A unit Weibull loss distribution with quantile regression and practical applications to actuarial science

• Autorzy: Abubakari Abdul Ghaniyyu , Nasiru Suleman , Chesneau Christophe

Identyfikatory

DOI

10.37190/ord240401

• Języki publikacji: EN

Abstrakty

EN

A new bounded distribution called the unit Weibull loss distribution has been studied. The corresponding probability density function plots reveal that it is suitable to analyze data that exhibit right skewness, left skewness, and approximately symmetric and decreasing shapes. Furthermore, the corresponding hazard rate function plots indicate that it is adequate to fit data that have J, bathtub, and modified bathtub hazard rate shapes. This makes the new distribution suitable for modeling data with complex characteristics. Statistical properties such as the quantile, moments, and moment-generating function are determined. Risk measures, including the value-at-risk, tail value-at-risk, and tail variance are also calculated. Furthermore, different principles are derived for the computation of insurance premiums. The parameters of the distribution are estimated using different methods, and their performance is assessed via Monte Carlo simulations. The accuracy of the estimates is thus empirically demonstrated. A quantile regression model with responses following the unit distribution is developed. Applications of the proposed distribution and its corresponding regression model to three insurance data sets are carried out, with their performance compared with other models. The results show that they outperform the competitors. Thus, the new methodology can serve as an alternative option to analyze insurance data.

Słowa kluczowe

EN

loss model; claim; risk measure; premium principles; regression model

• Wydawca: Oficyna Wydawnicza Politechniki Wrocławskiej
• Czasopismo: Operations Research and Decisions
• Rocznik: 2024
• Tom: Vol. 34, No. 4
• Strony: 1--39
• Opis fizyczny: Bibliogr. 39 poz., rys.

Twórcy

autor

Abubakari Abdul Ghaniyyu

• Department of Statistics and Actuarial Science, School of Mathematical Sciences, C. K. Tedam University of Technology and Applied Sciences, Ghana

• https://orcid.org/0000-0003-0423-8465

autor

Nasiru Suleman

• Department of Statistics and Actuarial Science, School of Mathematical Sciences, C. K. Tedam University of Technology and Applied Sciences, Ghana

• https://orcid.org/0000-0001-6652-4251

autor

Chesneau Christophe

• Department of Mathematics, LMNO, University of Caen, 14032 Caen, France

• https://orcid.org/0000-0002-1522-9292

Bibliografia

• [1] Abubakari, A. G., Luguterah, A., and Nasiru, S. Unit exponentiated Fréchet distribution: Actuarial measures, quantile regression and applications. Journal of the Indian Society for Probability and Statistics 23, 2 (2022), 387–424.
• [2] Afify, A. Z., Nassar, M., Kumar, D., and Cordeiro, G. M. A new unit distribution: properties, inference, and applications. Electronic Journal of Applied Statistical Analysis 15, 2 (2022), 460–484.
• [3] Ahmad, Z., Mahmoudi, E., and Hamedani, G. G. A family of loss distributions with an application to the vehicle insurance loss data. Pakistan Journal of Statistics and Operation Research 15, 3 (2019), 731–744.
• [4] Al-Omari, A. I., Alanzi, A. R. A., and Alshqaq, S. S. The unit two parameters Mirra distribution: Reliability analysis, properties, estimation and applications. Alexandria Engineering Journal 92 (2024), 238–253.
• [5] Altun, E., and Cordeiro, G. M. The unit-improved second-degree Lindley distribution: inference and regression modeling. Computational Statistics 35, 1 (2020), 259–279.
• [6] Anderson, T. W., and Darling, D. A. Asymptotic theory of certain "goodness of fit" criteria based on stochastic processes. The Annals of Mathematical Statistics 23, 2 (1952), 193–212.
• [7] Bantan, R. A. R., Chesneau, C., Jamal, F., Elgarhy, M., Tahir, M. H., Ali, A., Zubair, M., and Anam, S. Some new facts about the unit-Rayleigh distribution with applications. Mathematics 8, 11 (2020), 1954.
• [8] Bantan, R. A. R., Jamal, F., Chesneau, C., and Elgarhy, M. Theory and applications of the unit gamma/Gompertz distribution. Mathematics 9, 16 (2021), 1850.
• [9] Cheng, R., and Amin, N. Maximum product of spacings estimation with application to the lognormal distribution. Mathematical report 79-1, University of Wales IST, Cardif, UK, 1979.
• [10] Dutang, C., and Charpentier, A. (11 December 2020) Package ‘CASdatasets’ (accessed on 26 November 2022).
• [11] Ferrari, S., and Cribari-Neto, F. Beta regression for modelling rates and proportions. Journal of Applied Statistics 31, 7 (2004), 799–815.
• [12] Furman, E., and Landsman, Z. Tail variance premium with applications for elliptical portfolio of risks. ASTIN Bulletin: The Journal of the IAA 36, 2 (2006), 433–462.
• [13] Gómez-Déniz, E., Sordo, M. A., and Calderín-Ojeda, E. The Log–Lindley distribution as an alternative to the beta regression model with applications in insurance. Insurance: Mathematics and Economics 54 (2014), 49–57.
• [14] Haj Ahmad, H., Almetwally, E. M., Elgarhy, M., and Ramadan, D. A. On unit exponential Pareto distribution for modeling the recovery rate of COVID-19. Processes 11, 1 (2023), 232.
• [15] Jodrá, P., and Jiménez-Gamero, M. D. A quantile regression model for bounded responses based on the exponentialgeometric distribution. REVSTAT – Statistical Journal 18, 4 (2020), 415–436 .
• [16] Kao, J. H. K. A graphical estimation of mixed Weibull parameters in life test electron tube. Technometrics 1, 4 (1959), 389–407.
• [17] Korkmaz, M. Ç. A new heavy-tailed distribution defined on the bounded interval: the logit slash distribution and its application. Journal of Applied Statistics 47, 12 (2020), 2097–2119.
• [18] Korkmaz, M. Ç., Altun, E., Chesneau, C., and Yousof, H. M. On the Unit-Chen distribution with associated quantile regression and applications. Mathematica Slovaca 72, 3 (2022), 765–786.
• [19] Korkmaz, M. Ç., and Chesneau, C. On the unit Burr-XII distribution with the quantile regression modeling and applications. Computational and Applied Mathematics 40, (2021), 29.
• [20] Korkmaz, M. Ç., Chesneau, C., and Korkmaz, Z. S. The unit folded normal distribution: A new unit probability distribution with the estimation procedures, quantile regression modeling and educational attainment applications. Journal of Reliability and Statistical Studies 15, 1 (2022), 261–298.
• [21] Korkmaz, M. Ç., and Korkmaz, Z. S. The unit log–log distribution: a new unit distribution with alternative quantile regression modeling and educational measurements applications. Journal of Applied Statistics 50, 4 (2023), 889–908.
• [22] Krishna, A., Maya, R., Chesneau, C., and Irshad, M. R. The unit Teissier distribution and its applications. Mathematical and Computational Applications 27, 1 (2022), 12.
• [23] Kumaraswamy, P. A generalized probability density function for double-bounded random processes. Journal of Hydrology 46,1-2 (1980), 79–88.
• [24] Macdonald, P. D. M. Comments and Queries Comment on “An Estimation Procedure for Mixtures of Distributions” by Choi and Bulgren. Journal of the Royal Statistical Society: Series B (Methodological) 33, 2 (1971), 326–329.
• [25] Mazucheli, J., Menezes, A. F. B., and Dey, S. The unit-Birnbaum-Saunders distribution with applications. Chilean Journal of Statistics 9, 1 (2018), 47–57.
• [26] Mazucheli, J., Menezes, A. F. B., and Ghitany, M. E. The unit-Weibull distribution and associated inference. Journal of Applied Probability and Statistics 13, 2 (2018), 1–22.
• [27] Mazucheli, J., Menezes, A. F. B., and Chakraborty, S. On the one parameter unit-Lindley distribution and its associated regression model for proportion data. Journal of Applied Statistics 46, 4 (2019), 700–714.
• [28] Mazucheli, J., Menezes, A. F. B., Fernandes, L. B., de Oliveira, R. P., and Ghitany, M. E. The unit-Weibull distribution as an alternative to the Kumaraswamy distribution for the modeling of quantiles conditional on covariates. Journal of Applied Statistics 47, 6 (2020), 954–974.
• [29] Menezes, A. F. B., Mazucheli, J., and Dey, S. The unit-logistic distribution: Different methods of estimation. Pesquisa Operacional 38, 3 (2018), 555–578.
• [30] Mitnik, P. A., and Baek, S. The Kumaraswamy distribution: median-dispersion re-parameterizations for regression modeling and simulation-based estimation. Statistical Papers 54, 1 (2013), 177–192.
• [31] Nasiru, S., Abubakari, A. G., and Chesneau, C. The arctan power distribution: Properties, quantile and modal regressions with applications to biomedical data. Mathematical and Computational Applications 28, 1 (2023), 25.
• [32] Nasiru, S., Chesneau, C., Abubakari, A. G., and Angbing, I. D. Generalized unit half-logistic geometric distribution: Properties and regression with applications to insurance. Analytics 2, 2 (2023), 438–462.
• [33] Nasiru, S., Chesneau, C., and Ocloo, S. K. The log-cosine-power unit distribution: A new unit distribution for proportion data analysis. Decision Analytics Journal 10 (2024), 100397.
• [34] Peña-Ramírez, F. A., Guerra, R. R., and Mafalda, C. P. The unit ratio-extended Weibull family and the dropout rate in Brazilian undergraduate courses. PLOS ONE 18, 11 (2023), e0290885.
• [35] Ranneby, B. The maximum spacing method: An estimation method related to the maximum likelihood method. Scandinavian Journal of Statistics 11, 2 (1984), 93–112.
• [36] Swain, J. J., Venkatraman, S., and Wilson, J. R. Least-Squares estimation of distribution functions in Johnson’s translation system. Journal of Statistical Computation and Simulations 29, 4 (1988), 271–297.
• [37] Topp, C. W., and Leone, F. C. A family of J-shaped frequency functions. Journal of the American Statistical Association 50, 269 (1955), 209–219.
• [38] Wolny-Dominiak, A., and Trze¸siok, M. (4 September 2014) insurancedata: A collection of insurance datasets useful in risk classification in non-life insurance. R package version 1.0 (accessed on 12 October 2022).
• [39] Yıldırım, E., Ilıkkan, E. S., Gemeay, A. M., Makumi, N., Bakr, M. E., and Balogun, O. S. Power unit Burr-XII distribution: Statistical inference with applications. AIP Advances 13, 10 (2023), 105107.

Uwagi

Opracowanie rekordu ze środków MNiSW, umowa nr POPUL/SP/0154/2024/02 w ramach programu "Społeczna odpowiedzialność nauki II" - moduł: Popularyzacja nauki (2025).