Dynamic Multivariate Quantile Residual Life in Reliability Theory

We extend the univariate α-quantile residual life function to multivariate setting preserving its dynamic feature. Principal attributes of this function are derived and their relationship to the dynamic multivariate hazard rate function is discussed. A corresponding ordering, namely, α-quantile residual life order, for random vectors of lifetimes is introduced and studied. Based on the proposed ordering, a notion of positive dependency is presented. Finally, a discussion about conditions characterizing the class of decreasing multivariate α-quantile residual life functions is pointed out.

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Some new results on bathtub-shaped hazard rate models

Mathematical Biosciences and Engineering ◽

10.3934/mbe.2022057 ◽

2021 ◽

Vol 19 (2) ◽

pp. 1239-1250

Author(s):

Mohamed Kayid ◽

Keyword(s):

Order Statistics ◽

Hazard Rate ◽

Residual Life ◽

Change Points ◽

Residual Lifetime ◽

Lifetime Distributions ◽

Residual Life Function ◽

Basic Model ◽

Quantile Residual Life ◽

Life Function

<abstract><p>The most common non-monotonic hazard rate situations in life sciences and engineering involves bathtub shapes. This paper focuses on the quantile residual life function in the class of lifetime distributions that have bathtub-shaped hazard rate functions. For this class of distributions, the shape of the $ \alpha $-quantile residual lifetime function was studied. Then, the change points of the $ \alpha $-quantile residual life function of a general weighted hazard rate model were compared with the corresponding change points of the basic model in terms of their location. As a special weighted model, the order statistics were considered and the change points related to the order statistics were compared with the change points of the baseline distribution. Moreover, some comparisons of the change points of two different order statistics were presented.</p></abstract>

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MEAN RESIDUAL LIFE FUNCTION FOR ADDITIVE AND MULTIPLICATIVE HAZARD RATE MODELS

Probability in the Engineering and Informational Sciences ◽

10.1017/s0269964815000388 ◽

2015 ◽

Vol 30 (2) ◽

pp. 281-297 ◽

Cited By ~ 1

Author(s):

Ramesh C. Gupta

Keyword(s):

Hazard Rate ◽

Residual Life ◽

Turning Points ◽

Sufficient Conditions ◽

Mean Residual Life ◽

Residual Life Function ◽

Mean Residual Life Function ◽

Hazard Rate Models ◽

The Mean ◽

Life Function

This paper deals with the mean residual life function (MRLF) and its monotonicity in the case of additive and multiplicative hazard rate models. It is shown that additive (multiplicative) hazard rate does not imply reduced (proportional) MRLF and vice versa. Necessary and sufficient conditions are obtained for the two models to hold simultaneously. In the case of non-monotonic failure rates, the location of the turning points of the MRLF is investigated in both the cases. The case of random additive and multiplicative hazard rate is also studied. The monotonicity of the mean residual life is studied along with the location of the turning points. Examples are provided to illustrate the results.

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Inference on quantile residual life function under right-censored data

Journal of Nonparametric Statistics ◽

10.1080/10485252.2016.1190841 ◽

2016 ◽

Vol 28 (3) ◽

pp. 617-643 ◽

Cited By ~ 2

Author(s):

Cunjie Lin ◽

Li Zhang ◽

Yong Zhou

Keyword(s):

Censored Data ◽

Residual Life ◽

Right Censored Data ◽

Residual Life Function ◽

Quantile Residual Life ◽

Life Function

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

Journal of Quality and Reliability Engineering ◽

10.1155/2013/190437 ◽

2013 ◽

Vol 2013 ◽

pp. 1-13 ◽

Cited By ~ 7

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.

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Hazard rate ordering of order statistics and systems

Journal of Applied Probability ◽

10.1017/s0021900200001716 ◽

2006 ◽

Vol 43 (02) ◽

pp. 391-408 ◽

Cited By ~ 16

Author(s):

Jorge Navarro ◽

Moshe Shaked

Keyword(s):

Order Statistics ◽

Random Vector ◽

Hazard Rate ◽

Rate Function ◽

Reliability Theory ◽

Hazard Rate Function ◽

Coherent Systems ◽

Hazard Rate Order ◽

Limiting Behaviour ◽

Hazard Rate Ordering

LetX= (X1,X2, …,Xn) be an exchangeable random vector, and writeX(1:i)= min{X1,X2, …,Xi}, 1 ≤i≤n. In this paper we obtain conditions under whichX(1:i)decreases iniin the hazard rate order. A result involving more general (that is, not necessarily exchangeable) random vectors is also derived. These results are applied to obtain the limiting behaviour of the hazard rate function of the lifetimes of various coherent systems in reliability theory. The notions of the Samaniego signatures and the minimal signatures of such systems are extensively used in the paper. An interesting relationship between these two signatures is obtained. The results are illustrated in a series of examples.

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Hazard rate ordering of order statistics and systems

Journal of Applied Probability ◽

10.1239/jap/1152413730 ◽

2006 ◽

Vol 43 (2) ◽

pp. 391-408 ◽

Cited By ~ 59

Author(s):

Jorge Navarro ◽

Moshe Shaked

Keyword(s):

Order Statistics ◽

Random Vector ◽

Hazard Rate ◽

Rate Function ◽

Reliability Theory ◽

Hazard Rate Function ◽

Coherent Systems ◽

Hazard Rate Order ◽

Limiting Behaviour ◽

Hazard Rate Ordering

Let X = (X1, X2, …, Xn) be an exchangeable random vector, and write X(1:i) = min{X1, X2, …, Xi}, 1 ≤ i ≤ n. In this paper we obtain conditions under which X(1:i) decreases in i in the hazard rate order. A result involving more general (that is, not necessarily exchangeable) random vectors is also derived. These results are applied to obtain the limiting behaviour of the hazard rate function of the lifetimes of various coherent systems in reliability theory. The notions of the Samaniego signatures and the minimal signatures of such systems are extensively used in the paper. An interesting relationship between these two signatures is obtained. The results are illustrated in a series of examples.

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Modeling Extreme Values Utilizing an Asymmetric Probability Function

Symmetry ◽

10.3390/sym13091730 ◽

2021 ◽

Vol 13 (9) ◽

pp. 1730

Author(s):

Mohammed M. A. Almazah ◽

Muqrin A. Almuqrin ◽

Mohamed. S. Eliwa ◽

Mahmoud El-Morshedy ◽

Haitham M. Yousof

Keyword(s):

Hazard Rate ◽

Extreme Values ◽

Residual Life ◽

Real Data ◽

Moment Generating Function ◽

Rate Function ◽

Estimation Methods ◽

Model Parameters ◽

Hazard Rate Function ◽

Data Application

In this article, a new flexible probability density function with three parameters is proposed for modeling asymmetric data (positive and negative) with different types of kurtosis (mesokurtic, leptokurtic and platykurtic). Some of its statistical and reliability properties, including hazard rate function, moments, moment generating function, incomplete moments, mean deviations, moment of the residual life, moment of the reversed residual life, and order statistics are derived. Its hazard rate function can be either constant, increasing-constant, decreasing-constant, U shape, upside down shape or upside down-U shape. Seven classical estimation methods are considered to estimate the unknown model parameters. Monte Carlo simulation experiments are performed to compare the performance of the seven different estimation methods. Finally, a distinctive asymmetric real data application is analyzed for illustrating the flexibility of the new model.

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