# A New K-Product Generalized Transformation: Investigating Weibull Distribution as the Baseline

## Keywords:

G-class of probability distributions, Weibull distribution, hazard function, entropy measure, maximum likelihood estimation## Abstract

This article is about defining and studying an improved technique of parameter induction to a continuous probability distribution through a new G-class of probability models. In particular, the Weibull distribution is used in the defined technique and it is named as KP-W distribution. The importance of this generalization of Weibull distribution comes from its ability to model various kinds of hazard functions such as ascending, descending, first decreasing and then increasing, or constant hazard rate functions. Different properties of this generalized modified model have been deliberated along with raw moments and functions which can generate moments, quantiles, hazard function, Rényi entropy, stress-strength parameter, order statistics, the average time to wait until served and average remaining life. Maximum likelihood (ML) estimation of the proposed G class and its sub-model, the KP-W is also presented. Finally, the KP-W model is judged for its goodness to fit using data sets from different fields to showcase its practical applications.

## References

M. Nassar, M. A. Alzaatreh, , M. Mead, and Abo-Kasem, O. Alpha power Weibull distribution: Properties and applications. Communications in Statistics-Theory and Methods, Vol. 46(20), pp. 10236–10252, 2017.

R. C. Gupta, R. D. Gupta, and P. L. Gupta, P. L. Modeling failure time data by Lehman alternatives', Communications in Statistics Theory and Methods, Vol. 27, pp. 887-904, 1998

W. T. Shaw and I. R. and Buckley. The alchemy of probability distributions: beyond Gram-Charlier expansions, and a skew-kurtotic-normal distribution from a rank transmutation map. arXiv preprint arXiv:0901.0434.2009

D. Kumar, U. Singh and S. K. Singh. A method of proposing new distribution and its application to Bladder cancer patients’ data. Journal of Statistics Applications & Probability, Vol. 2(3), pp. 235-245, 2015.

D. Kumar, U. Singh and S. K. Singh. A new distribution using sine function-its application to bladder cancer patients’ data. Journal of Statistics Applications & Probability, Vol. 4(3), pp. 417-427, 2015.

D. Kumar, U. Singh and U. Singh, U. Life time distributions: Derived from some minimum guarantee distribution. Sohag Journal of Mathematics, Vol.4(1), pp. 7-11, 2017.

S. K. Maurya, A. Kaushik, R. K. Singh, S. K. Singh, and U. Singh, U. A new method of proposing distribution and its application to real data. Imperial Journal of Interdisciplinary Research, Vol. 2(6), pp. 1331– 1338, 2016.

S. K. Maurya, A. Kaushik, S. K. Singh, and U. Singh. A new class of distribution having decreasing, increasing, and bathtub-shaped failure rate. Communications in Statistics-Theory and Methods, Vol. 46(20), pp. 10359–10372, 2017

Z. Mahmood and C Chesneau. A new sine-G family of distributions: properties and applications. hal-02079224, 2019.

N. Kyurkchiev. A new transmuted cumulative distribution function based on the Verhulst logistic function with application in population dynamics. Biomath Communications, Vol. 4(1), 2017 DOI: 10.11145/bmc.2017.05.132

A. N. Marshall and I. Olkin. A new method for adding a parameter to a family of distributions with applications to the exponential and Weibull families, Biometrika, Vol. 84, pp. 641-552,1997

L. Souza, W. R. Junior, C. C. R. de Brito, C. Chesneau, R. L. Fernandes and T. A. E. Ferreira, Tan-G class of trigonometric distributions and its applications. CUBO, A Mathematical Journal, Vol. 23(1), pp. 01–20, 2021.

I. Elbatal, S. M. Alghamdi, F. Jamal, S. Khan, E. M. Almetwally and M. Elgarhy., Kavya-Manoharan Weibull-G family of distributions: Statistical inference under progressive type-II censoring scheme, Vol. 87(2),pp. 191-223, 2023

A. Shafiq, S. A. Lone, T. N. Sindhu, Y. El-Khatib, Q. M. Al-Mdallal, and T. Muhammad. A new modified Kies Fréchet distribution: Applications of mortality rate of Covid-19, Results in Physics, 28:104638. doi:10.1016/j.rinp.2021.104638, 2021.

T. N. Sindhu and A. Atangana. Reliability analysis incorporating exponentiated inverse Weibull distribution and inverse power law. Quality and Reliability Engineering International, Vol. 37(6), pp. 2399-2422, 2021.

T. N. Sindhu, A. Shafiq and Q.M. Al-Mdallal. Exponentiated transformation of gumbel type-II distribution for modeling COVID-19 data. Alexandria Engineering Journal, Vol. 60 (1), pp. 671–689, 2021.

T. N. Sindhu, A. Shafiq, and Q. M. Al-Mdallal. On the analysis of number of deaths due to Covid- 19 outbreak data using a new class of distributions. Results in Physics, Vol. 21, 03747, 2021.

T. N. Sindhu, A. Shafiq and Z. Hussain. Generalized exponentiated unit Gompertz distribution for modeling arthritic pain relief times data: classical approach to statistical inference. Journal of Biopharmaceutical Statistics,2023

Nassar, M. and Nada, N. The beta generalized Pareto distribution. Journal of Statistics: Advances in Theory and Applications, Vol. 6(1/2), pp. 436–452, 2011.

Lawless, J. F. Statistical models and methods for lifetime data, volume 362. John Wiley & Sons, 2003..

## Downloads

## Published

## How to Cite

*University of Wah Journal of Science and Technology (UWJST)*,

*7*, 26–50. Retrieved from https://uwjst.org.pk/index.php/uwjst/article/view/200