An Inferential Aptness of a Weibull Generated Distribution and Application

  • Brijesh P. Singh Department of Statistics, Institute of Science, Banaras Hindu University, Varanasi-221005, India
  • Utpal Dhar Das Department of Statistics, Institute of Science, Banaras Hindu University, Varanasi-221005, India
Keywords: Bonferroni and Gini coefficient, K-S test, MLE, Moments, MRLF, Rényi and Shannon entropy

Abstract

In this article an attempt has been made to develop a flexible single parameter continuous distribution using Weibull distribution. The Weibull distribution is most widely used lifetime distributions in both medical and engineering sectors. The exponential and Rayleigh distribution is particular case of Weibull distribution. Here in this study we use these two distributions for developing a new distribution. Important statistical properties of the proposed distribution is discussed such as moments, moment generating and characteristic function. Various entropy measures like Rényi, Shannon and cumulative entropy are also derived. The kthkt⁢h order statistics of pdf and cdf also obtained. The properties of hazard function and their limiting behavior is discussed. The maximum likelihood estimate of the parameter is obtained that is not in closed form, thus iteration procedure is used to obtain the estimate. Simulation study has been done for different sample size and MLE, MSE, Bias for the parameter λλ has been observed. Some real data sets are used to check the suitability of model over some other competent distributions for some data sets from medical and engineering science. In the tail area, the proposed model works better. Various model selection criterion such as -2LL, AIC, AICc, BIC, K-S and A-D test suggests that the proposed distribution perform better than other competent distributions and thus considered this as an alternative distribution. The proposed single parameter distribution is found more flexible as compare to some other two parameter complicated distributions for the data sets considered in the present study.

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Author Biographies

Brijesh P. Singh, Department of Statistics, Institute of Science, Banaras Hindu University, Varanasi-221005, India

Brijesh P. Singh, is currently working as Professor in the Department of Statistics, Institute of Science, Banaras Hindu University, Varanasi, India. He has obtained Ph. D. degree in Statistics from Banaras Hindu University, Varanasi and has more than 20 years’ experience of teaching and research in the area of Statistical Demography and modeling. His research interests are in statistical modeling and analysis of demographic data specially fertility, mortality, reproductive health and domestic violence with its reason and consequences.

Utpal Dhar Das, Department of Statistics, Institute of Science, Banaras Hindu University, Varanasi-221005, India

Utpal Dhar Das, is presently working as research scholar in the Department of Statistics, Institute of Science, Banaras Hindu University, Varanasi, India. He is a bright fellow in Mathematics and Statistics and was awarded gold medal in M. Sc. (Statistics) from Assam University, Silchar. He has published 12 research articles in reputed both international and national journals. His research interests are in the areas of Generalized Probability distributions, transformed probability distributions.

References

A. A., Albabtain. A New Extended Rayleigh Distribution, Journal of King Saud University-Science, 2576–2581, 2020.

A. A., Essam, A. H., Mohamed. A weighted three-parameter Weibull distribution, J. Applied Sci. Res, 9, 6627–6635, 2013.

A. Azzalini. A class of distributions which includes the normal ones, Scandinavian journal of statistics, 171–178, 1985.

A. M., Abouammoh, R., Ahmad, A., Khalique. On new renewal better than used classes of life distributions, Statistics & Probability Letters, 48(2), 189–194, 2000.

A., Rényi. On measures of entropy and information. In Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability, Volume 1: Contributions to the Theory of Statistics. The Regents of the University of California, 1960, 547–561, 1961.

Brijesh P., Singh, S., Singh, U. D., Das. A general class of new continuous mixture distribution and application. J. Math. Comput. Sci., 11(1): 585–602, 2020.

Brijesh P., Singh, U. D., Das, S., Singh, A Compounded Probability Model for Decreasing Hazard and its Inferential Properties, Reliability: Theory & Applications, 16(2 (62)): 230–246, 2021.

C. E., Bonferroni, Elementi di Statistica Generale. Seeber. Firenze, 1930.

C. E., Shannon. A mathematical theory of communication, The Bell system technical journal, 27(3): 379–423, 1948.

D.N.P., Murthy, M., Xie, R., Jiang. Weibull Models (Vol. 505), USA, John Wiley and Sons, 2004.

D., Kundu, M. Z., Raqab. Generalized Rayleigh distribution: different methods of estimations, Computational statistics & data analysis, 49(1): 187–200, 2005.

F., Merovci, I., Elbatal. Weibull-Rayleigh distribution: theory and applications, Appl. Math. Inf. Sci, 9(5): 1–11, 2015.

G. S., Rao, S., Mbwambo. Exponentiated inverse Rayleigh distribution and an application to coating weights of iron sheets data, Journal of Probability and Statistics, 2019.

H., Ahmad Sartawi, M. S., Abu-Salih. Bayesian prediction bounds for the Burr type X model, Communications in Statistics-Theory and Methods, 20(7): 2307–2330, 1991.

H. J., Kim. A class of weighted multivariate normal distributions and its properties, Journal of Multivariate Analysis, 99(8): 1758–1771, 2008.

J. G., Surles, W. J., Padgett. Inference for reliability and stress-strength for a scaled Burr Type X distribution, Lifetime Data Analysis, 7(2): 187–200, 2001.

J. X., Kersey. Weighted inverse Weibull and beta-inverse Weibull distribution, M.Sc. Thesis, Georgia Southern University, Statesboro, Georgia, 2010.

M. A., Hussian. A weighted inverted exponential distribution, International Journal of Advanced Statistics and Probability, 1(3): 142–150, 2013.

M., Mahdy. A class of weighted Weibull distributions and its properties, Studies in Mathematical Sciences, 6(1): 35–45, 2013.

M. Z., Raqab. Order statistics from the Burr type X model, Computers & Mathematics with Applications, 36(4): 111–120, 1998.

P. E., Oguntunde, E. A., Owoloko, O. S., Balogun. On a new weighted exponential distribution: theory and application, Asian Journal of Applied Sciences, 9(1): 1–12, 2016.

R.B., Wallace, D.N.P., Murthy. Reliability Wiley, New York, 2000.

R. D., Gupta, D., Kundu. Exponentiated exponential family: an alternative to gamma and Weibull distributions, Biometrical Journal: Journal of Mathematical Methods in Biosciences, 43(1): 117–130, 2001.

S., Nadarajah, S., Kotz. The exponentiated type distributions, Acta Applicandae Mathematica, 92(2): 97–111, 2006.

S., Shahbaz, M. Q., Shahbaz, N. S., Butt. A class of weighted Weibull distribution, Pak. j. stat. oper. Res, 6(1): 53–59, 2010.

T., Bjerkedal. Acquisition of Resistance in Guinea Pies infected with Different Doses of Virulent Tubercle Bacilli, American Journal of Hygiene, 72(1): 130–48, 1960.

Z. F., Jaheen. Empirical Bayes estimation of the reliability and failure rate functions of the Burr type X failure model, Journal of Applied Statistical Science, 3(4): 281–288, 1996.

Published
2021-12-18
Section
Articles