This paper proposes a new three-parameter inverse Weibull-type distribution via a similar way of constructing a skew normal distribution. The additional shape parameter increases the distribution’s modelling capability as other three-parameter inverse Weibull-type distributions. The mathematical properties of this inverse extended Weibull distribution are studied and proven. In the empirical study of four insurance datasets from Australia, Denmark and the USA, we show that the proposed distribution performs very well compared with other three-parameter competitors.

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References

1. W. Weibull , “ A statistical distribution function of wide applicability”, Journal of Applied Mechanics 18 (1951) 293–297. Crossref, Web of Science, Google Scholar
2. S. Pasari and O. Dikshit , “ Stochastic earthquake interevent time modeling from exponentiated Weibull distributions”, Natural Hazards 90 (2018) 823–842. Crossref, Web of Science, Google Scholar
3. A. Keller, M. Goblin, M , and N. Farnworth , “ Reliability analysis of commercial vehicle engines”, Reliability Engineering 10 (1985) 15–25. Crossref, Google Scholar
4. R. L. Smith and J. C. Naylor , “ A comparison of maximum likelihood and Bayesian estimators for the three-parameter Weibull distribution”, Journal of the Royal Statistical Society, Series C (Applied Statistics) 36 (1987) 358–369. Web of Science, Google Scholar
5. R. P. Yatigammana, S. T. B. Choy and J. S. K. Chan , “ Autoregressive conditional duration model with an extended Weibull error distribution”, in Causal Inference in Econometrics (Springer International Publishing, 2015), pp. 83–107. Google Scholar
6. B. Oluyede and T. Yang , Generalizations of the inverse Weibull and related distributions with applications”, Electronic Journal of Applied Statistical Analysis 7 (2014) 94–116. Google Scholar
7. S. Alkarni, A. Afify, I. Elbatal and M. Elgarhy , “ The extended inverse Weibull distribution: Properties and applications”, Complexity (2020) 1–11. Crossref, Web of Science, Google Scholar
8. S. Nadarajah and S. Kotz , “ On some recent modifications of Weibull distribution, IEEE Transactions on Reliability 54 (2005) 561–562. Crossref, Web of Science, Google Scholar
9. C. D. Lai and M. Xie , “ Weibull related distributions”, in Stochastic Ageing and Dependence for Reliability (Springer, New York, 2006), pp. 139–166. Google Scholar
10. G. Mudholkar and D. Srivastava , “ Exponentiated Weibull family for analyzing bathtub failure-rate data”, IEEE Transactions on Reliability 42 (1993) 299–302. Crossref, Web of Science, Google Scholar
11. A. Choudhury , “ A simple derivation of moments of the exponentiated Weibull distribution”, Metrika 62 (2005) 17–22. Crossref, Web of Science, Google Scholar
12. G. S. Mudholkar, D. K. Srivastava and G. D. Kollia , “ A Generalization of the Weibull distribution with application to the analysis of survival Data”, Journal of the American Statistical Association 91 (1996) 1575–1583. Crossref, Web of Science, Google Scholar
13. M. M. Nassar and F. H. Eissa , “ On the Exponentiated Weibull Distribution”, Communications in Statistics. Theory and Methods 32 (2003) 1317–1336. Crossref, Web of Science, Google Scholar
14. M. Pal, M. M Ali and J. Woo , “ Exponentiated Weibull distribution”, Statistica 66 (2006) 139–147. Google Scholar
15. A. Flaih, H. Elsalloukh, E. Mendi and M. Milanova , “ The exponentiated inverted Weibull distribution”, Appl. Math. Inf. Sci. 6 (2012) 167–171. Google Scholar
16. A. Marshall and I. Olkin , “ A new method for adding a parameter to a family of distributions with application to the exponential and Weibull families”, Biometrika 84 (1997) 641–652. Crossref, Web of Science, Google Scholar
17. M. Cordeiro and A. J. Lemonte, “On the Marshall-Olkin extended Weibull distribution”, Statistical Papers 54 (2012) 333–353. Google Scholar
18. M. Eling , “ Fitting insurance claims to skewed distributions: Are the skew-normal and skew-Student good models?” Insurance, Mathematics & Economics 51 (2012) 239–248. Crossref, Web of Science, Google Scholar
19. P. de Jong and G. Z. Heller . Generalized Linear Models for Insurance Data (Cambridge University Press, UK, 2008). Crossref, Google Scholar
20. K. Cooray and M. Ananda , “ Modeling actuarial data with a composite lognormal-Pareto model”, Scandinavian Actuarial Journal (2005) 321–334. Crossref, Google Scholar
21. J. S. K. Chan, S. T. B. Choy, U. E. Makov and Z. Landsman , “ Modelling insurance losses using contaminated generalised beta type-II distribution”, ASTIN Bulletin 48 (2018) 871–904. Crossref, Web of Science, Google Scholar
22. J. D. Cummins and L. R. Freifelder , “ A comparative analysis of alternative maximum probable yearly aggregate loss estimators”, The Journal of Risk and Insurance 45 (1978) 27–52. Crossref, Web of Science, Google Scholar