some majorization (cf.…”

Section: Remarksupporting

confidence: 65%

“…Many papers have appeared on the stochastic comparison of order statistics arising from i.ni.d. exponential random variables including [19,20,12,7,[9][10][11]5,13,14,17,18,[22][23][24][25][26][27]. The gamma distribution has been used extensively in reliability and survival analysis due to its flexibility in shape and some nice distributional properties; for more details on this distribution, one may refer to Johnson et al [8].…”

Section: Introductionmentioning

confidence: 99%

“…, Y n be a random sample of size n from an exponential distribution with hazard rateλ =  n i=1 λ i /n, the arithmetic mean of λ ′ i s, and denote by Y n:n the corresponding largest order statistic. Dykstra et al [7] then showed that…”

Section: Introductionmentioning

confidence: 99%

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Hazard rate comparison of parallel systems with heterogeneous gamma components

Zhao

2013

Journal of Multivariate Analysis

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a b s t r a c tWe compare the hazard rate functions of the largest order statistic arising from independent heterogeneous gamma random variables and that arising from i.i.d. gamma random variables. Specifically, let X 1 , . . . , X n be independent gamma random variables with X i having shape parameter 0 < r ≤ 1 and scale parameter λ i , i = 1, . . . , n. Denote by Y n:n the largest order statistic arising from i.i.d. gamma random variables Y 1 , . . . , Y n with Y i having shape parameter r and scale parameterλ = (  n i=1 λ i ) 1/n , the geometric mean of λ ′ i s. It is shown that X n:n is stochastically larger than Y n:n in terms of hazard rate order. The result derived here strengthens and generalizes some of the results known in the literature and leads to a sharp upper bound on the hazard rate function of the largest order statistic from heterogeneous gamma variables in terms of that of the largest order statistic from i.i.d. gamma variables. A numerical example is finally provided to illustrate the main result established here.

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Section: Remarksupporting

confidence: 65%

Section: Introductionmentioning

confidence: 99%

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Hazard rate comparison of parallel systems with heterogeneous gamma components

Zhao

2013

Journal of Multivariate Analysis

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“…Some interesting partial ordering results on order statistics and spacings from independent but non-identically random variables have been obtained by Pledger and Proschan (1971), Proschan and Sethuraman (1976), Bapat and Kochar (1994), Boland, El-Neweihi, and Proschan (1994 ), Kochar and Kirmani (1995), Kochar and Korwar (1996), Kochar and Rojo (1996), Dykstra, Kochar, and Rojo (1997), Kochar and Ma (1999), Bon and Paltanea (1999), Kochar (1999), Khaledi and Kochar (1999), , and Khaledi and Kochar (2001).…”

Section: Introductionmentioning

confidence: 77%

Stochastic Orderings among Order Statistics and Sample Spacings

Khaledi¹,

Kochar

2002

Uncertainty and Optimality

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In this paper we review some of the results obtained recently in the area of stochastic comparisons of order statistics and sample spacings. We consider the cases when the parent observations are identically as well as non-identically distributed. But most of the time we shall be assuming that the observations are independent. The case of independent exponentials with unequal scale parameters is discussed in detail.

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“…Such comparisons are studied with exponential, gamma, Weibull, generalized exponential or Fréchet distributed components with unbounded support. One may refer to Dykstra et al [2], Misra and Misra [16], Zhao and Balakrishnan ( [23]), Torrado and Kochar [20], Kundu and Chowdhury [8], Kundu et al [9], Gupta et al [7] and the references there in. Moreover, not much attention has been paid so far to the stochastic comparison of two systems having finite range distributed components.…”

Section: Introductionmentioning

confidence: 99%