M-Estimations of Shape and Scale Parameters by Order Statistics in Least Informative Distributions on q-deformed logarithm

Abstract

The maximum logq likelihood estimation (MLqE) method is used to estimate robustly parameters recently. In robust estimation method, the least informative distribution (LID) proposed by Huber is a convex combination of two probability density functions 𝑓0 and 𝑓1. In this study, the recently proposed least informative distributions (LIDs) in MLqE are used to estimate parameters. This paper also studies on the objective functions proposed by maximum logq-likelihood principle (MLqE) originally derived by logq-likelihood. The role and capability of order statistics in LIDs in MLqE are examined while getting the estimates of shape and scale parameters. The distance measure for evaluation of fitting performance is given to choose a value for the parameter 𝑞 in logq when the objective functions derived from MLqE are used. The simulation and real data application are given. Thus, we compare the fitting performance of objective functions constructed by MLE on log, MLqE on logq and LIDs with order statistics in MLqE. We observed that order statistic chosen for density 𝑓1 in LID in MLqE has a new objective function to fit the data sets. In the simulation, we make two contaminations into artificial data sets. The first contamination is inliers derived by order statistics and the second one is outliers. Thus, we observe that the new objective function can give satisfactory results.

Keywords

• Distribution
• efficiency
• estimation
• order statistics
• robustness

References

1. Andrews DF, Hampel FR, 2015. Robust estimates of location: Survey and advances. Princeton University Press.
2. Arnold BC, Balakrishnan N, Nagaraja HN, 1992. A first course in order statistics (Vol. 54). Siam.
3. Bozdogan H, 1987. Model selection and Akaike's information criterion (AIC): The general theory and its analytical extensions. Psychometrika 52(3):345-370.
4. Csaki F, 1981. Second international symposium on information theory. Académiai Kiadó, Budapest.
5. Çankaya MN, Korbel J, 2017. On statistical properties of Jizba–Arimitsu hybrid entropy. Physica A: Statistical Mechanics and its Applications 475: 1-10.
6. Çankaya MN, Korbel J, 2018. Least informative distributions in maximum q-log-likelihood estimation. Physica A: Statistical Mechanics and its Applications 509: 140-150.
7. Çankaya MN, 2018. Asymmetric bimodal exponential power distribution on the real line. Entropy 20(1): 1-23.
8. Elze HT, 2004. Introduction: Quantum Theory and Beneath? In Decoherence and Entropy in Complex Systems. Springer. Berlin, Heidelberg, 119-124.

9. Ferrari D, Yang Y, 2010. Maximum Lq-likelihood estimation. The Annals of Statistics 38(2): 753-783.
10. Giuzio M, Ferrari D, Paterlini S, 2016. Sparse and robust normal and t-portfolios by penalized Lq-likelihood minimization. European Journal of Operational Research 250(1): 251-261.
11. Gelfand I, Fomin S, 1963. Calculus of Variations. Prentice-Hall Inc. Englewood Cliffs. NJ.
12. Godambe VP, 1960. An optimum property of regular maximum likelihood estimation. The Annals of Mathematical Statistics 31(4):1208-1211.
13. Hampel FR, Ronchetti EM, Rousseeuw PJ, Stahel WA, 2011. Robust statistics: the approach based on influence functions. John Wiley & Sons. Vol. 196.
14. Huber-Carol, C, Balakrishnan, N, Nikulin M, Mesbah M. (Eds.), 2012. Goodness-of-fit tests and model validity. Springer Science & Business Media.
15. Huber PJ, 1981. Ronchetti EM. Robust statistics. John Wiley & Sons. New York.
16. Jizba P, 2004. Information theory and generalized statistics. In Decoherence and Entropy in Complex Systems (pp. 362-376). Springer, Berlin, Heidelberg.
17. Jizba P, Korbel J, 2016. On q-non-extensive statistics with non-Tsallisian entropy. Physica A: Statistical Mechanics and its Applications 444: 808-827.
18. Malik SC, Arora S, 1992. Mathematical analysis. New Age International.
19. Ni XS, Huo X, 2009. Another look at Huber's estimator: A new minimax estimator in regression with stochastically bounded noise. Journal of statistical planning and inference 139(2):503-515.
20. Örkcü HH, Özsoy VS, Aksoy E, Dogan MI, 2015. Estimating the parameters of 3-p Weibull distribution using particle swarm optimization: A comprehensive experimental comparison. Applied Mathematics and Computation 268: 201-226.
21. Proschan F, 1963. Theoretical explanation of observed decreasing failure rate. Technometrics 5(3), 375-383.
22. Prudnikov AP, Brychkov, IA, Marichev OI, 1986. Integrals and series: special functions (Vol. 2). CRC Press.
23. Rinne, H, 2008. The Weibull distribution: A handbook. CRC press.
24. Shevlyakov G, Morgenthaler S, Shurygin A, 2008. Redescending M-estimators. Journal of Statistical Planning and Inference 138(10): 2906-2917.
25. Suyari H, 2006. Mathematical structures derived from the q-multinomial coefficient in Tsallis statistics. Physica A: Statistical Mechanics and its Applications 368(1): 63-82.
26. Tsallis C, 1988. Possible generalization of Boltzmann-Gibbs statistics. Journal of statistical physics 52(1-2): 479-487.