scholarly journals Stochastic orders for a multivariate Pareto distribution

2021 ◽  
Vol 29 (1) ◽  
pp. 53-69
Author(s):  
Luigi-Ionut Catana
Keyword(s):  
Pareto Distribution ◽  
Random Variables ◽  
Stochastic Orders ◽  
Distribution Family ◽  
Theoretical Results

Abstract In this article we give some theoretical results for equivalence between different stochastic orders of some kind multivariate Pareto distribution family. Weak multivariate orders are equivalent or imply different stochastic orders between extremal statistics order of two random variables sequences. The random variables in this article are not neccesary independent.

scholarly journals SOME UNIFIED RESULTS ON COMPARING LINEAR COMBINATIONS OF INDEPENDENT GAMMA RANDOM VARIABLES

2012 ◽  
Vol 26 (3) ◽  
pp. 393-404 ◽  
Author(s):  
Subhash Kochar ◽  
Maochao Xu
Keyword(s):  
Random Variables ◽  
Stochastic Orders ◽  
Sufficient Condition ◽  
Linear Combinations

In this paper, a new sufficient condition for comparing linear combinations of independent gamma random variables according to star ordering is given. This unifies some of the newly proved results on this problem. Equivalent characterizations between various stochastic orders are established by utilizing the new condition. The main results in this paper generalize and unify several results in the literature including those of Amiri, Khaledi, and Samaniego [2], Zhao [18], and Kochar and Xu [9].

scholarly journals Estimation of a System Performance in Pareto Distribution with Two Independent Random Variables

2014 ◽  
Vol 14 (21) ◽  
pp. 2854-2856 ◽  
Author(s):  
Medhat Ahmed El Damsesy ◽  
Mohammed Mohammed E ◽  
Ahmed Mohammed El Gazar
Keyword(s):  
System Performance ◽  
Pareto Distribution ◽  
Random Variables ◽  
Independent Random Variables

scholarly journals Non-Comparability with respect to the convex transform order with applications

2020 ◽  
Vol 57 (4) ◽  
pp. 1339-1348
Author(s):  
Idir Arab ◽  
Milto Hadjikyriakou ◽  
Paulo Eduardo Oliveira
Keyword(s):  
Random Variables ◽  
Negative Answer ◽  
Stochastic Orders ◽  
Convex Transform Order

AbstractIn the literature of stochastic orders, one rarely finds results characterizing non-comparability of random variables. We prove simple tools implying the non-comparability with respect to the convex transform order. The criteria are used, among other applications, to provide a negative answer for a conjecture about comparability in a much broader scope than its initial statement.

scholarly journals On Occurrences of F-S Strings in Linearly and Circularly Ordered Binary Sequences

2010 ◽  
Vol 47 (1) ◽  
pp. 157-178 ◽  
Author(s):  
Frosso S. Makri
Keyword(s):  
Waiting Time ◽  
Applied Research ◽  
Random Variables ◽  
Upper Bounds ◽  
Binary Sequences ◽  
Mean Values ◽  
Numerical Examples ◽  
Binary Trials ◽  
Binary Random Variables ◽  
Theoretical Results

Consider a sequence of exchangeable or independent binary trials ordered on a line or on a circle. The statistics denoting the number of times an F-S string of length (at least) k1 + k2, that is, (at least) k1 failures followed by (at least) k2 successes in n such trials, are studied. The associated waiting time for the rth occurrence of an F-S string of length (at least) k1 + k2 in linearly ordered trials is also examined. Exact formulae, lower/upper bounds and approximations are derived for their distributions. Mean values and variances of the number of occurrences of F-S strings are given in exact formulae too. Particular exchangeable and independent sequences of binary random variables, used in applied research, combined with numerical examples clarify further the theoretical results.

scholarly journals The Residual Lifetime of Surviving Components of Coherent System under Periodical Inspections

Mathematics ◽  
2020 ◽  
Vol 8 (12) ◽  
pp. 2181
Author(s):  
Zhouxia Guo ◽  
Jiandong Zhang ◽  
Rongfang Yan
Keyword(s):  
Reliability Function ◽  
Stochastic Orders ◽  
Residual Lifetime ◽  
Coherent System ◽  
Numerical Examples ◽  
Aging Properties ◽  
Theoretical Results

In this manuscript, we gain a mixture representation for reliability function of the residual lifetime of unfailed components in a coherent system under periodical inspections, given that the number of failed components before time t1 is r(≥0), but the system is still operating at time t1, and the system eventually failed at time t2(>t1). Some aging properties and stochastic orders of the residual lifetime on survival components are also established. Finally, some numerical examples and graphs are given in order to confirm the theoretical results.

scholarly journals Complete convergence for arrays of ratios of order statistics

2019 ◽  
Vol 17 (1) ◽  
pp. 439-451
Author(s):  
Yu Miao ◽  
Huanhuan Ma ◽  
Shoufang Xu ◽  
Andre Adler
Keyword(s):  
Order Statistics ◽  
Order Statistic ◽  
Complete Convergence ◽  
Pareto Distribution ◽  
Random Variables ◽  
Independent Random Variables

Abstract Let {Xn,k, 1 ≤ k ≤ mn, n ≥ 1} be an array of independent random variables from the Pareto distribution. Let Xn(k) be the kth largest order statistic from the nth row of the array and set Rn,in,jn = Xn(jn)/Xn(in) where jn < in. The aim of this paper is to study the complete convergence of the ratios {Rn,in,jn, n ≥ 1}.

The total time on test transform and the excess wealth stochastic orders of distributions

2002 ◽  
Vol 34 (04) ◽  
pp. 826-845 ◽  
Author(s):  
Subhash C. Kochar ◽  
Xiaohu Li ◽  
Moshe Shaked
Keyword(s):  
Statistical Theory ◽  
Stochastic Order ◽  
Random Variables ◽  
Distribution Functions ◽  
Stochastic Orders ◽  
Right Spread Order

For nonnegative random variables X and Y we write X ≤TTT Y if ∫0 F -1(p)(1-F(x))dx ≤ ∫0 G -1(p)(1-G(x))dx all p ∈ (0,1), where F and G denote the distribution functions of X and Y respectively. The purpose of this article is to study some properties of this new stochastic order. New properties of the excess wealth (or right-spread) order, and of other related stochastic orders, are also obtained. Applications in the statistical theory of reliability and in economics are included.

scholarly journals Extension of the past lifetime and its connection to the cumulative entropy

2015 ◽  
Vol 52 (04) ◽  
pp. 1156-1174 ◽  
Author(s):  
Antonio Di Crescenzo ◽  
Abdolsaeed Toomaj
Keyword(s):  
Random Variables ◽  
Independent Random Variables ◽  
Stochastic Orders ◽  
Absolutely Continuous ◽  
The Past ◽  
Probability Densities ◽  
The Mean ◽  
Weighted Probability ◽  
Series Systems ◽  
Cumulative Entropy

Given two absolutely continuous nonnegative independent random variables, we define the reversed relevation transform as dual to the relevation transform. We first apply such transforms to the lifetimes of the components of parallel and series systems under suitably proportionality assumptions on the hazard rates. Furthermore, we prove that the (reversed) relevation transform is commutative if and only if the proportional (reversed) hazard rate model holds. By repeated application of the reversed relevation transform we construct a decreasing sequence of random variables which leads to new weighted probability densities. We obtain various relations involving ageing notions and stochastic orders. We also exploit the connection of such a sequence to the cumulative entropy and to an operator that is dual to the Dickson-Hipp operator. Iterative formulae for computing the mean and the cumulative entropy of the random variables of the sequence are finally investigated.

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