Estimation of Hazard Rate and Mean Residual Life Ordering for Fuzzy Random Variable

L2-metric is used to find the distance between triangular fuzzy numbers. The mean and variance of a fuzzy random variable are also determined by this concept. The hazard rate is estimated and its relationship with mean residual life ordering of fuzzy random variable is investigated. Additionally, we have focused on deriving bivariate characterization of hazard rate ordering which explicitly involves pairwise interchange of two fuzzy random variablesXandY.

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Dynamic multivariate mean residual life functions

Journal of Applied Probability ◽

10.1017/s0021900200042467 ◽

1991 ◽

Vol 28 (03) ◽

pp. 613-629 ◽

Cited By ~ 2

Author(s):

Moshe Shaked ◽

J. George Shanthikumar

Keyword(s):

Hazard Rate ◽

Residual Life ◽

Stochastic Ordering ◽

Partial Ordering ◽

Mean Residual Life ◽

Positive Dependence ◽

Basic Properties ◽

Rate Functions ◽

Hazard Rate Ordering ◽

Weak Notion

In this paper we introduce and study a dynamic notion of mean residual life (mrl) functions in the context of multivariate reliability theory. Basic properties of these functions are derived and their relationship to the multivariate conditional hazard rate functions is studied. A partial ordering, called the mrl ordering, of non-negative random vectors is introduced and its basic properties are presented. Its relationship to stochastic ordering and to other related orderings (such as hazard rate ordering) is pointed out. Using this ordering it is possible to introduce a weak notion of positive dependence of random lifetimes. Some properties of this positive dependence notion are given. Finally, using the mrl ordering, a dynamic notion of multivariate DMRL (decreasing mean residual life) is introduced and studied. The relationship of this multivariate DMRL notion to other notions of dynamic multivariate aging is highlighted in this paper.

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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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An Efficient Estimation of the Mean Residual Life Function with Length-Biased Right-Censored Data

Mathematical Problems in Engineering ◽

10.1155/2014/937397 ◽

2014 ◽

Vol 2014 ◽

pp. 1-5 ◽

Cited By ~ 3

Author(s):

Hongping Wu ◽

Yihui Luan

Keyword(s):

Monte Carlo ◽

Censored Data ◽

Residual Life ◽

Random Variable ◽

Efficient Estimation ◽

Mean Residual Life ◽

Right Censored Data ◽

Monte Carlo Simulation Study ◽

The Mean ◽

Model Setup

The mean residual life (MRL) function for a lifetime random variableT0is one of the basic parameters of interest in survival analysis. In this paper, we propose a new estimator of the MRL function with length-biased right-censored data and evaluate its performance through a small Monte Carlo simulation study. The results of the simulations show that the proposed estimator outperforms the existing one referred to in Data and Model Setup Section in terms of Monte Carlo bias and mean square error, especially when the censoring rate is heavy. We also show that the proposed estimator converges in distribution under some conditions.

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ORDERING CONVOLUTIONS OF HETEROGENEOUS EXPONENTIAL AND GEOMETRIC DISTRIBUTIONS REVISITED

Probability in the Engineering and Informational Sciences ◽

10.1017/s026996481000001x ◽

2010 ◽

Vol 24 (3) ◽

pp. 329-348 ◽

Cited By ~ 7

Author(s):

Tiantian Mao ◽

Taizhong Hu ◽

Peng Zhao

Keyword(s):

Hazard Rate ◽

Residual Life ◽

Sufficient Conditions ◽

Random Variables ◽

Mean Residual Life ◽

Reversed Hazard Rate ◽

Exponential Random Variables ◽

The Mean ◽

Geometric Variables ◽

Fail Safe

Let Sn(a1, …, an) be the sum of n independent exponential random variables with respective hazard rates a1, …, an or the sum of n independent geometric random variables with respective parameters a1, …, an. In this article, we investigate sufficient conditions on parameter vectors (a1, …, an) and $(a_{1}^{\ast},\ldots,a_{n}^{\ast})$ under which Sn(a1, …, an) and $S_{n}(a_{1}^{\ast},\ldots,a_{n}^{\ast})$ are ordered in terms of the increasing convex and the reversed hazard rate orders for both exponential and geometric random variables and in terms of the mean residual life order for geometric variables. For the bivariate case, all of these sufficient conditions are also necessary. These characterizations are used to compare fail-safe systems with heterogeneous exponential components in the sense of the increasing convex and the reversed hazard rate orders. The main results complement several known ones in the literature.

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The -mean squared dispersion associated with a fuzzy random variable

Fuzzy Sets and Systems ◽

10.1016/s0165-0114(97)00389-8 ◽

2000 ◽

Vol 111 (3) ◽

pp. 307-317 ◽

Cited By ~ 36

Author(s):

Marı́a Asunción Lubiano ◽

Marı́a Angeles Gil ◽

Miguel López-Dı́az ◽

Marı́a Teresa López

Keyword(s):

Fuzzy Random Variable ◽

Random Variable ◽

The Mean ◽

Fuzzy Random

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Dynamic multivariate mean residual life functions

Journal of Applied Probability ◽

10.2307/3214496 ◽

1991 ◽

Vol 28 (3) ◽

pp. 613-629 ◽

Cited By ~ 24

Author(s):

Moshe Shaked ◽

J. George Shanthikumar

Keyword(s):

Hazard Rate ◽

Residual Life ◽

Stochastic Ordering ◽

Partial Ordering ◽

Mean Residual Life ◽

Positive Dependence ◽

Basic Properties ◽

Rate Functions ◽

Hazard Rate Ordering ◽

Weak Notion

In this paper we introduce and study a dynamic notion of mean residual life (mrl) functions in the context of multivariate reliability theory. Basic properties of these functions are derived and their relationship to the multivariate conditional hazard rate functions is studied.A partial ordering, called the mrl ordering, of non-negative random vectors is introduced and its basic properties are presented. Its relationship to stochastic ordering and to other related orderings (such as hazard rate ordering) is pointed out. Using this ordering it is possible to introduce a weak notion of positive dependence of random lifetimes. Some properties of this positive dependence notion are given.Finally, using the mrl ordering, a dynamic notion of multivariate DMRL (decreasing mean residual life) is introduced and studied. The relationship of this multivariate DMRL notion to other notions of dynamic multivariate aging is highlighted in this paper.

Download Full-text