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.

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.

Shock models leading to increasing failure rate and decreasing mean residual life survival

1982 ◽  
Vol 19 (01) ◽  
pp. 158-166 ◽  
Author(s):  
Malay Ghosh ◽  
Nader Ebrahimi
Keyword(s):  
Poisson Process ◽  
Failure Rate ◽  
Residual Life ◽  
Mean Residual Life ◽  
Increasing Failure Rate ◽  
Shock Models

Shock models leading to various univariate and bivariate increasing failure rate (IFR) and decreasing mean residual life (DMRL) distributions are discussed. For proving the IFR properties, shocks are not necessarily assumed to be governed by a Poisson process.

ROLLER-COASTER FAILURE RATES AND MEAN RESIDUAL LIFE FUNCTIONS WITH APPLICATION TO THE EXTENDED GENERALIZED INVERSE GAUSSIAN MODEL

2010 ◽  
Vol 25 (1) ◽  
pp. 103-118 ◽  
Author(s):  
Ramesh C. Gupta ◽  
Weston Viles
Keyword(s):  
Failure Rate ◽  
Generalized Inverse ◽  
Residual Life ◽  
Gaussian Model ◽  
Change Points ◽  
Mean Residual Life ◽  
Additional Parameter ◽  
Inverse Gaussian ◽  
Generalized Inverse Gaussian ◽  
The Relationship

The investigation in this article was motivated by an extended generalized inverse Gaussian (EGIG) distribution, which has more than one turning point of the failure rate for certain values of the parameters. In order to study the turning points of a failure rate, we appeal to Glaser's eta function, which is much simpler to handle. We present some general results for studying the reationship among the change points of Glaser's eta function, the failure rate, and the mean residual life function (MRLF). Additionally we establish an ordering among the number of change points of Glaser's eta function, the failure rate, and the MRLF. These results are used to investigate, in detail, the monotonicity of the three functions in the case of the EGIG. The EGIG model has one additional parameter, δ, than the generalized inverse Gaussian (GIG) model's three parameters; see Jorgensen [7]. It has been observed that the EGIG model fits certain datasets better than the GIG of Jorgensen [7]. Thus, the purpose of this article is to present some general results dealing with the relationship among the change points of the three functions described earlier. The EGIG model is used as an illustration.

Shock models leading to increasing failure rate and decreasing mean residual life survival

1982 ◽  
Vol 19 (1) ◽  
pp. 158-166 ◽  
Author(s):  
Malay Ghosh ◽  
Nader Ebrahimi
Keyword(s):  
Poisson Process ◽  
Failure Rate ◽  
Residual Life ◽  
Mean Residual Life ◽  
Increasing Failure Rate ◽  
Shock Models

Shock models leading to various univariate and bivariate increasing failure rate (IFR) and decreasing mean residual life (DMRL) distributions are discussed. For proving the IFR properties, shocks are not necessarily assumed to be governed by a Poisson process.

scholarly journals Reversed Preservation Properties for Series and Parallel Systems

2007 ◽  
Vol 44 (4) ◽  
pp. 928-937 ◽  
Author(s):  
Félix Belzunce ◽  
Helena Martínez-Puertas ◽  
José M. Ruiz
Keyword(s):  
Failure Rate ◽  
Residual Life ◽  
Parallel Systems ◽  
Mean Residual Life ◽  
Convex Order ◽  
Increasing Failure Rate ◽  
Decreasing Failure Rate ◽  
New Better Than Used ◽  
Better Than

Recently Li and Yam (2005) studied which ageing properties for series and parallel systems are inherited for the components. In this paper we provide new results for the increasing convex and concave orders, the increasing mean residual life (IMRL), decreasing failure rate (DFR), the new worse than used in expectation (NWUE), the increasing failure rate in average (IFRA), the decreasing failure rate in average (DFRA), and the new better than used in the convex order (NBUC) ageing classes.

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.

scholarly journals Reversed Preservation Properties for Series and Parallel Systems

2007 ◽  
Vol 44 (04) ◽  
pp. 928-937
Author(s):  
Félix Belzunce ◽  
Helena Martínez-Puertas ◽  
José M. Ruiz
Keyword(s):  
Failure Rate ◽  
Residual Life ◽  
Parallel Systems ◽  
Mean Residual Life ◽  
Convex Order ◽  
Increasing Failure Rate ◽  
Decreasing Failure Rate ◽  
New Better Than Used ◽  
Better Than

Recently Li and Yam (2005) studied which ageing properties for series and parallel systems are inherited for the components. In this paper we provide new results for the increasing convex and concave orders, the increasing mean residual life (IMRL), decreasing failure rate (DFR), the new worse than used in expectation (NWUE), the increasing failure rate in average (IFRA), the decreasing failure rate in average (DFRA), and the new better than used in the convex order (NBUC) ageing classes.

scholarly journals A Distribution for Instantaneous Failures

Stats ◽  
2019 ◽  
Vol 2 (2) ◽  
pp. 247-258 ◽  
Author(s):  
Pedro L. Ramos ◽  
Francisco Louzada
Keyword(s):  
Residual Life ◽  
Likelihood Estimation ◽  
Estimation Methods ◽  
Mean Residual Life ◽  
Parameter Distribution ◽  
Coverage Probabilities ◽  
Instantaneous Failures ◽  
The Relationship ◽  
Mean Variance ◽  
New Distribution

A new one-parameter distribution is proposed in this paper. The new distribution allows for the occurrence of instantaneous failures (inliers) that are natural in many areas. Closed-form expressions are obtained for the moments, mean, variance, a coefficient of variation, skewness, kurtosis, and mean residual life. The relationship between the new distribution with the exponential and Lindley distributions is presented. The new distribution can be viewed as a combination of a reparametrized version of the Zakerzadeh and Dolati distribution with a particular case of the gamma model and the occurrence of zero value. The parameter estimation is discussed under the method of moments and the maximum likelihood estimation. A simulation study is performed to verify the efficiency of both estimation methods by computing the bias, mean squared errors, and coverage probabilities. The superiority of the proposed distribution and some of its concurrent distributions are tested by analyzing four real lifetime datasets.

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