On some composite Kies families: distributional properties and saturation in Hausdorff sense
Volume 10, Issue 3 (2023), pp. 287–312
Pub. online: 21 March 2023
Type: Research Article
Open Access
Open Access
Received
9 November 2022
9 November 2022
Revised
7 January 2023
7 January 2023
Accepted
12 March 2023
12 March 2023
Published
21 March 2023
21 March 2023
Abstract
The stochastic literature contains several extensions of the exponential distribution which increase its applicability and flexibility. In the present article, some properties of a new power modified exponential family with an original Kies correction are discussed. This family is defined as a Kies distribution which domain is transformed by another Kies distribution. Its probabilistic properties are investigated and some limitations for the saturation in the Hausdorff sense are derived. Moreover, a formula of a semiclosed form is obtained for this saturation. Also the tail behavior of these distributions is examined considering three different criteria inspired by the financial markets, namely, the VaR, AVaR, and expectile based VaR. Some numerical experiments are provided, too.
References
Afify, A.Z., Gemeay, A.M., Alfaer, N.M., Cordeiro, G.M., Hafez, E.H.: Power-modified Kies-exponential distribution: Properties, classical and Bayesian inference with an application to engineering data. Entropy 24(7), 883 (2022). MR4467767. https://doi.org/10.3390/e24070883
Al-Babtain, A.A., Shakhatreh, M.K., Nassar, M., Afify, A.Z.: A new modified Kies family: Properties, estimation under complete and type-II censored samples, and engineering applications. Mathematics 8(8), 1345 (2020). MR4199201. https://doi.org/10.3934/math.2021176
Alsubie, A.: Properties and applications of the modified Kies–Lomax distribution with estimation methods. J. Math., 2021(2), 1–18 (2021). MR4346604. https://doi.org/10.1155/2021/1944864
Berger, M.-H., Jeulin, D.: Statistical analysis of the failure stresses of ceramic fibres: Dependence of the weibull parameters on the gauge length, diameter variation and fluctuation of defect density. J. Mater. Sci. 38(13), 2913–2923 (2003). https://doi.org/10.1023/A:1024405123420
Bhatti, F.A., Ahmad, M.: On a new family of kies burr iii distribution: Development, properties, characterizations, and applications. Sci. Iran. 27(5), 2555–2571 (2020). ISSN 1026-3098. . URL http://scientiairanica.sharif.edu/article_21382.html
Gupta, R.C., Bradley, D.M.: Representing the mean residual life in terms of the failure rate. Math. Comput. Model. 37(12–13), 1271–1280 (2003). MR1996036. https://doi.org/10.1016/S0895-7177(03)90038-0
Hansen, L.P.: Large sample properties of generalized method of moments estimators. Econometrica: Journal of the econometric society, 50(4) 1029–1054 (1982). MR0666123. https://doi.org/10.2307/1912775
Koenker, R.W.: Quantile regression. Cambridge University Press, (2005). ISBN 9780521845731. MR2268657. https://doi.org/10.1017/CBO9780511754098
Kumar, C.S., Dharmaja, S.H.S.: On some properties of Kies distribution. Metron 72(1), 97–122 (2014). MR3176964. https://doi.org/10.1007/s40300-013-0018-8
Kumar, C.S., Dharmaja, S.H.S.: The exponentiated reduced Kies distribution: Properties and applications. Commun. Stat., Theory Methods 46(17), 8778–8790 (2017). MR3680792. https://doi.org/10.1080/03610926.2016.1193199
Kumar, C.S., Dharmaja, S.H.S.: On modified Kies distribution and its applications. J. Stat. Res. 51(1), 41–60 (2017). MR3702285. https://doi.org/10.47302/jsr.2017510103
Lai, C.-D.: Generalized Weibull distributions. In: Generalized Weibull Distributions, pp. 23–75. Springer, (2014). MR3115122. https://doi.org/10.1007/978-3-642-39106-4_2
Matsushita, S., Hagiwara, K., Shiota, T., Shimada, H., Kuramoto, K., Toyokura, Y.: Lifetime data analysis of disease and aging by the weibull probability distribution. J. Clin. Epidemiol. 45(10), 1165–1175 (1992). https://www.sciencedirect.com/science/article/pii/089543569290157I. https://doi.org/10.1016/0895-4356(92)90157-I
McCool, J.I.: Using the Weibull distribution: reliability, modeling, and inference, vol. 950. John Wiley & Sons, (2012). MR3014584. https://doi.org/10.1002/9781118351994
Prabhakar Murthy, D.N., Xie, M., Jiang, R.: Weibull models. John Wiley & Sons, (2004). MR2013269
Newey, W.K., Powell, J.L.: Asymmetric least squares estimation and testing. Econometrica: Journal of the Econometric Society 55(4), 819–847 (1987). MR0906565. https://doi.org/10.2307/1911031
Rinne, H.: The Weibull distribution: a handbook. Chapman and Hall/CRC, (2008). MR2477856
Sanku, D.E.Y., Nassarn, M., Kumar, D.: Moments and estimation of reduced Kies distribution based on progressive type-II right censored order statistics. Hacet. J. Math. Stat. 48(1), 332–350 (2019). MR3976180. https://doi.org/10.15672/hjms.2018.611
Seguro, J.V., Lambert, T.W.: Modern estimation of the parameters of the weibull wind speed distribution for wind energy analysis. J. Wind Eng. Ind. Aerodyn. 85(1), 75–84 (2000). https://www.sciencedirect.com/science/article/pii/S0167610599001221. https://doi.org/10.1016/S0167-6105(99)00122-1
Sendov, B.: Hausdorff approximations, vol. 50. Springer, (1990). MR1078632. https://doi.org/10.1007/978-94-009-0673-0
Shafiq, A., Lone, S.A., Naz Sindhu, T., El Khatib, Y., Al-Mdallal, Q.M., Muhammad, T.: A new modified Kies Fréchet distribution: Applications of mortality rate of Covid-19. Results Phys. 28, 104638 (2021). https://www.sciencedirect.com/science/article/pii/S2211379721007294. https://doi.org/10.1016/j.rinp.2021.104638
Sobhi, M.M.A.: The modified Kies–Fréchet distribution: properties, inference and application. AIMS Math. 6, 4691–4714 (2021). MR4220431. https://doi.org/10.3934/math.2021276
Soulimani, A., Benjillali, M., Chergui, H., da Costa, D.B.: Multihop weibull-fading communications: Performance analysis framework and applications. J. Franklin Inst. 358(15), 8012–8044 (2021). https://www.sciencedirect.com/science/article/pii/S0016003221004701. MR4319388. https://doi.org/10.1016/j.jfranklin.2021.08.004
Weibull, W.: A statistical distribution function of wide applicability. J. Appl. Mech. 18(3), 293–297 (1951). https://doi.org/10.1115/1.4010337
Wilks, D.S.: Rainfall intensity, the weibull distribution, and estimation of daily surface runoff. J. Appl. Meteorol. Climatol. 28(1), 52–58 (1989). https://doi.org/10.1175/1520-0450(1989)028<0052:RITWDA>2.0.CO;2
Yazhou, J., Molin, W., Zhixin, J.: Probability distribution of machining center failures. Reliab. Eng. Syst. Saf. 50(1), 121–125 (1995). https://www.sciencedirect.com/science/article/pii/095183209500070I. https://doi.org/10.1016/0951-8320(95)00070-I
Zaevski, T.S., Kyurkchiev, N.: Some notes on the four-parameters Kies distribution. Comptes rendus de l’Académie bulgare des Sciences 75(10), 1403–1409 (2022). MR4504780
