Some results on the relative ageing of two life distributions

1994 ◽  
Vol 31 (4) ◽  
pp. 991-1003 ◽  
Author(s):  
Debasis Sengupta ◽  
Jayant V. Deshpande
Keyword(s):  
Hazard Ratio ◽  
Partial Ordering ◽  
Closure Properties ◽  
Partial Orderings ◽  
Life Distributions ◽  
Crossing Hazards ◽  
Moment Bounds

Kalashnikov and Rachev (1986) have proposed a partial ordering of life distributions which is equivalent to an increasing hazard ratio, when the ratio exists. This model can represent the phenomenon of crossing hazards, which has received considerable attention in recent years. In this paper we study this and two other models of relative ageing. Their connections with common partial orderings in the reliability literature are discussed. We examine the closure properties of the three orderings under several operations. Finally, we give reliability and moment bounds for a distribution when it is ordered with respect to a known distribution.

Some results on the relative ageing of two life distributions

1994 ◽  
Vol 31 (04) ◽  
pp. 991-1003 ◽  
Author(s):  
Debasis Sengupta ◽  
Jayant V. Deshpande
Keyword(s):  
Hazard Ratio ◽  
Partial Ordering ◽  
Closure Properties ◽  
Partial Orderings ◽  
Life Distributions ◽  
Crossing Hazards ◽  
Moment Bounds

Kalashnikov and Rachev (1986) have proposed a partial ordering of life distributions which is equivalent to an increasing hazard ratio, when the ratio exists. This model can represent the phenomenon of crossing hazards, which has received considerable attention in recent years. In this paper we study this and two other models of relative ageing. Their connections with common partial orderings in the reliability literature are discussed. We examine the closure properties of the three orderings under several operations. Finally, we give reliability and moment bounds for a distribution when it is ordered with respect to a known distribution.

CLASSIFICATION OF MULTIVARIATE LIFE DISTRIBUTIONS BASED ON PARTIAL ORDERING

2002 ◽  
Vol 16 (1) ◽  
pp. 129-137 ◽  
Author(s):  
Dilip Roy
Keyword(s):  
Present Article ◽  
Failure Rate ◽  
Partial Ordering ◽  
Increasing Failure Rate ◽  
Multivariate Generalization ◽  
Partial Orderings ◽  
Life Distributions ◽  
New Better Than Used ◽  
Better Than

Barlow and Proschan presented some interesting connections between univariate classifications of life distributions and partial orderings where equivalent definitions for increasing failure rate (IFR), increasing failure rate average (IFRA), and new better than used (NBU) classes were given in terms of convex, star-shaped, and superadditive orderings. Some related results are given by Ross and Shaked and Shanthikumar. The introduction of a multivariate generalization of partial orderings is the object of the present article. Based on that concept of multivariate partial orderings, we also propose multivariate classifications of life distributions and present a study on more IFR-ness.

Families of life distributions characterized by two moments

1990 ◽  
Vol 27 (03) ◽  
pp. 720-725 ◽  
Author(s):  
Manish C. Bhattacharjee ◽  
Jayaram Sethuraman
Keyword(s):  
Exponential Distribution ◽  
Partial Ordering ◽  
Life Distribution ◽  
Partial Orderings ◽  
Life Distributions ◽  
The Moment

We consider several classical notions of partial orderings among life distributions which have been used to describe ageing properties and tail domination. We show that if a distribution G dominates another distribution F in one of these partial orderings introduced here, and if two moments of G agree with those of F, including the moment that describes this partial ordering, then G = F. This leads to a characterization of the exponential distribution among HNBUE and HNWUE life distribution classes, and thus extends the results of Basu and Bhattacharjee (1984) and rectifies an error in that paper.

Families of life distributions characterized by two moments

1990 ◽  
Vol 27 (3) ◽  
pp. 720-725 ◽  
Author(s):  
Manish C. Bhattacharjee ◽  
Jayaram Sethuraman
Keyword(s):  
Exponential Distribution ◽  
Partial Ordering ◽  
Life Distribution ◽  
Partial Orderings ◽  
Life Distributions ◽  
The Moment

We consider several classical notions of partial orderings among life distributions which have been used to describe ageing properties and tail domination. We show that if a distribution G dominates another distribution F in one of these partial orderings introduced here, and if two moments of G agree with those of F, including the moment that describes this partial ordering, then G = F. This leads to a characterization of the exponential distribution among HNBUE and HNWUE life distribution classes, and thus extends the results of Basu and Bhattacharjee (1984) and rectifies an error in that paper.

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.

Natural Partial Orders

1968 ◽  
Vol 20 ◽  
pp. 535-554 ◽  
Author(s):  
R. A. Dean ◽  
Gordon Keller
Keyword(s):  
Partial Orders ◽  
Partial Ordering ◽  
Finite Set ◽  
Partial Orderings

Let n be an ordinal. A partial ordering P of the ordinals T = T(n) = {w: w < n} is called natural if x P y implies x ⩽ y.A natural partial ordering, hereafter abbreviated NPO, of T(n) is thus a coarsening of the natural total ordering of the ordinals. Every partial ordering of a finite set 5 is isomorphic to a natural partial ordering. This is a consequence of the theorem of Szpielrajn (5) which states that every partial ordering of a set may be refined to a total ordering. In this paper we consider only natural partial orderings. In the first section we obtain theorems about the lattice of all NPO's of T(n).

Partial Orderings of Distributions Based on Right-Spread Functions

1998 ◽  
Vol 35 (1) ◽  
pp. 221-228 ◽  
Author(s):  
J. M. Fernandez-Ponce ◽  
S. C. Kochar ◽  
J. Muñoz-Perez
Keyword(s):  
Probability Distributions ◽  
Partial Ordering ◽  
Dispersion Measure ◽  
Dispersive Ordering ◽  
Partial Orderings

In this paper we introduce a quantile dispersion measure. We use it to characterize different classes of ageing distributions. Based on the quantile dispersion measure, we propose a new partial ordering for comparing the spread or dispersion in two probability distributions. This new partial ordering is weaker than the well known dispersive ordering and it retains most of its interesting properties.

A new ordering for stochastic majorization: theory and applications

1992 ◽  
Vol 24 (03) ◽  
pp. 604-634 ◽  
Author(s):  
Cheng-Shang Chang
Keyword(s):  
Sample Path ◽  
Stochastic Ordering ◽  
Partial Ordering ◽  
Scheduling Problems ◽  
Unified Approach ◽  
Finite Buffers ◽  
Routing And Scheduling ◽  
Partial Orderings ◽  
Stochastic Load ◽  
Stochastic Majorization

In this paper, we develop a unified approach for stochastic load balancing on various multiserver systems. We expand the four partial orderings defined in Marshall and Olkin, by defining a new ordering based on the set of functions that are symmetric, L-subadditive and convex in each variable. This new partial ordering is shown to be equivalent to the previous four orderings for comparing deterministic vectors but differs for random vectors. Sample-path criteria and a probability enumeration method for the new stochastic ordering are established and the ordering is applied to various fork-join queues, routing and scheduling problems. Our results generalize previous work and can be extended to multivariate stochastic majorization which includes tandem queues and queues with finite buffers.

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.

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