Generalised M-Transformation Weibull Distribution: Statistical Properties and Reliability Applications

Authors

  • Francis Kwame Arpoh Department of Statistics and Actuarial Science, Kwame Nkrumah University of Science and Technology, Kumasi, Ghana
  • Gabriel Asare Okyere Department of Statistics and Actuarial Science, Kwame Nkrumah University of Science and Technology, Kumasi, Ghana
  • Isaac Akpor Adjei Department of Statistics and Actuarial Science, Kwame Nkrumah University of Science and Technology, Kumasi, Ghana

DOI:

https://doi.org/10.23755/rm.v56i0.1759

Abstract

Many real-world datasets show complex features that classical life-time models cannot capture. This limitation highlights the need for new distributions with greater flexibility in skewness and kurtosis, as well as improved fit to observed data. In this study, we introduce the Generalised M-Transformation Weibull (GMTW) distribution to meet these needs. We derive its main statistical properties, including the quantile function and reliability measures. The GMTW distribution can represents diverse hazard rate shapes including bathtub, unimodal, and monotonic forms, making it suitable for reliability studies. Parameters are estimated using the maximum likelihood method, and the long run behaviour of the estimators is checked through extensive Monte Carlo simulations. The model’s usefulness is shown with three datasets: industrial reliability, glass fibre tensile strength, and bladder cancer remission times. Likelihood Ratio Tests and goodness-of-fit criteria confirm that the GMTW distribution performs better than its related forms and other well-known lifetime distributions.

References

Aarset, M. V. (1987). How to identify a bathtub hazard rate. IEEE Transactions on Reliability, 36(1), 106–108.

Alzaatreh, A., Lee, C., & Famoye, F. (2013). A new method for generating families of continuous distributions. Metron, 71(1), 63–79.

Bebbington, M., Lai, C. D., & Zitikis, R. (2007). A flexible Weibull extension. Reliability Engineering & System Safety, 92(6), 719–726.

Bowley, A. L. (1920). Elements of Statistics (4th ed.). London: P.S. King & Son.

David, H. A., & Nagaraja, H. N. (2003). Order Statistics (3rd ed.). John Wiley & Sons.

Gradshteyn, I. S., & Ryzhik, I. M. (2014). Table of Integrals, Series, and Products (8th ed.). Academic Press.

Hyndman, R. J., & Fan, Y. (1996). Sample quantiles in statistical packages. The American Statistician, 50(4), 361–365.

Khan, M. S., & King, R. (2013). Transmuted modified Weibull distribution: A generalization of the modified Weibull probability distribution. European Journal of Pure and Applied Mathematics, 6(1), 66–88.

Kumar, D., Singh, U., Singh, S. K., & Mukherjee, S. (2017). The new probability distribution: An aspect to a life time distribution. Mathematical Sciences Letters, 6(1), 35–42.

Lee, E. T., & Wang, J. W. (2003). Statistical Methods for Survival Data Analysis (3rd ed.). New Jersey: John Wiley & Sons.

Moors, J. J. A. (1988). A quantile alternative for kurtosis. Journal of the Royal Statistical Society: Series D (The Statistician), 37(1), 25–32.

Mudholkar, G. S., & Srivastava, D. K. (1993). Exponentiated Weibull family for analyzing bathtub failure-rate data. IEEE Transactions on Reliability, 42(2), 299–302.

Nocedal, J., & Wright, S. J. (2006). Numerical Optimization (2nd ed.). New York, NY: Springer.

Smith, R. L., & Naylor, J. C. (1987). A comparison of maximum likelihood and Bayesian estimators for the Weibull distribution. Journal of the Royal Statistical Society: Series C (Applied Statistics), 36(3), 358–369.

Virtanen, P., et al. (2020). SciPy 1.0: Fundamental algorithms for scientific computing in Python. Nature Methods, 17, 261–272.

Weibull, W. (1951). A statistical distribution function of wide applicability. Journal of Applied Mechanics, 18, 293–297.

Downloads

Published

2026-06-30

Issue

Vol. 56 (2026)

Section

Volume 56-2026

License

This journal provides immediate open access to its content on the principle that making research freely available to the public supports a greater global exchange of knowledge. The journal is committed to real and immediate open access for academic work. All of the journal articles are free to access immediately from the date of publication. There are no charge for readers to download articles and reviews for their own scholarly use. Benefits of open access for authors, include:

  • Free access for all users worldwide
  • Increased visibility and readership
  • Rapid publication
  • No spatial constraints
  • Authors can share their research papers on their personal web-pages (i.e. ResearchGate, etc.)

Ratio Mathematica publishes open access articles under the terms of the Creative Commons Attribution (CC BY) License. The Creative Commons Attribution License (CC-BY) allows users to copy, distribute and transmit an article, adapt the article and make commercial use of the article. The CC BY license permits commercial and non-commercial re-use of an open access article, as long as the author is properly attributed. Copyright on any research article published by Ratio Mathematica is retained by the author(s). Authors grant Ratio Mathematica a license to publish the article and identify itself as the original publisher. Authors also grant any third party the right to use the article freely as long as its original authors, citation details and publisher are identified. 
For further information visit: https://creativecommons.org/licenses/by/3.0/. The following logo will appear in any paper that will be published from the volume number 31/2016 onwards.