### The Marshall-Olkin Weibull Truncated Negative Binomial Distribution and its Applications

#### Abstract

The Weibull distribution is one of the widely known lifetime distribution that has been extensively used for modelling data in reliability and survival analysis. A generalization of both the Marshall-OlkinWeibull distribution and the Weibull truncated negative binomial distribution is introduced and studied in this article. Various distributional properties of the new distribution are derived. Estimation of model parameters using the method of maximum likelihood is discussed. Applications to a real data set is provided to show the flexibility and potentiality of the new distribution over other Weibull models. The first order autoregressive minification process with the new distribution as marginal is also developed. We hope that the new model will serve as a good alternative to other models available in the literature for modeling positive real data in several areas.

#### Keywords

#### Full Text:

PDF (English)#### References

M. V. AARSET (1987). How to identify a bathtub hazard rate. IEEE Transactions on Reliability, 36, no. 1, pp. 106–108.

T. ALICE, K. JOSE (2005). Marshall–Olkin semi–Weibull minification processes. Recent Advances in Statistical Theory and Applications, 1, pp. 6–17.

M. BABU (2016). On a generalization of Weibull distribution and its applications. International Journal of Statistics and Applications, 6, pp. 168–176.

I. ELBATAL, G. ARYAL (2013). On the transmuted additive Weibull distribution. Austrian Journal of Statistics, 42, no. 2, pp. 117–132.

I. ELBATAL, M.MANSOUR, M.AHSANULLAH (2016). The additive Weibull–Geometric distribution: Theory and applications. Journal of Statistical Theory and Applications, 15, no. 2, pp. 125–141.

M. GHITANY, F. AL-AWADHI, L. ALKHALFAN (2007). Marshall–Olkin extended Lomax distribution and its application to censored data. Communications in Statistics - Theory and Methods, 36, no. 10, pp. 1855–1866.

M. GHITANY, E. AL-HUSSAINI, R. AL-JARALLAH (2005). Marshall–Olkin extended Weibull distribution and its application to censored data. Journal of Applied Statistics, 32, no. 10, pp. 1025–1034.

K. JAYAKUMAR, T.MATHEW (2008). On a generalization to Marshall–Olkin scheme and its application to Burr type XII distribution. Statistical Papers, 49, no. 3, pp. 421–439.

K. JAYAKUMAR, K. SANKARAN (2016a). A generalization of additive Weibull distribution and its properties. Journal of the Kerala Statistical Association, 27, pp. 22–34.

K. JAYAKUMAR, K. SANKARAN (2016b). On a generalisation of uniform distribution and its properties. Statistica, 76, no. 1, pp. 83–91.

K. JAYAKUMAR, K. SANKARAN (2017). Generalized exponential truncated negative binomial distribution. American Journal of Mathematical and Management Sciences, 36, no. 2, pp. 98–111.

K. JOSE, S. R. NAIK, M. M. RISTIC (2010). Marshall–Olkin q–Weibull distribution and max–min processes. Statistical Papers, 51, no. 4, pp. 837–851.

B. KRISHNAN, D. GEORGE (2017). On a generalization of Marshall-Olkin Weibull distribution and its applications. Journal of the Kerala Statistical Association, 28, pp. 46–67.

A.W. MARSHALL, I. OLKIN (1997). A new method for adding a parameter to a family of distributions with application to the exponential and Weibull families. Biometrika, 84, no. 3, pp. 641–652.

S. NADARAJAH, K. JAYAKUMAR, M. M. RISTIC (2013). A newfamily of lifetime models. Journal of Statistical Computation and Simulation, 83, no. 8, pp. 1389–1404.

M. M. RISTIC, D. KUNDU (2015). Marshall–Olkin generalized exponential distribution. Metron, 73, no. 3, pp. 317–333.

P. SANKARAN, K. JAYAKUMAR (2008). On proportional odds models. Statistical Papers, 49, no. 4, pp. 779–789.

M. H. TAHIR, S. NADARAJAH (2015). Parameter induction in continuous univariate distributions: Well-established G families. Anais da Academia Brasileira de Ciências, 87, no. 2, pp. 539–568.

A. THOMAS, K. JOSE (2003). Marshall–Olkin Pareto processes. Far East Journal of Theoretical Statistics, 9, no. 2, pp. 117–132.

DOI: 10.6092/issn.1973-2201/7496