On the Robust Estimations of Location and Scale Parameters for Least Informative Distributions

Year 2020, Volume: 15 Issue: 2, 71 - 78, 24.09.2020

Mehmet Niyazi Çankaya

Abstract

M-estimation as generalization of maximum likelihood estimation (MLE) method is well-known approach to get the robust estimations of location and scale parameters in objective function ρ especially. Maximum log_q likelihood estimation (MLqE) method uses different objective function called as ρ_(log_q ). These objective functions are called as M-functions which can be used to fit data set. The least informative distribution (LID) is convex combination of two probability density functions f_0 and f_1. In this study, the location and scale parameters in any objective functions ρ_log, ρ_(log_q ) and ψ_(log_q ) (f_0,f_1 ) which are from MLE, MLqE and LIDs in MLqE are estimated robustly and simultaneously. The probability density functions which are f_0 and f_1 underlying and contamination distributions respectively are chosen from exponential power (EP) distributions, since EP has shape parameter α to fit data efficiently. In order to estimate the location μ and scale σ parameters, Huber M-estimation, MLE of generalized t (Gt) distribution are also used. Finally, we test the fitting performance of objective functions by using a real data set. The numerical results showed that ψ_(log_q ) (f_0,f_1 ) is more resistance values of estimates for μ and σ when compared with other ρ functions.

Keywords

: Least informative distributions, maximum log_q likelihood estimation method, robustness

References

• [1] Ni, X. S., & 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.
• [2] Huber, P. J. (1992). Robust estimation of a location parameter. In Breakthroughs in statistics (pp. 492-518). Springer, New York, NY.
• [3] Huber, P. J., & Ronchetti, E. M. (1981). Robust statistics john wiley & sons. New York, 1(1).
• [4] Shevlyakov, G., Morgenthaler, S., & Shurygin, A. (2008). Redescending M-estimators. Journal of Statistical Planning and Inference, 138(10), 2906-2917.
• [5] Ferrari, D., & Yang, Y. (2010). Maximum Lq-likelihood estimation. The Annals of Statistics, 38(2), 753-783.
• [6] 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.
• [7] Andrews, D. F., & Hampel, F. R. (2015). Robust estimates of location: Survey and advances. Princeton University Press.
• [8] Hampel, F. R., Ronchetti, E. M., Rousseeuw, P. J., & Stahel, W. A. (2011). Robust statistics: the approach based on influence functions (Vol. 196). John Wiley & Sons.
• [9] Tsallis, C. (1988). Possible generalization of Boltzmann-Gibbs statistics. Journal of statistical physics, 52(1-2), 479-487.
• [10] Çankaya, M. N., & Korbel, J. (2018). Least informative distributions in maximum q-log-likelihood estimation. Physica A: Statistical Mechanics and its Applications, 509, 140-150.
• [11] Malik, S. C., & Arora, S. (1992). Mathematical analysis. New Age International.
• [12] Örkcü, H. H., Özsoy, V. S., Aksoy, E., & Dogan, M. I. (2015). Estimating the parameters of 3-p Weibull distribution using particle swarm optimization: A comprehensive experimental comparison. Applied Mathematics and Computation, 268, 201-226.
• [13] Machado, J. T. (2014). Fractional order generalized information. Entropy, 16(4), 2350-2361.
• [14] 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.
• [15] Hadjiagapiou, I. A. (2011). The random field Ising model with an asymmetric and anisotropic bimodal probability distribution. Physica A: Statistical Mechanics and its Applications, 390(20), 3204-3215.
• [16] Hadjiagapiou, I. A. (2012). The random field Ising model with an asymmetric and anisotropic trimodal probability distribution. Physica A: Statistical Mechanics and its Applications, 391(13), 3541-3555.
• [17] Arslan, O., & Genc, A. I. (2009). The skew generalized t distribution as the scale mixture of a skew exponential power distribution and its applications in robust estimation. Statistics, 43(5), 481-498.
• [18] Çankaya, M. N., & Korbel, J. (2017). On statistical properties of Jizba–Arimitsu hybrid entropy. Physica A: Statistical Mechanics and its Applications, 475, 1-10.
• [19] Korbel, J. (2017). Rescaling the nonadditivity parameter in Tsallis thermostatistics. Physics Letters A, 381(32), 2588-2592.
• [20] Jizba, P., Korbel, J., & Zatloukal, V. (2017). Tsallis thermostatics as a statistical physics of random chains. Physical Review E, 95(2), 022103.
• [21] Elze, H. T. (2004). Introduction: Quantum Theory and Beneath?. In Decoherence and Entropy in Complex Systems (pp. 119-124). Springer, Berlin, Heidelberg.
• [22] Jizba, P., & Korbel, J. (2016). On q-non-extensive statistics with non-Tsallisian entropy. Physica A: Statistical Mechanics and its Applications, 444, 808-827.
• [23] Akaike, H., Petrov, B. N., & Csaki, F. (1973). Second international symposium on information theory.
• [24] Bozdogan, H. (1987). Model selection and Akaike's information criterion (AIC): The general theory and its analytical extensions. Psychometrika, 52(3), 345-370.
• [25] M.N. Cankaya, Y.M. Bulut, F.Z. Dogru and O. Arslan, A bimodal extension of the generalized gamma distribution, Revista Colombiana de Estadistica, 38 (2015), no. 2, 353-370.
• [26] Çankaya, M. N. (2018). Asymmetric bimodal exponential power distribution on the real line. Entropy, 20(1), 23.
• [27] Çankaya, M. N., & Arslan, O. (2020). On the robustness properties for maximum likelihood estimators of parameters in exponential power and generalized T distributions. Communications in Statistics-Theory and Methods, 49(3), 607-630.
• [28] Ronchetti, E. (1997). Robustness aspects of model choice. Statistica Sinica, 327-338.