Bivariate mean residual life

Nair, K. R.; Nair, N. Unnikrishnan

doi:10.1109/24.44183

Cited by 26 publications

(8 citation statements)

References 10 publications

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“…Arnold and Zahedi 419885 provided some general characterization properties of a multivariate mean residual life function and established its relationship with the hazard gradient. Nair and Nair (1989) proved similar results for the bivariate case. Kulkarni and Rattihalli (1996) obtained characterizing conditions on a two-dimensional vector function to be the bivariate mean residual life function of an absolutely continuous bivariate life time distribution and introduced some bivariate probability models based on the form of a bivariate mean residual life function.…”

Section: Introductionsupporting

confidence: 80%

Nonparametric Estimation of a Bivariate Mean Residual Life Function

Kulkarni

Rattihalli

2002

Journal of the American Statistical Association

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In many statistical studies involving failure data, mean residual life function is of prime importance. The bivariate mean residual life function has received relatively less attention in the literature. In this article we use a simple nonparametric estimator for a bivariate mean residual life function. The estimator is shown to be uniformly strongly consistent and, on proper normalization, converges weakly to a zero-mean bivariate Gaussian process. Numerical studies demonstrate that the estimator performs well even for moderate sample sizes. Results are applied to a real dataset related to cancer recurrence. A few supporting results in connection with weak convergence proved in Appendix C may be of independent interest.

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Section: Introductionsupporting

confidence: 80%

Nonparametric Estimation of a Bivariate Mean Residual Life Function

Kulkarni

Rattihalli

2002

Journal of the American Statistical Association

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“…That is, it is not clear that there exists a survival function S * n for which (1.4) holds with e 1 (x, y), e 2 (x, y) replaced by e * n1 (x, y), e * n2 (x, y). Nair and Nair [10] and Kulkarni and Rattihalli [6] provided necessary and sufficient conditions for a nonnegative functioñ m(x, y) to be a proper mrl function. The latter authors note that Nair and Nair's results are incomplete as their conditions are not necessary and sufficient.…”

Section: The One-sample Problemmentioning

confidence: 99%

Estimation of two ordered bivariate mean residual life functions

Rojo

Ghebremichael

2006

Journal of Multivariate Analysis

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Situations occur frequently in which the mean residual life (mrl) functions of two populations must be ordered. For example, if a mechanical device is improved, the mrl function for the improved device should not be less than that of the original device. Also, mrl functions for medical patients should often be ordered depending on the status of concomitant variables. This paper proposes nonparametric estimators of the bivariate mrl function under a mrl ordering. The estimators are shown to be asymptotically unbiased, strongly uniformly consistent and weakly convergent to a bivariate Gaussian process. The estimators are shown to be the projections, in a sense to be made precise, of the empirical mrl function onto an appropriate convex set of mrl functions. In the one-sample problem, the new estimators dominate the empirical mrl function in terms of risk with respect to a wide class of loss functions.

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“…The mrl functions (3.4)-(3.6) should be contrasted with the mrl functions studied by Arnold and Zahedi (1988) and Nair and Nair (1989). The latter functions determine uniquely the joint distribution F, but they do not have dynamics in time like the functions (3.4)-(3.6).…”

Section: Multivariate Conditional Mrl Functionsmentioning

confidence: 99%

Dynamic multivariate mean residual life functions

Shaked

Shanthikumar

1991

Journal of Applied Probability

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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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Bivariate mean residual life

Cited by 26 publications

References 10 publications

Nonparametric Estimation of a Bivariate Mean Residual Life Function

Nonparametric Estimation of a Bivariate Mean Residual Life Function

Estimation of two ordered bivariate mean residual life functions

Dynamic multivariate mean residual life functions

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