Multivariate conditional hazard rates and the MIFRA and MIFR properties

1988 ◽  
Vol 25 (01) ◽  
pp. 150-168 ◽  
Author(s):  
Moshe Shaked ◽  
J. George Shanthikumar
Keyword(s):  
Failure Rate ◽  
Sufficient Conditions ◽  
Load Sharing ◽  
Hazard Rates ◽  
Reliability Model ◽  
Increasing Failure Rate ◽  
Imperfect Repair ◽  
The Relationship

If T = (T 1, · ··, Tn ) is a vector of random lifetimes then its distribution can be determined by a set λof multivariate conditional hazard rates. In this paper, sufficient conditions on λare found which imply that T is Block–Savits MIFRA (multivariate increasing failure rate average) or Savits MIFR (multivariate increasing failure rate). Applications for a multivariate reliability model of Ross and for load-sharing models are given. The relationship between Shaked and Shanthikumar model of multivariate imperfect repair and the MIFRA property is also discussed.

Multivariate conditional hazard rates and the MIFRA and MIFR properties

1988 ◽  
Vol 25 (1) ◽  
pp. 150-168 ◽  
Author(s):  
Moshe Shaked ◽  
J. George Shanthikumar
Keyword(s):  
Failure Rate ◽  
Sufficient Conditions ◽  
Load Sharing ◽  
Hazard Rates ◽  
Reliability Model ◽  
Increasing Failure Rate ◽  
Imperfect Repair ◽  
The Relationship

If T = (T1, · ··, Tn) is a vector of random lifetimes then its distribution can be determined by a set λof multivariate conditional hazard rates. In this paper, sufficient conditions on λare found which imply that T is Block–Savits MIFRA (multivariate increasing failure rate average) or Savits MIFR (multivariate increasing failure rate). Applications for a multivariate reliability model of Ross and for load-sharing models are given. The relationship between Shaked and Shanthikumar model of multivariate imperfect repair and the MIFRA property is also discussed.

Failure distributions of shock models

1980 ◽  
Vol 17 (03) ◽  
pp. 745-752 ◽  
Author(s):  
Gary Gottlieb
Keyword(s):  
Failure Rate ◽  
Sufficient Conditions ◽  
Cumulative Damage ◽  
Shock Model ◽  
Life Distribution ◽  
Increasing Failure Rate ◽  
Shock Models ◽  
Damage Process

A single device shock model is studied. The device is subject to some damage process. Under the assumption that as the cumulative damage increases, the probability that any additional damage will cause failure increases, we find sufficient conditions on the shocking process so that the life distribution will be increasing failure rate.

The hazard and vitality measures of ageing

1989 ◽  
Vol 26 (03) ◽  
pp. 532-542 ◽  
Author(s):  
Joseph Kupka ◽  
Sonny Loo
Keyword(s):  
Sudden Death ◽  
Failure Rate ◽  
Residual Life ◽  
Ageing Process ◽  
Mean Residual Life ◽  
Absolutely Continuous ◽  
Intrinsic Interest ◽  
Increasing Failure Rate ◽  
Time Period ◽  
The Relationship

A new measure of the ageing process called the vitality measure is introduced. It measures the ‘vitality' of a time period in terms of the increase in average lifespan which results from surviving that time period. Apart from intrinsic interest, the vitality measure clarifies the relationship between the familiar properties of increasing hazard and decreasing mean residual life. The main theorem asserts that increasing hazard is equivalent to the requirement that mean residual life decreases faster than vitality. It is also shown for general (i.e. not necessarily absolutely continuous) distributions that the properties of increasing hazard, increasing failure rate, and increasing probability of ‘sudden death' are all equivalent.

Multivariate hazard rates and stochastic ordering

1987 ◽  
Vol 19 (1) ◽  
pp. 123-137 ◽  
Author(s):  
Moshe Shaked ◽  
J. George Shanthikumar
Keyword(s):  
Stochastic Ordering ◽  
Load Sharing ◽  
Hazard Rates ◽  
Imperfect Repair

Properties of the conditional hazard rates of X1, · ··, Xn and Y1, · ··, Yn, which imply (X1, · ··, Xn) (Y1, · ··, Yn), are found. These are used to find conditions on the hazard rates of T = (T1, · ··, Tn) which ensure that T has the MIHR | property of Arjas (1981a) and the ‘weakened by failure’ property of Arjas and Norros (1984). Applications for load-sharing model and multivariate imperfect repair are given.

Multivariate hazard rates and stochastic ordering

1987 ◽  
Vol 19 (01) ◽  
pp. 123-137 ◽  
Author(s):  
Moshe Shaked ◽  
J. George Shanthikumar
Keyword(s):  
Stochastic Ordering ◽  
Load Sharing ◽  
Hazard Rates ◽  
Simple Property ◽  
Imperfect Repair

Properties of the conditional hazard rates of X 1, · ··, Xn and Y 1, · ··, Yn, which imply (X 1, · ··, Xn ) (Y 1, · ··, Yn ), are found. These are used to find conditions on the hazard rates of T = (T 1, · ··, Tn ) which ensure that T has the MIHR | property of Arjas (1981a) and the ‘weakened by failure’ property of Arjas and Norros (1984). Applications for load-sharing model and multivariate imperfect repair are given.

On the Closure of Aging Classes Under the Formation of Coherent Structures

2021 ◽  
pp. 2140002
Author(s):  
Ioannis S. Triantafyllou
Keyword(s):  
Failure Rate ◽  
Coherent Structures ◽  
Sufficient Conditions ◽  
Failure Criteria ◽  
Closure Property ◽  
Coherent Systems ◽  
Increasing Failure Rate ◽  
Independent And Identically Distributed

In this paper, we study the closure property of the Increasing Failure Rate (IFR) class under the formation of coherent systems. Sufficient conditions for the nonpreservation of the IFR attribute for reliability structures consisting of [Formula: see text] independent and identically distributed ([Formula: see text] components are provided. More precisely, we deal with the IFR preservation (or nonpreservation) under the formation of structures with two common failure criteria by the aid of their signature vectors.

Failure distributions of shock models

1980 ◽  
Vol 17 (3) ◽  
pp. 745-752 ◽  
Author(s):  
Gary Gottlieb
Keyword(s):  
Failure Rate ◽  
Sufficient Conditions ◽  
Cumulative Damage ◽  
Shock Model ◽  
Life Distribution ◽  
Increasing Failure Rate ◽  
Shock Models ◽  
Damage Process

A single device shock model is studied. The device is subject to some damage process. Under the assumption that as the cumulative damage increases, the probability that any additional damage will cause failure increases, we find sufficient conditions on the shocking process so that the life distribution will be increasing failure rate.

On the first-passage times of pure jump processes

1988 ◽  
Vol 25 (3) ◽  
pp. 501-509 ◽  
Author(s):  
Moshe Shaked ◽  
J. George Shanthikumar
Keyword(s):  
Failure Rate ◽  
Sufficient Conditions ◽  
Jump Process ◽  
First Passage Times ◽  
Jump Processes ◽  
First Passage ◽  
Increasing Failure Rate ◽  
Passage Times

Let Tx be the time it takes for a pure jump process, which starts at 0, to cross a threshold x > 0. Sufficient conditions on the parameters of this process under which Tx has increasing failure rate average (IFRA), increasing failure rate (IFR) or logconcave density (PF2) are identified. The conditions for IFRA are weaker than those of Drosen (1986). Sufficient conditions on the parameter of a pure jump process for Tx to the IFR or PF2 are not available in the literature.

scholarly journals On increasing-failure-rate random variables

2005 ◽  
Vol 42 (03) ◽  
pp. 797-809 ◽  
Author(s):  
Sheldon M. Ross ◽  
J. George Shanthikumar ◽  
Zegang Zhu
Keyword(s):  
Markov Chain ◽  
Failure Rate ◽  
Renewal Process ◽  
Counting Process ◽  
Random Number ◽  
Sufficient Conditions ◽  
Random Variables ◽  
Random Variable ◽  
Increasing Failure Rate ◽  
Random Threshold

We provide sufficient conditions for the following types of random variable to have the increasing-failure-rate (IFR) property: sums of a random number of random variables; the time at which a Markov chain crosses a random threshold; the time until a random number of events have occurred in an inhomogeneous Poisson process; and the number of events of a renewal process, and of a general counting process, that have occurred by a randomly distributed time.

The hazard and vitality measures of ageing

1989 ◽  
Vol 26 (3) ◽  
pp. 532-542 ◽  
Author(s):  
Joseph Kupka ◽  
Sonny Loo
Keyword(s):  
Sudden Death ◽  
Failure Rate ◽  
Residual Life ◽  
Ageing Process ◽  
Mean Residual Life ◽  
Absolutely Continuous ◽  
Intrinsic Interest ◽  
Increasing Failure Rate ◽  
Time Period ◽  
The Relationship

A new measure of the ageing process called the vitality measure is introduced. It measures the ‘vitality' of a time period in terms of the increase in average lifespan which results from surviving that time period. Apart from intrinsic interest, the vitality measure clarifies the relationship between the familiar properties of increasing hazard and decreasing mean residual life. The main theorem asserts that increasing hazard is equivalent to the requirement that mean residual life decreases faster than vitality. It is also shown for general (i.e. not necessarily absolutely continuous) distributions that the properties of increasing hazard, increasing failure rate, and increasing probability of ‘sudden death' are all equivalent.

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