Partial orders and minimization of records in a sequence of independent random variables

1999 ◽  
Vol 36 (04) ◽  
pp. 965-973
Author(s):  
Raúl Gouet ◽  
Jaime San Martín
Keyword(s):  
Partial Order ◽  
Likelihood Ratio ◽  
Hazard Rate ◽  
Random Variables ◽  
Partial Orders ◽  
Parametric Models ◽  
Independent Random Variables ◽  
Expected Number ◽  
Monotone Likelihood Ratio ◽  
Hazard Rate Ordering

Given independent random variables X 1,…,X n , with continuous distributions F 1,…,F n , we investigate the order in which these random variables should be arranged so as to minimize the number of upper records. We show that records are stochastically minimized if the sequence F 1,…,F n decreases with respect to a partial order, closely related to the monotone likelihood ratio property. Also, the expected number of records is shown to be minimal when the distributions are comparable in terms of a one-sided hazard rate ordering. Applications to parametric models are considered.

Partial orders and minimization of records in a sequence of independent random variables

1999 ◽  
Vol 36 (4) ◽  
pp. 965-973
Author(s):  
Raúl Gouet ◽  
Jaime San Martín
Keyword(s):  
Partial Order ◽  
Likelihood Ratio ◽  
Hazard Rate ◽  
Random Variables ◽  
Partial Orders ◽  
Parametric Models ◽  
Independent Random Variables ◽  
Expected Number ◽  
Monotone Likelihood Ratio ◽  
Hazard Rate Ordering

Given independent random variables X1,…,Xn, with continuous distributions F1,…,Fn, we investigate the order in which these random variables should be arranged so as to minimize the number of upper records. We show that records are stochastically minimized if the sequence F1,…,Fn decreases with respect to a partial order, closely related to the monotone likelihood ratio property. Also, the expected number of records is shown to be minimal when the distributions are comparable in terms of a one-sided hazard rate ordering. Applications to parametric models are considered.

scholarly journals Stochastic Order Relations Among Parallel Systems from Weibull Distributions

2015 ◽  
Vol 52 (01) ◽  
pp. 102-116 ◽  
Author(s):  
Nuria Torrado ◽  
Subhash C. Kochar
Keyword(s):  
Likelihood Ratio ◽  
Hazard Rate ◽  
Stochastic Order ◽  
Random Variables ◽  
Parallel Systems ◽  
Parallel System ◽  
Weibull Distributions ◽  
Scale Parameters ◽  
Order Relations ◽  
Stochastic Order Relations

Let X λ1 , X λ2 , …, X λ n be independent Weibull random variables with X λ i ∼ W(α, λ i ), where λ i > 0 for i = 1, …, n. Let X n:n λ denote the lifetime of the parallel system formed from X λ1 , X λ2 , …, X λ n . We investigate the effect of the changes in the scale parameters (λ1, …, λ n ) on the magnitude of X n:n λ according to reverse hazard rate and likelihood ratio orderings.

scholarly journals On the Convolution of Heterogeneous Bernoulli Random Variables

2011 ◽  
Vol 48 (3) ◽  
pp. 877-884 ◽  
Author(s):  
Maochao Xu ◽  
N. Balakrishnan
Keyword(s):  
Likelihood Ratio ◽  
Hazard Rate ◽  
Random Variables ◽  
Likelihood Ratio Order ◽  
Hazard Rate Order ◽  
Reversed Hazard Rate ◽  
Bernoulli Random Variables

In this paper, some ordering properties of convolutions of heterogeneous Bernoulli random variables are discussed. It is shown that, under some suitable conditions, the likelihood ratio order and the reversed hazard rate order hold between convolutions of two heterogeneous Bernoulli sequences. The results established here extend and strengthen the previous results of Pledger and Proschan (1971) and Boland, Singh and Cukic (2002).

ON PARALLEL SYSTEMS WITH HETEROGENEOUS GAMMA COMPONENTS

2011 ◽  
Vol 25 (3) ◽  
pp. 369-391 ◽  
Author(s):  
Peng Zhao
Keyword(s):  
Order Statistics ◽  
Likelihood Ratio ◽  
Hazard Rate ◽  
Shape Parameter ◽  
Scale Parameter ◽  
Random Variables ◽  
Parallel Systems ◽  
Likelihood Ratio Order ◽  
Maximum Order ◽  
Hazard Rate Order

In this article, we study ordering properties of lifetimes of parallel systems with two independent heterogeneous gamma components in terms of the likelihood ratio order and the hazard rate order. LetX1andX2be two independent gamma random variables withXihaving shape parameterr>0 and scale parameter λi,i=1, 2, and letX*1andX*2be another set of independent gamma random variables withX*ihaving shape parameterrand scale parameter λ*i,i=1, 2. Denote byX2:2andX*2:2the corresponding maximum order statistics, respectively. It is proved that, among others, if (λ1, λ2) weakly majorize (λ*1, λ*2), thenX2:2is stochastically greater thanX*2:2in the sense of likelihood ratio order. We also establish, among others, that if 0<r≤1 and (λ1, λ2) isp-larger than (λ*1, λ*2), thenX2:2is stochastically greater thanX*2:2in the sense of hazard rate order. The results derived here strengthen and generalize some of the results known in the literature.

Order-Preserving Shock Models

1994 ◽  
Vol 8 (1) ◽  
pp. 125-134 ◽  
Author(s):  
Y. Kebir
Keyword(s):  
Likelihood Ratio ◽  
Hazard Rate ◽  
Stochastic Ordering ◽  
Shock Models ◽  
Likelihood Ratio Ordering ◽  
Hazard Rate Ordering

To date, research in shock models has been primarily concerned with the classpreserving properties of certain shock and damage kernels. This article focuses on the order-preserving properties of those kernels. We show that they preserve stochastic ordering, hazard rate ordering, backward hazard rate ordering, and likelihood ratio ordering.

Hazard rate ordering of the largest order statistics from geometric random variables

2018 ◽  
Vol 55 (2) ◽  
pp. 652-658
Author(s):  
Bara Kim ◽  
Jeongsim Kim
Keyword(s):  
Order Statistics ◽  
Open Problem ◽  
Hazard Rate ◽  
Random Variables ◽  
Hazard Rate Order ◽  
Two Samples ◽  
Hazard Rate Ordering

Abstract Mao and Hu (2010) left an open problem about the hazard rate order between the largest order statistics from two samples of n geometric random variables. Du et al. (2012) solved this open problem when n = 2, and Wang (2015) solved for 2 ≤ n ≤ 9. In this paper we completely solve this problem for any value of n.

STOCHASTIC ORDERINGS BETWEEN p-SPACINGS OF GENERALIZED ORDER STATISTICS FROM TWO SAMPLES

2006 ◽  
Vol 20 (3) ◽  
pp. 465-479 ◽  
Author(s):  
Taizhong Hu ◽  
Weiwei Zhuang
Keyword(s):  
Order Statistics ◽  
Likelihood Ratio ◽  
Hazard Rate ◽  
Random Variables ◽  
Generalized Order Statistics ◽  
Unified Approach ◽  
Stochastic Orderings ◽  
Two Samples ◽  
Likelihood Ratio Ordering ◽  
Generalized Order

The concept of generalized order statistics was introduced as a unified approach to a variety of models of ordered random variables. The purpose of this article is to investigate conditions on the distributions and the parameters on which the generalized order statistics are based to establish the likelihood ratio ordering of general p-spacings and the hazard rate and the dispersive orderings of (normalizing) simple spacings from two samples. We thus strengthen and complement some results in Franco, Ruiz, and Ruiz [7] and Belzunce, Mercader, and Ruiz [5]. This article is a continuation of Hu and Zhuang [10].

STOCHASTIC ARRANGEMENT INEQUALITIES FOR SOME SYSTEMS WITH HAZARD-RATE ORDERED COMPONENTS

1994 ◽  
Vol 01 (02) ◽  
pp. 185-196
Author(s):  
HAIJUN LI
Keyword(s):  
Hazard Rate ◽  
Optimal Allocation ◽  
Random Variables ◽  
Monotonicity Property ◽  
Hazard Rate Ordering ◽  
Parametrized Family

Using an arrangement monotonicity property of the parametrized family of hazard rate ordered random variables, and a bivariate characterization of the hazard rate ordering, we obtain some new stochastic arrangement inequalities for the random variables that are hazard rate ordered. Some applications of optimal allocation and assembly are also discussed.

scholarly journals Asymptotic Conditional Distribution of Exceedance Counts

2012 ◽  
Vol 44 (01) ◽  
pp. 270-291 ◽  
Author(s):  
Michael Falk ◽  
Diana Tichy
Keyword(s):  
Asymptotic Distribution ◽  
Conditional Distribution ◽  
Random Variables ◽  
Extremal Index ◽  
Stationary Sequence ◽  
Independent Random Variables ◽  
Fragility Index ◽  
Expected Number

We investigate the asymptotic distribution of the number of exceedances among d identically distributed but not necessarily independent random variables (RVs) above a sequence of increasing thresholds, conditional on the assumption that there is at least one exceedance. Our results enable the computation of the fragility index, which represents the expected number of exceedances, given that there is at least one exceedance. Computed from the first d RVs of a strictly stationary sequence, we show that, under appropriate conditions, the reciprocal of the fragility index converges to the extremal index corresponding to the stationary sequence as d increases.

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