Maximum Likelihood Estimation for the Inverted Kumaraswamy Distribution Based on the Extreme Ranked Set Sampling

Abstract

This study investigates the use of extreme ranked set sampling (ERSS) for parameter estimation in the Inverted Kumaraswamy (IK) distribution. The Kumaraswamy distribution has wide applications in fields such as reliability testing, environmental studies, financial analysis, and survival analysis. The paper emphasizes the efficiency advantages of ERSS, particularly in capturing extreme values, which are crucial for distributions with heavy tails or skewed data. By incorporating ERSS, this research demonstrates that more accurate and efficient parameter estimates can be obtained compared to traditional sampling methods like simple random sampling (SRS). A simulation study is also performed to demonstrate the performance of the proposed estimators. Finally, a real data set is presented for illustrative purposes. The findings suggest that ERSS outperforms SRS in terms of precision, particularly in contexts where extreme values play a significant role. This work contributes to the advancement of sampling techniques for extreme value contexts, with potential applications in various fields, including environmental research, finance, and reliability analysis.

Keywords

• Inverted Kumaraswamy Distribution
• Ranked Set Sampling
• Extreme Ranked Set Sampling
• Maximum Likelihood Estimator

Ethical Statement

The study is complied with research and publication ethics.

References

1. A. I. Al-Omari et al., “Reliability estimation of inverse Lomax distribution using extreme ranked set sampling,” Adv. Math. Phys., vol. 2021, no. 1, pp. 4599872, 2021.
2. D. S. Bhoj and G. Chandra, “Weighted Ranked Set Sampling for Skewed Distributions,” Mathematics, vol. 12, no. 13, pp. 2023, 2024.
3. H. Jiang and W. Gui, “Bayesian inference for the parameters of Kumaraswamy distribution via ranked set sampling,” Symmetry, vol. 13, no. 7, pp. 1170, 2021.
4. N. A. Mode, L. L. Conquest and D. A. Marker, “Ranked set sampling for ecological research: accounting for the total costs of sampling,” Environmetrics, vol. 10, no. 2, pp. 179–194, 1999.
5. A. A. Yavuz and D. Öz, “A comparative study of Ranked Set Sampling (RSS) and Simple Random Sampling (SRS) in agricultural studies: A case study on the walnut tree,” Eurasian J. Forest Sci., vol. 8, no. 1, pp. 94–108, 2020.
6. J. Gillariose et al., “Reliability test plan for the Marshall-Olkin extended inverted Kumaraswamy distribution,” Reliability: Theory & Applications, vol. 16, no. 3(63), pp. 26–36, 2021.
7. K. Bağcı, “Parameter Estimation of the Inverted Kumaraswamy Distribution by Using L-Moments Method: An Application on Precipitation Data,” Cumhuriyet Sci. J., vol. 45, no. 3, pp. 629–635, 2024.
8. W. Tian et al., “Change point analysis for Kumaraswamy distribution,” Mathematics, vol. 11, no. 3, pp. 553, 2023.

9. J. K. Pokharel et al., “Probability Distributions for Modeling Stock Market Returns—An Empirical Inquiry,” Int. J. Financial Stud., vol. 12, no. 2, pp. 43, 2024.
10. A. I. Ishaq et al., “Log-Kumaraswamy distribution: its features and applications,” Front. Appl. Math. Stat., vol. 9, pp. 1258961, 2023.
11. J. Bengalath and B. Punathumparambath, “Harris extended inverted Kumaraswamy distribution: Properties and applications to COVID-19 data,” Int. J. Data Sci. Anal., pp. 1–23, 2024.
12. A. Al-Nasser, A. Al-Omari and M. Al-Rawwash, “Monitoring the process mean based on quality control charts using on folded ranked set sampling,” Pak. J. Stat. Oper. Res., pp. 79–92, 2013.
13. P. K. Sahu and N. Gupta, “On general weighted extropy of extreme ranked set sampling,” Commun. Stat. Theory Methods, pp. 1–19, 2024.
14. N. Alotaibi et al., “Statistical inference for the Kavya–Manoharan Kumaraswamy model under ranked set sampling with applications,” Symmetry, vol. 15, no. 3, pp. 587, 2023.
15. A. Ali et al., “Estimation of population mean by using a generalized family of estimators under classical ranked set sampling,” Res. Math. Stat., vol. 8, no. 1, 2021.
16. H. Samawi, L. Yu and J. Yin, “On Cox proportional hazards model performance under different sampling schemes,” PLoS ONE, vol. 18, no. 4, pp. e0278700, 2023.
17. H. M. Samawi, M. S. Ahmed and W. Abu‐Dayyeh, “Estimating the population mean using extreme ranked set sampling,” Biometrical J., vol. 38, no. 5, pp. 577–586, 1996.
18. E. Castillo, A. S. Hadi, N. Balakrishnan and J. M. Sarabia, Extreme Value and Related Models with Applications in Engineering and Science, Hoboken, NJ: Wiley, 2005.