Abhimanyu Singh Yadav https://orcid.org/0000-0002-2411-5190 , S. K. Singh , Umesh Singh

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The aim of this paper is to introduce a new weighted probability distribution to model the non-monotone failure rate pattern for survival data. The proposed distribution is generalized by considering inverted Rayleigh distribution as a baseline distribution called an extended weighted inverted Rayleigh distribution. Different statistical properties such as moment, quantile function, moment generating function, entropy measurement, Bonferroni and Lorenz curve, stochastic ordering and order statistics have been derived. Different estimation procedures have also been discussed to estimate the unknown parameters of the proposed probability distribution. The Monte Carlo simulation study has been conducted to compare the performances of the proposed estimators obtained through various methods of estimation. Finally, two real data sets have been used to show the applicability of the proposed model in a real-life scenario.


moments and inverse moments, entropy measurements, order statistics, classical methods of estimation


AZZALINI, A., (1985). A class of distributions which includes the normal ones, Scandinavian Journal of Statistics, 12, pp. 171–178.

AHMAD, A., AHMAD, S. P., AHMED, A., (2014). Characterization and Estimation of Double Weighted Rayleigh Distribution, Journal of Agriculture and Life Sciences, 1(2), pp. 2375–4222.

BONFERRONI, C. E., ( 1930). Elementi di Statistica General, Seeber, Firenze.

CHENG, R. C. H., AMIN, N. A. K., (1979). Maximum product of spacings estimation with applications to the lognormal distribution, University of Wales IST, Math Report, pp. 79–1.

COOLEN, F. P. A., NEWBY, M. J., (1990). A note on the use of the product of spacings in Bayesian inference, Memorandum COSOR; Vol. 9035), Eindhoven: Technische Universiteit Eindhoven.

DAVID, H. A., (1970). Order Statistics, New York, Wiley.

EFRON, B., (1988). Logistic regression, survival analysis, and the Kaplan-Meier curve, Journal of American Statistical Association, 83, pp. 414–425.

FATIMA, K., AHMAD, S. P., (2017). Weighted Inverse Rayleigh Distribution, International Journal of Statistics and Systems, 12(1), pp. 119–137.

GHOSH, K., JAMMALAMADAKA, S., R., (2001). A general estimation method using spacings, Journal of Statistical Planning and Inference, 93.

GUPTA, R. D., KUNDU, D., (1999). Generalized exponential distributions, Australia and New Zealand Journal of Statistics, 41, pp. 173–188.

GUPTA, R. D., KUNDU, D., (2001). Generalized exponential distributions: different methods of estimation, Journal of Statistical Computation and Simulation, 69, pp. 315–338.

GUPTA, R. D., KUNDU, D. (2009,). A new class of weighted exponential distributions, Statistics, 43, pp. 621–634.

KAO, J. H. K., (1958). Computer methods for estimating Weibull parameters in reliability studies, IRE Transactions on Reliability and Quality Control, 13, pp. 15–22.

KAO, J. H. K., (1959). A graphical estimation of mixed Weibull parameters in life testing electron tube, Technometrics, 1, pp. 389–407.

KUNDU, D., RAQAB, M. Z., (2005). Generalized Rayleigh distribution: different methods of estimation, Computational Statistics and Data Analysis, 49, pp. 187–200.

LAWLESS, J. F., (1982). Statistical Models and Methods for Lifetime Data, New York: John Wiley & Sons.

MAKKAR, P., SRIVASTAVA, P. K. , SINGH R. S., UPADHYAY, S. K., (2014). Bayesian survival analysis of head and neck cancer data using log normal model, Communications in Statistics - Theory and Methods, 43, pp. 392–407.

NADARAJAH, S., HAGHIGHI, F., (2011). An extension of the exponential distribution, Statistics: Journal of Theoretical and Applied Statistics, 45(6), pp. 543–558.

NADARAJAH, S., KOTZ, S., (2006). The exponentiated type distributions, Acta Applicandae Mathematicae, 92(2), pp. 97–111.

RAYLEIGH, J. W. S., (1880). On the resultant of a large number of vibrations of the some pitch and of arbitrary phase. Philosophical Magazine, 5-th Series, 10, pp. 73–78.

RANNEBY, B., (1984). The Maximum Spacings Method. An Estimation Method Related to the Maximum Likelihood Method, Scandinavian Journal of Statistics, 11, pp. 93– 112.

RENYI, A., (1961). On measures of entropy and information, in: Proceedings of the 4th Berkeley Symposium on Mathematical Statistics and Probability, University of California Press, Berkeley.

SINGH, U., SINGH, S. K., SINGH, R. K., (2014). Comparative study of traditional estimation method and maximum product spacing method in Generalized inverted exponential Distribution, Journal of Statistics Application and Probability, 3(2), pp. 1–17.

SWAIN, J. J., VENKATARAMAN, S., WILSON, J. R., (1988). Least squares estimation of distribution functions in Johnson’s translation system, Journal of Statistical Computation and Simulation, 29, pp.271–297.

SHAKED, M., SHANTHIKUMAR, J. G., (1994). Stochastic Orders and Their Applications, New York, Academic Press.

VODA, V. G., (1972). On the inverse Rayleigh random variable, Rep. Statistical Application Res. Jues, 19 (4), pp. 13–21.

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