# Multivariate Variance Residual Life in Discrete Time

## DOI:

https://doi.org/10.6092/issn.1973-2201/6856## Keywords:

Multivariate variance residual life, Geometric, Waring and negative hyper geometric distributions, Increasing (decreasing) variance residual life classes, Multivariate equilibrium models## Abstract

Among various characteristics of residual life, the concept of variance residual life in the univariate case has been extensively discussed in reliability literature. In the present work we extend this notion to the discrete multivariate case and study its properties. Different versions of classes of multivariate distributions based on the monotonicity of variance residual life are also presented along with some characterizations.

## References

A. M. ABOUAMMOH, A. KANJO, A. KHALIQUE (1990). On aspects of variance residual life. Microelectron Reliability, 30, pp. 751–760.

A. ADATIA, A. G. LAW, Q. WANG (1991). Characterization of a mixture of gamma distributions via conditional finite moments. Communications in Statistics-Theory and Methods, 20, no. 5-6, pp. 1937–1949.

B. AL-ZAHRANI, J. G. DAR, A. DAGHASTINI (2013). A note on closure properties of a class of discrete lifetime distributions. International Journal of Statistics and Probability, 2, pp. 42–46.

B. C. ARNOLD, H. ZAHEDI (1988). On multivariate mean remaining life functions. Journal of Multivariate Analysis, 25, no. 1, pp. 1–9.

Y. Y. CHEN, M. HOLLANDER, N. A. LANGBERG (1983). Tests for monotone mean residual life, using randomly censored data. Biometrics, 39, pp. 119–127.

A. C. DALLAS (1981). A characterization using conditional variance. Metrika, 28, pp. 151–153.

S. EL-ARISHY (2005). A conditional variance characterization of some discrete probability distributions. Statistical Papers, 46, no. 1, pp. 31–45.

R. C. GUPTA (1987). On the monotonic properties of the residual variance and their applications in reliability. Journal of Statistical Planning and Inference, 16, pp. 329–335.

R. C. GUPTA (2006). Variance residual life function in reliability studies. Metron, 54, pp. 343–345.

R. C. GUPTA, S. N. U. A. KIRMANI (2000). Residual coefficient of variation and some characterization results. Journal of Statistical Planning and Inference, 91, pp. 23–31.

R. C. GUPTA, S. N. U. A. KIRMANI (2004). Moments of residual life and some characterization results. Journal of Applied Statistical Science, 13, pp. 155–167.

R. C. GUPTA, S. N. U. A. KIRMANI, R. L. LAUNER (1987). On life distributions having monotone residual variance. Probability in the Engineering and Informational Sciences, 1, no. 3, pp. 299–307.

R. P. GUPTA, P. G. SANKARAN (1998). Bivariate equilibrium distributions and its applications to reliability. Communications in Statistics-Theory and Methods, 27, pp. 385–394.

W. J.HALL, J.A.WELLNER (1981). Mean residual life. In M.CSORGO, D.A.DAWSON, J.N. K.RAO, E. SALEH (eds.), Statistics and Related Topics, North Holland Publishing Company, North Holland, Amsterdam, pp. 169–184.

N. HITHA, N. U. NAIR (1989). Characterization of some discrete models by properties of residual life function. Calcutta Statistical Association Bulletin, 38, pp. 151–154.

S. KARLIN (1982). Some results on optimal partitioning of variance and monotonicity with truncation level. In G. KALLIANPUR, P. R. KRISHNAIAH, J. K. GHOSH (eds.), Statistics and Probability, Essays in Honour of C. R. Rao, North Holland Publishing Company, North Holland, Amsterdam, pp. 375–382.

M. KHORASHADIZADEH, A. H. R. ROKNABADI, G. R. M. BORZADARAN (2010). Variance residual life function in discrete random ageing. Metron, 68, no. 1, pp. 67–75.

K. R. M. NAIR, N. U. NAIR (1988). On characterizing the bivariate exponential and geometric distributions. Annals of the Institute of Statistical Mathematics, 40, no. 2, pp. 261–267.

N. U. NAIR, G. ASHA (1997). Some classes of multivariate life distributions in discrete time. Journal of Multivariate Analysis, 62, no. 2, pp. 181–189.

N.U.NAIR, M. PREETH (2008). Multivariate equilibrium distribution of order n. Statistics and Probability Letters, 78, pp. 3312–3320.

N. U. NAIR, K. K. SUDHEESH (2010). Characterization of continuous distributions by properties of conditional variance. Statistical Methodology, 7, no. 1, pp. 30–40.

J. NAVARRO, J. M. SARABIA (2010). Alternative definitions of bivariate equilibrium distributions. Journal of Statistical Planning and Inference, 140, no. 7, pp. 2046–2056.

D. ROY (2005). Characterization of bivariate discrete distributions based on residual life properties. Brazilian Journal of Probability and Statistics, 19, pp. 53–64.

W. E. STEIN, R. DATTERO (1999). Bondisson functions in reliability theory. Applied Stochastic Models in Business and Industry, 15, pp. 103–109.

J. STOYANOV, M. H. M. AL-SADI (2004). Properties of classes of distributions based on conditional variances. Journal of Applied Probability, 41, pp. 953–960.

K. K. SUDHEESH, N. U. NAIR (2010). Characterization of discrete distributions by conditional variance. Metron, 68, no. 1, pp. 77–85.

H. ZAHEDI (1985). Some new classes of multivariate survival distribution functions. Journal of Statistical Planning and Inference, 11, no. 2, pp. 171–188.

## Downloads

## Published

## How to Cite

*Statistica*,

*77*(3), 181–205. https://doi.org/10.6092/issn.1973-2201/6856

## Issue

## Section

## License

Copyright (c) 2018 Statistica

This journal is licensed under a Creative Commons Attribution 3.0 Unported License (full legal code).

Authors accept to transfer their copyrights to the journal.

See also our Open Access Policy.