On min- and max-Kies families: distributional properties and saturation in Hausdorff sense
Volume 11, Issue 3 (2024), pp. 265–288
Pub. online: 9 January 2024
Type: Research Article
Open Access
Open Access
Received
14 August 2023
14 August 2023
Revised
3 November 2023
3 November 2023
Accepted
2 January 2024
2 January 2024
Published
9 January 2024
9 January 2024
Abstract
The purpose of this paper is to explore two probability distributions originating from the Kies distribution defined on an arbitrary domain. The first one describes the minimum of several Kies random variables whereas the second one is for their maximum – they are named min- and max-Kies, respectively. The properties of the min-Kies distribution are studied in details, and later some duality arguments are used to examine the max variant. Also the saturations in the Hausdorff sense are investigated. Some numerical experiments are provided.
References
Afify, A., Gemeay, A., Alfaer, N., Cordeiro, G., Hafez, E.: 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
Ahmad, Z., Mahmoudi, E., Roozegar, R., Alizadeh, M., Afify, A.: A new exponential-X family: modeling extreme value data in the finance sector. Math. Probl. Eng. 2021, 1–14 (2021). https://doi.org/10.1155/2021/2394931
Al-Babtain, A., Shakhatreh, M., Nassar, M., Afify, A.: 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/10.3390/math8081345
Alsubie, A.: Properties and applications of the modified Kies–Lomax distribution with estimation methods. J. Math. 2021, 1–18 (2021). MR4346604. https://doi.org/10.1155/2021/1944864
Gupta, R., Bradley, D.: 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
He, W., Ahmad, Z., Afify, A., Goual, H.: The arcsine exponentiated-X family: validation and insurance application. Complexity 2020, 1–18 (2020). MR4218892. https://doi.org/10.3390/e22060603
Kumar, C.S., Dharmaja, 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.: 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.: 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
McCool, J.: Using the Weibull Distribution: Reliability, Modeling, and Inference, vol. 950. John Wiley & Sons (2012). MR3014584. https://doi.org/10.1002/9781118351994
Rinne, H.: The Weibull Distribution: A Handbook. Chapman and Hall/CRC (2008). MR2477856
Sanku, D., 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
Sendov, B.: Hausdorff Approximations, vol. 50. Springer (1990). MR1078632. https://doi.org/10.1007/978-94-009-0673-0
Sobhi, 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
Weibull, W.: A statistical distribution function of wide applicability. J. Appl. Mech. 18(3), 293–297 (1951). https://doi.org/10.1115/1.4010337
Zaevski, T., Kyurkchiev, N.: Some notes on the four-parameters Kies distribution. C. R. Acad. Bulg. Sci. 75(10), 1403–1409 (2022). MR4504780
Zaevski, T., Kyurkchiev, N.: On some composite Kies families: distributional properties and saturation in Hausdorff sense. Mod. Stoch. Theory Appl. 10(3), 287–312 (2023). MR4608189. https://doi.org/10.15559/23-vmsta227
Zaevski, T., Nedeltchev, D.: From BASEL III to BASEL IV and beyond: Expected shortfall and expectile risk measures. Int. Rev. Financ. Anal. 87, 102645 (2023). https://doi.org/10.1016/j.irfa.2023.102645
Zhenwu, Y., Ahmad, Z., Almaspoor, Z., Khosa, S.: On the genesis of the Marshall-Olkin family of distributions via the T-X family approach: Statistical modeling. Comput. Mater. Continua 67(1), 753–760 (2021). MR4417175. https://doi.org/10.5269/bspm.53071
