ON A CLASS OF GENERALIZED MARSHALL–OLKIN BIVARIATE DISTRIBUTIONS AND SOME RELIABILITY CHARACTERISTICS

2013 ◽  
Vol 27 (2) ◽  
pp. 261-275 ◽  
Author(s):  
Ramesh C. Gupta ◽  
S.N.U.A. Kirmani ◽  
N. Balakrishnan
Keyword(s):  
Residual Life ◽  
Bivariate Distribution ◽  
Point Of View ◽  
Mean Residual Life ◽  
Bivariate Distributions ◽  
Bivariate Exponential Distribution ◽  
Bivariate Model ◽  
Reliability Characteristics ◽  
Bivariate Exponential ◽  
Special Cases

We consider here a general class of bivariate distributions from reliability point of view, and refer to it as generalized Marshall–Olkin bivariate distributions. This class includes as special cases the Marshall–Olkin bivariate exponential distribution and the class of bivariate distributions studied recently by Sarhan and Balakrishnan [25]. For this class, the reliability, survival, hazard, and mean residual life functions are all derived, and their monotonicity is discussed for the marginal as well as the conditional distributions. These functions are also studied for the series and parallel systems based on this bivariate distribution. Finally, the Clayton association measure for this bivariate model is derived in terms of the hazard gradient.

A generalized bivariate exponential distribution

1967 ◽  
Vol 4 (2) ◽  
pp. 291-302 ◽  
Author(s):  
Albert W. Marshall ◽  
Ingram Olkin
Keyword(s):  
Poisson Process ◽  
Exponential Distribution ◽  
Residual Life ◽  
Waiting Times ◽  
Practical Importance ◽  
Moment Generating Function ◽  
Bivariate Exponential Distribution ◽  
Bivariate Exponential ◽  
Shock Models ◽  
Bivariate Poisson Process

In a previous paper (Marshall and Olkin (1966)) the authors have derived a multivariate exponential distribution from points of view designed to indicate the applicability of the distribution. Two of these derivations are based on “shock models” and one is based on the requirement that residual life is independent of age.The practical importance of the univariate exponential distribution is partially due to the fact that it governs waiting times in a Poisson process. In this paper, the distribution of joint waiting times in a bivariate Poisson process is investigated. There are several ways to define “joint waiting time”. Some of these lead to the bivariate exponential distribution previously obtained by the authors, but others lead to a generalization of it. This generalized bivariate exponential distribution is also derived from shock models. The moment generating function and other properties of the distribution are investigated.

On bounds for some optimal policies in reliability

2002 ◽  
Vol 39 (3) ◽  
pp. 491-502 ◽  
Author(s):  
Jie Mi
Keyword(s):  
Lower Bound ◽  
Residual Life ◽  
Rate Function ◽  
Replacement Policy ◽  
Mean Residual Life ◽  
Failure Rate Function ◽  
Underlying Distribution ◽  
Bathtub Shape ◽  
Reliability Characteristics ◽  
Quality Of Products

Often in the study of reliability and its applications, the goal is to maximize or minimize certain reliability characteristics or some cost functions. For example, burn-in is a procedure used to improve the quality of products before they are used in the field. A natural question which arises is how long the burn-in procedure should last in order to maximize the mean residual life or the conditional survival probability. In the literature, an upper bound for the optimal burn-in time is obtained by assuming that the underlying distribution of the products has a bathtub-shaped failure rate function; however, no lower bound is available. A similar question arises in studying replacement policy, warranty policy, and inspection models. This article gives a lower bound for the optimal burn-in time, and lower and upper bounds for the optimal replacement and warranty policies, under the same bathtub-shape assumption.

scholarly journals On the Mean Residual Life Function and Stress and Strength Analysis under Different Loss Function for Lindley Distribution

2013 ◽  
Vol 2013 ◽  
pp. 1-13 ◽  
Author(s):  
Sajid Ali
Keyword(s):  
Residual Life ◽  
Reliability Theory ◽  
Loss Functions ◽  
Strength Analysis ◽  
Mean Residual Life ◽  
Lindley Distribution ◽  
Residual Life Function ◽  
Mean Residual Life Function ◽  
Reliability Characteristics ◽  
Life Function

Purpose. Mathematical properties of Lindley distribution are derived under different loss functions. These properties include mean residual life function, Lorenz curve, stress and strength characteristic, and their respective posterior risk via simulation scheme. Methodology. Bayesian approach is used for the reliability characteristics. Results are compared on the basis of posterior risk. Findings. Using prior information on the parameter of Lindley distribution, Bayes estimates for reliability characteristics are compared under different loss functions. Practical Implications. Since Lindley distribution is a mixture of gamma and exponential distribution, so Bayesian estimation of reliability characteristics will have a great implication in reliability theory. Originality. A real life application to waiting time data at the bank is also described for the developed procedures. This study is useful for researcher and practitioner in reliability theory.

An Improved Estimation of Parameter of Morgenstern-Type Bivariate Exponential Distribution Using Ranked Set Sampling

2022 ◽  
pp. 1-25
Author(s):  
Vishal Mehta
Keyword(s):  
Exponential Distribution ◽  
Mean Squared Error ◽  
Ranked Set Sampling ◽  
Error Estimator ◽  
Bivariate Exponential Distribution ◽  
Ranked Set Sample ◽  
Minimum Mean Squared Error ◽  
Squared Error ◽  
Bivariate Exponential ◽  
Special Cases

In this chapter, the authors suggest some improved versions of estimators of Morgenstern type bivariate exponential distribution (MTBED) based on the observations made on the units of ranked set sampling (RSS) regarding the study variable Y, which is correlated with the auxiliary variable X, where (X,Y) follows a MTBED. In this chapter, they firstly suggested minimum mean squared error estimator for estimation of 𝜃2 based on censored ranked set sample and their special case; further, they have suggested minimum mean squared error estimator for best linear unbiased estimator of 𝜃2 based on censored ranked set sample and their special cases; they also suggested minimum mean squared error estimator for estimation of 𝜃2 based on unbalanced multistage ranked set sampling and their special cases. Efficiency comparisons are also made in this work.

Preservation results for life distributions based on comparisons with asymptotic remaining life under replacements

2000 ◽  
Vol 37 (4) ◽  
pp. 999-1009 ◽  
Author(s):  
M. C. Bhattacharjee ◽  
A. M. Abouammoh ◽  
A. N. Ahmed ◽  
A. M. Barry
Keyword(s):  
Residual Life ◽  
Point Of View ◽  
Mean Residual Life ◽  
Remaining Life ◽  
Repairable Systems ◽  
Repair Strategy ◽  
Aging Properties ◽  
Life Distributions ◽  
The Mean ◽  
Life Function

We investigate some preservation properties of two nonparametric classes of survival distributions and their duals, under appropriate reliability operations. The aging properties defining these nonparametric classes are based on comparing the mean life of a new unit to the mean residual life function of the asymptotic remaining survival time of the unit under repeated perfect repairs. They are motivated from a point of view that realistic notions of degradation, applicable to repairable systems, should be based on contrasting some aspect of the remaining life of a repairable unit (under a given repair strategy, such as renewals) to the life of a new unit.

scholarly journals Properties and inference for a new class of XGamma distributions with an application

2019 ◽  
Vol 13 (4) ◽  
pp. 335-346
Author(s):  
Hayrinisa Demirci Biçer
Keyword(s):  
Monte Carlo ◽  
Hazard Rate ◽  
Residual Life ◽  
Moment Generating Function ◽  
Distribution Model ◽  
Mean Residual Life ◽  
Survival Functions ◽  
Data Set ◽  
New Class ◽  
Special Cases

Abstract The current paper introduces a new flexible probability distribution model called transmuted XGamma distribution which pullulates from the XGamma distribution and possesses the characteristics of XGamma distribution in special cases. In the paper, we obtain the explicit expressions for some important statistical properties of the introduced distribution such as hazard rate and survival functions, mean residual life, moment-generating function, moments, skewness, kurtosis, distribution of its order statistics, Lorenz and Bonferroni curves. Besides obtaining the various effective estimators for the parameters of the distribution, estimation performances of these estimators are comparatively examined with a series of Monte Carlo simulations. Furthermore, to demonstrate the modeling ability of the proposed distribution on real-world phenomena, an illustrative example is performed by using an actual data set in connection with the field of the lifetime.

Preservation results for life distributions based on comparisons with asymptotic remaining life under replacements

2000 ◽  
Vol 37 (04) ◽  
pp. 999-1009 ◽  
Author(s):  
M. C. Bhattacharjee ◽  
A. M. Abouammoh ◽  
A. N. Ahmed ◽  
A. M. Barry
Keyword(s):  
Residual Life ◽  
Point Of View ◽  
Mean Residual Life ◽  
Remaining Life ◽  
Repairable Systems ◽  
Repair Strategy ◽  
Aging Properties ◽  
Life Distributions ◽  
The Mean ◽  
Life Function

We investigate some preservation properties of two nonparametric classes of survival distributions and their duals, under appropriate reliability operations. The aging properties defining these nonparametric classes are based on comparing the mean life of a new unit to the mean residual life function of the asymptotic remaining survival time of the unit under repeated perfect repairs. They are motivated from a point of view that realistic notions of degradation, applicable to repairable systems, should be based on contrasting some aspect of the remaining life of a repairable unit (under a given repair strategy, such as renewals) to the life of a new unit.

MEAN RESIDUAL LIFE AND OTHER PROPERTIES OF WEIBULL RELATED BATHTUB SHAPE FAILURE RATE DISTRIBUTIONS

2004 ◽  
Vol 11 (02) ◽  
pp. 113-132 ◽  
Author(s):  
C. D. LAI ◽  
LINGYUN ZHANG ◽  
M. XIE
Keyword(s):  
Failure Rate ◽  
Residual Life ◽  
Weibull Model ◽  
Parameters Estimation ◽  
Mean Residual Life ◽  
Bathtub Shape ◽  
Reliability Characteristics ◽  
Rate Functions ◽  
Life Distributions ◽  
The Relationship

The two-parameter Weibull distribution is widely used in reliability analysis. Because of its monotonic ageing behaviour, its applicability is hampered in certain reliability situations. Several generalizations and extensions of the Weibull model have been proposed in the literature to overcome this limitation but their properties have not yet been described in a unified manner. In this paper, graphical displays of the mean residual life curves of several families of Weibull related life distributions are given together with their corresponding failure rate functions. The relationship between these two functions are visibly demonstrated. We focus our attention on the Weibull related families that have bathtub or modified bathtub shape failure rates. Important reliability characteristics such as burn-in, change point and flatness of bathtub of these families are examined. Model selection and parameters estimation are also discussed.

SHARP BOUNDS FOR SURVIVAL PROBABILITY WHEN AGEING IS NOT MONOTONE

2018 ◽  
Vol 33 (2) ◽  
pp. 205-219 ◽  
Author(s):  
Ruhul Ali Khan ◽  
Murari Mitra
Keyword(s):  
Change Point ◽  
Residual Life ◽  
Mean Residual Life ◽  
Survival Functions ◽  
Sharp Bounds ◽  
Residual Life Function ◽  
Mean Residual Life Function ◽  
Special Cases ◽  
The Mean ◽  
Life Function

We exploit a novel bounding argument to obtain sharp bounds for survival functions belonging to the Increasing initially then Decreasing Mean Residual Life (IDMRL) class introduced by Guess, Hollander and Proschan (1986) [8]. The bounds obtained are in terms of the mean, change point and pinnacle of the mean residual life function. The bounds for the monotonic ageing classes Decreasing Mean Residual Life (DMRL) and Increasing Mean Residual Life (IMRL) are obtained as special cases. Discussions on the bounds as well as two concrete illustrative examples are included.

On bounds for some optimal policies in reliability

2002 ◽  
Vol 39 (03) ◽  
pp. 491-502 ◽  
Author(s):  
Jie Mi
Keyword(s):  
Lower Bound ◽  
Residual Life ◽  
Rate Function ◽  
Replacement Policy ◽  
Mean Residual Life ◽  
Failure Rate Function ◽  
Underlying Distribution ◽  
Bathtub Shape ◽  
Reliability Characteristics ◽  
Quality Of Products

Often in the study of reliability and its applications, the goal is to maximize or minimize certain reliability characteristics or some cost functions. For example, burn-in is a procedure used to improve the quality of products before they are used in the field. A natural question which arises is how long the burn-in procedure should last in order to maximize the mean residual life or the conditional survival probability. In the literature, an upper bound for the optimal burn-in time is obtained by assuming that the underlying distribution of the products has a bathtub-shaped failure rate function; however, no lower bound is available. A similar question arises in studying replacement policy, warranty policy, and inspection models. This article gives a lower bound for the optimal burn-in time, and lower and upper bounds for the optimal replacement and warranty policies, under the same bathtub-shape assumption.

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