Journal of the Iranian Statistical Society

Quantile based Past Geometric Vitality Function of Order Statistics

Document Type : Original Article

Authors

Department of Statistics, University of Kerala, Kariavattom, Trivandrum, India-695581.

Abstract

Nair and Rajesh (2000) introduced the geometric vitality function, which explains the failure pattern of components or systems based on the component's geometric mean of the remaining lifetime. Recently quantile-based studies have found greater interest among researchers as an alternative method of measuring the uncertainty of a random variable. The quantile-based measures possess some unique properties to the distribution function approach. The present paper introduces a quantile-based past geometric vitality function of order statistics and its properties. Finally, we provide an application for the new measure based on some distributions which are useful in lifetime data analysis.

Keywords

Aswin, I. C., Sankaran, P. G. and Sunoj, S. M.(2020), A Class of Distributions with Quadratic Hazard Quantile Function. J Indian Soc Probab Stat, 21, 409-426.
Cover, T. M., and Thomas, J. A. (2006), Elements of Information Theory.Wiley.
Di Crescenzo, Antonio, and Maria Longobardi (2002), Entropy-Based Measure of Uncertainty in Past Lifetime Distributions. Journal of Applied Probability, 39(2), 434-440.
Dileep Kumar, M., and Sankaran, P. G. (2000), A new family of quantile functions and its applications. Communications in Statistics-Theory and Methods, 50(18), 4216-4235.
Gayathri, R., and Sathar, E. I. A. (2021), On past geometric vitality function of order statistics. Metrika, 84, 263-280.
Gilchrist,W. (2000), Statistical modelling with quantile function. CRC Press, Florida.
Hankin, R. K. S., and Lee, A. (2006), A new family of non-negative distributions. Australian and New Zealand Journal of Statistics, 8(1), 67-78.
Jiang, R., and Murthy, D. (1998), Two sectional models involving three weibull distributions. Quality and Reliability Engineering International, 13(2), 83-96.
Kayal, S., and Tripathy, M. R. (2018), A quantile-based Tsallis alpha divergence. Physica A: Statistical Mechanics and its Applications, 492, 496-505.
Kotz, S., and Shanbhag, D. N. (1980), Some new approaches to probability distributions. Advances in Applied Probability, 12(4), 903-921.
Krishnan, A. S., Sankaran, P. G., and Sunoj, S. M. (2020), Some reliability properties of extropy for residual and past lifetime random variables. Journal of the Korean Statistical Society, 49, 457-474.
Krishnan, A. S., Sunoj, S. M., and Sankaran, P. G. (2019), Quantile-based reliability aspects of cumulative Tsallis entropy in past lifetime. Metrika, 82(1), 17-38.
Nair, K. R. M., and Rajesh, G. (2000), Geometric vitality function and its applications to reliability. IAPQR TRANSACTIONS, 25, 1-8.
Nair, N. U., Sankaran, P. G., and Vinesh Kumar, B. (2011), Modelling lifetimes by quantile function using parzen’s score function. A Journal of Theoretical and Applied Statistics, 46(6), 1-13.
Nair, N. U., Sankaran, P. G., and Balakrishnan, N. (2013), Quantile-based reliability analysis. New York: Springer.
Rajesh, G., Sathar, E. I. A., Maya, R. and Nair, K. R. Muraleedharan. (2014), Nonparametric estimation of the geometric vitality function. Communications in Statistics-Theory and Methods, 43(1), 115-130.
Rajesh, G., and Sunoj, S. M. (2019), Some properties of cumulative Tsallis entropy of order alpha. Statistical Papers, Springer, 60(3), 933-943.
Ramberg, J. S., and Schmeiser, B. W. (1974), An approximate method for generating asymetric random variables. Communication of the ACM, 17, 78-82.
Sankaran, P. G., and Nair, N. U. (2009), Nonparametric estimation of the hazard quantile function. Journal of Nonparametric Statistics, 21(6), 757-767.
Sathar, E. I. A. , Rajesh, G., and Nair, K. R. M.(2010), Bivariate geometric vitality function and some characterization results. Calcutta Statistical Association Bulletin, Springer, 62(3-4), 207-228.
Shaked, M., and Shanthikumar, J. G. (2007), Stochastic Orders. New York: Springer.
Sunoj, S. M., Krishnan,A. S., and Sankaran, P.G. (2018), Quantile-based study of cumulative residual Tsallis entropy measures. Physica A: Statistical Mechanics and its Applications, 494, 410-421.
Sunoj, S. M., Sankaran, P. G., and Asok K. Nanda. (2013), Quantile based entropy function in past lifetime. Statistics and Probability Letters, Springer, 83(1), 366-372.
Sunoj, S. M., Sankaran, P. G., and Maya, S. S. (2009), Characterizations of life distributions using conditional expectations of doubly (interval) truncated random variables.
Communications in Statistics-Theory and Methods, Springer, 38, 1441-1452. Vikas Kumar and Rekha (2018), A quantile approach of Tsallis entropy for order statistics. Physica A: Statistical Mechanics and its Applications, 503, 916-928.
Wang, S. (1998), An actuarial index of the right-tail risk. North American Actuarial Journal, 2, 88-101.
Journal of the Iranian Statistical Society
Volume 21, Issue 1
June 2022
Pages 55-79
  • Receive Date: 13 January 2022
  • Revise Date: 11 October 2022
  • Accept Date: 03 November 2022