MARSHALL-OLKIN YAPISINDA İKİ DEĞİŞKENLİ LOG-LOJİSTİK DAĞILIMI: PARAMETRE TAHMİN STRATEJİLERİNİN DEĞERLENDİRİLMESİ

Öz

Log-lojistik (LL) dağılımı, kalın kuyruklu yapısı ve esnek tehlike fonksiyonu nedeniyle güvenilirlik, aktüeryal, finansal risk ve gelir dağılımı çalışmalarında yaygın olarak kullanılmaktadır. Bu çalışmada, LL dağılımı Marshall-Olkin (MO) yapısı altında iki değişkenli bir forma genişletilmiş ve bağımlılık özellikleri kopula gösterimi aracılığıyla ele alınmıştır. Maksimum olabilirlik tahminine (MLE) ek olarak, aykırı değerlere karşı daha dayanıklı M ve L tahmin edicileri formüle edilerek sağlam tahmin yaklaşımları geliştirilmiştir. Kopula gösterimi, üst kuyruk bağımlılığı ve tekillik bileşeni teorik olarak türetilmiş ve tahmin edicilerin etkinliği simülasyonlarla karşılaştırılmıştır. Sonuçlar, M ve L tahmin edicilerinin küçük ve orta ölçekli örneklemlerde MLE'ye kıyasla daha düşük hata ve daha kararlı sapma sağladığını ortaya koymaktadır. Aynı zamanda, MLE'nin asimptotik verimliliği nedeniyle büyük örneklemlerde fark azalmaktadır. Dahası, sağlam tahmin ediciler MLE için güvenilir başlangıç değerleri sağlayarak optimizasyonun kararlılığını artırmaktadır. UEFA Şampiyonlar Ligi verilerine yapılan uygulamalar, marjinal uyum testlerinin LL hipotezini reddetmediğini, kopula tabanlı testlerin ise model uyumunu doğruladığını göstermiştir. En iyi sonuçlar, log-olasılık ve AIC kriterlerine dayalı L-tahmin edicisi ile elde edilmiştir. Bulgular, sağlam yöntemlerin yoğun kuyruklu ve sınırlı örneklemli veri kümelerinde, MLE'nin ise büyük örneklemlerde daha etkili olduğunu göstermektedir.

Anahtar Kelimeler

• Log-Lojistik Dağılım
• Marshall–Olkin
• Sağlam Tahmin
• M-Tahmin Edici
• L-Tahmin Edici

BIVARIATE LOG-LOGISTIC DISTRIBUTION IN MARSHALL–OLKIN STRUCTURE: ASSESSING THE PARAMETER ESTIMATION STRATEGIES

Öz

The log-logistic (LL) distribution is widely used in reliability, actuarial, financial risk, and income distribution studies due to its heavy-tailed structure and flexible hazard function. In this study, the LL distribution is extended to a bivariate form under the Marshall–Olkin (MO) structure, and its dependence properties are addressed through the copula representation. In addition to maximum likelihood estimation (MLE), robust estimation approaches are developed by formulating M- and L-estimators that are more robust to outliers. The copula representation, upper-tail dependence, and the singularity component are theoretically derived, and the effectiveness of the estimators is compared with simulations. The results reveal that the M- and L-estimators provide lower error and more stable deviation than the MLE in small and medium samples. At the same time, the difference decreases in large samples due to the asymptotic efficiency of the MLE. Furthermore, robust estimators increase the stability of the optimization by providing reliable starting values for the MLE. Applications to UEFA Champions League data showed that marginal fit tests did not reject the LL hypothesis, while copula-based tests confirmed model fit. The best results were obtained with the L-estimator based on log-likelihood and AIC criteria. The findings indicate that robust methods are more effective in heavy-tailed and limited-sample datasets, while MLE is more effective in large samples.

Anahtar Kelimeler

• Log-Logistic Distribution
• Marshall–Olkin
• Robust Estimation
• M-Estimator
• L-Estimator

Kaynakça

1. [1] Gui, W. (2013). Marshall-Olkin extended the log-logistic distribution and its application in minification processes. Applied Mathematical Sciences, 7, 3947–3961. https://doi.org/10.12988/ams.2013.35268
2. [2] Moakofi, T., Oluyede, B., & Makubate, B. (2021). Marshall-Olkin Lindley-Log-logistic distribution: Model, properties and applications. Mathematica Slovaca, 71(5), 1269–1290. https://doi.org/10.1515/ms-2021-0052
3. [3] Lepetu, L., Oluyede, B. O., Makubate, B., Foya, S., & Mdlongwa, P. (2021). Marshall-Olkin Log-Logistic Extended Weibull Distribution: Theory, Properties and Applications. Journal of Data Science, 15(4), 691–722. https://doi.org/10.6339/JDS.201710_15(4).00007
4. [4] Cakmakyapan, S., Ozel, G., Mousa Hussein El Gebaly, Y., & Hamedani, G. G. (2018). The Kumaraswamy Marshall-Olkin Log-Logistic Distribution with Application. Journal of Statistical Theory and Applications, 17(1), 59. https://doi.org/10.2991/jsta.2018.17.1.5
5. [5] Alshangiti, A. M., Kayid, M., & Alarfaj, B. (2016). Further results on the Marshall–Olkin log-logistic distribution: reliability analysis, parameter estimation, and applications. SpringerPlus, 5(1), 385. https://doi.org/10.1186/s40064-016-2007-x
6. [6] Mahmoud, M. R., & Mohammed, S. M. (2013). Estimatıon Of Parameters Of The Marshall-Olkın Extended Log-Logıstıc Dıstrıbutıon From Progressıvely Censored Samples. European Journal of Statistics and Probability, 2(3), 1–13.
7. [7] Yegen, D., & Özel, G. (2018). Marshall-Olkin Half Logistic Distribution with Theory and Applications. Alphanumeric Journal, 6(2), 407–416. https://doi.org/10.17093/alphanumeric.409992
8. [8] Jose, K. K. (2011). Marshall-Olkin Family of Distributions and their applications in reliability theory, time series modeling, and stress-strength analysis. 3918–3923.

9. [9] Popović, B. V., Genç, A. İ., & Domma, F. (2018). Copula-based properties of the bivariate Dagum distribution. Computational and Applied Mathematics, 37(5), 6230–6251. https://doi.org/10.1007/s40314-018-0682-7
10. [10] Santos-Neto, M., Bourguignon, M., Zea, L. M., Nascimento, A. D., & Cordeiro, G. M. (2014). The Marshall-Olkin extended Weibull family of distributions. Journal of Statistical Distributions and Applications, 1(1), 9. https://doi.org/10.1186/2195-5832-1-9
11. [11] Sarhan, A. M., & Balakrishnan, N. (2007). A new class of bivariate distributions and its mixture. Journal of Multivariate Analysis, 98(7), 1508–1527. https://doi.org/10.1016/j.jmva.2006.07.007
12. [12] Barreto-Souza, W., & Lemonte, A. J. (2013). Bivariate Kumaraswamy distribution: Properties and a new method to generate bivariate classes. Statistics, 47(6), 1321–1342. https://doi.org/10.1080/02331888.2012.694446
13. [13] Maronna, R. A., Martin, R. D., & Yohai, V. J. (2006). Robust Statistics: Theory and Methods. John Wiley & Sons.
14. [14] Elamir, E. A. H., & Seheult, A. H. (2004). Exact variance structure of sample L-moments. Journal of Statistical Planning and Inference, 124(2), 337–359. https://doi.org/10.1016/S0378-3758(03)00213-1
15. [15] Marshall, A. W., & Olkin, I. (1967). A Multivariate Exponential Distribution. Journal of the American Statistical Association, 62(317), 30–44.
16. [16] Nelsen, R. B. (2006). An Introduction to Copulas. Springer Science & Business Media.
17. [17] Huber, P. J., & Ronchetti, E. M. (2009). Robust Statistics (1st ed.). Wiley. https://doi.org/10.1002/9780470434697
18. [18] Serfling, R. J. (1980). Approximation Theorems of Mathematical Statistics (1st ed.). Wiley. https://doi.org/10.1002/9780470316481
19. [19] Hosking, J. R. M. (1990). L-Moments: Analysis and Estimation of Distributions Using Linear Combinations of Order Statistics. Journal of the Royal Statistical Society Series B: Statistical Methodology, 52(1), 105–124. https://doi.org/10.1111/j.2517-6161.1990.tb01775.x
20. [20] Meintanis, S. G. (2007). Test of fit for Marshall–Olkin distributions with applications. Journal of Statistical Planning and Inference, 137(12), 3954–3963. https://doi.org/10.1016/j.jspi.2007.04.013
21. [21] Kundu, D., & Gupta, R. D. (2009). Bivariate generalized exponential distribution. Journal of Multivariate Analysis, 100(4), 581–593. https://doi.org/10.1016/j.jmva.2008.06.012
22. [22] Berg, D. (2009). Copula goodness-of-fit testing: An overview and power comparison. The European Journal of Finance, 15(7–8), 675–701. https://doi.org/10.1080/13518470802697428
23. [23] Rüschendorf, L. (2009). On the distributional transform, Sklar’s theorem, and the empirical copula process. Journal of Statistical Planning and Inference, 139(11), 3921–3927. https://doi.org/10.1016/j.jspi.2009.05.030
24. [24] Genest, C., Rémillard, B., & Beaudoin, D. (2009). Goodness-of-fit tests for copulas: A review and a power study. Insurance: Mathematics and Economics, 44(2), 199–213. https://doi.org/10.1016/j.insmatheco.2007.10.005