Characterization of life distributions under some generalized stochastic orderings

1997 ◽  
Vol 34 (3) ◽  
pp. 711-719 ◽  
Author(s):  
Jun Cai ◽  
Yanhong Wu
Keyword(s):  
Generating Functions ◽  
Monotone Function ◽  
Laplace Transforms ◽  
Stochastic Orderings ◽  
Completely Monotone ◽  
Moment Generating Functions ◽  
Life Distributions ◽  
Stochastic Equality ◽  
Unified Method

In this paper we investigate the characterizations of life distributions under four stochastic orderings, < p, < (p), < (p) and < L, by a unified method. Conditions for the stochastic equality of two non-negative random variables under the four stochastic orderings are derived. Many previous results are consequences. As applications, we provide characterizations of life distributions by a single value of their Laplace transforms under orderings < p and < (p) and their moment generating functions under orderings < p and < (p). Under ordering < L, a characterization is given by the expected value of a strictly completely monotone function. The conditions for the stochastic equality of two non-negative vectors under the stochastic orderings < (p), < (p) and < L are presented in terms of the Laplace transforms and moment generating functions of their extremes and sample means. Characterizations of the exponential distribution among L and L life distribution classes are also given.

Characterization of life distributions under some generalized stochastic orderings

1997 ◽  
Vol 34 (03) ◽  
pp. 711-719
Author(s):  
Jun Cai ◽  
Yanhong Wu
Keyword(s):  
Generating Functions ◽  
Monotone Function ◽  
Laplace Transforms ◽  
Stochastic Orderings ◽  
Completely Monotone ◽  
Moment Generating Functions ◽  
Life Distributions ◽  
Stochastic Equality ◽  
Unified Method

In this paper we investigate the characterizations of life distributions under four stochastic orderings, &lt; p , &lt; (p), &lt; (p) and &lt; L, by a unified method. Conditions for the stochastic equality of two non-negative random variables under the four stochastic orderings are derived. Many previous results are consequences. As applications, we provide characterizations of life distributions by a single value of their Laplace transforms under orderings &lt; p and &lt; (p) and their moment generating functions under orderings &lt; p and &lt; (p). Under ordering &lt; L, a characterization is given by the expected value of a strictly completely monotone function. The conditions for the stochastic equality of two non-negative vectors under the stochastic orderings &lt; (p), &lt; (p) and &lt; L are presented in terms of the Laplace transforms and moment generating functions of their extremes and sample means. Characterizations of the exponential distribution among L and L life distribution classes are also given.

Bounds of moment generating functions of some life distributions

2005 ◽  
Vol 46 (4) ◽  
pp. 575-585 ◽  
Author(s):  
Ibrahim A. Ahmad ◽  
A. R. Mugdadi
Keyword(s):  
Generating Functions ◽  
Moment Generating Functions ◽  
Life Distributions

Stochastic Orders in Terms of Laplace Transforms

1995 ◽  
Vol 45 (3-4) ◽  
pp. 195-202 ◽  
Author(s):  
Asok K. Nanda
Keyword(s):  
Laplace Transform ◽  
Sufficient Conditions ◽  
Laplace Transforms ◽  
Necessary And Sufficient Conditions ◽  
Stochastic Orders ◽  
Partial Orderings ◽  
Life Distributions ◽  
Necessary And Sufficient

Recently s-FR and s-ST orderings have been defined in the literature. They are more general in the sense that most of the earlier known partial orderings reduce as particular cases of these orderings. Moreover, these orderings have helped in defining new and useful ageing criterion. In this paper, using Laplace transform, we characterize, by means of necessary and sufficient conditions. the property that two life distributions are ordered in the s-FR and s-ST sense. The characterization of LR, FR, MR, VR, STand HAMR orderings follow as particular cases.

scholarly journals A Finite-Interval Uniqueness Theorem for Bilateral Laplace Transforms

2007 ◽  
Vol 2007 ◽  
pp. 1-6 ◽  
Author(s):  
Patrick Chareka
Keyword(s):  
Laplace Transform ◽  
Uniqueness Theorem ◽  
Generating Functions ◽  
Finite Interval ◽  
Probability Distributions ◽  
Laplace Transforms ◽  
Complex Argument ◽  
Moment Generating Functions

Two or more bilateral Laplace transforms with a complex argument “s” may be equal in a finite vertical interval when, in fact, the transforms correspond to different functions. In this article, we prove that the existence of a bilateral Laplace transform in any finite horizontal interval uniquely determines the corresponding function. The result appears to be new as we could not find it in the literature. The novelty of the result is that the interval need not contain zero, the function need not be nonnegative and need not be integrable. The result has a potential to be useful in the context of fitting probability distributions to data using Laplace transforms or moment generating functions.

Minimal distributions in a stochastic partial ordering and bounds for Gaussian processes

1983 ◽  
Vol 20 (03) ◽  
pp. 529-536
Author(s):  
W. J. R. Eplett
Keyword(s):  
Gaussian Process ◽  
Gaussian Processes ◽  
Partial Ordering ◽  
Boundary Crossing ◽  
Conditional Probabilities ◽  
Life Distribution ◽  
Life Distributions ◽  
Bounded Below ◽  
Crossing Probabilities

A natural requirement to impose upon the life distribution of a component is that after inspection at some randomly chosen time to check whether it is still functioning, its life distribution from the time of checking should be bounded below by some specified distribution which may be defined by external considerations. Furthermore, the life distribution should ideally be minimal in the partial ordering obtained from the conditional probabilities. We prove that these specifications provide an apparently new characterization of the DFRA class of life distributions with a corresponding result for IFRA distributions. These results may be transferred, using Slepian's lemma, to obtain bounds for the boundary crossing probabilities of a stationary Gaussian process.

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