On the estimating equations and objective functions for parameters of exponential power distribution: Application for disorder

The efficient modeling for disorder in a phenomena depends on the chosen score and objective functions. The main parameters in modeling are location, scale and shape. The exponential power distribution known as generalized Gaussian is extensively used in modeling. In real world, the observations are member of different parametric models or disorder in a data set exists. In this study, estimating equations for the parameters of exponential power distribution are derived to have robust and also efficient M-estimators when the data set includes disorder or contamination. The robustness property of M-estimators for the parameters is examined. Fisher information matrices based on the derivative of score functions from $\log$, $\log_q$ and distorted log-likelihoods are proposed by use of Tsallis $q$-entropy in order to have variances of M-estimators. It is shown that matrices derived by score functions are positive semidefinite if conditions are satisfied. Information criteria inspired by Akaike and Bayesian are arranged by taking the absolute value of score functions. Fitting performances of score functions from estimating equations and objective functions are tested by applying volume, information criteria and mean absolute error which are essential tools in modeling to assess the fitting competence of the proposed functions. Applications from simulation and real data sets are carried out to compare the performance of estimating equations and objective functions. It is generally observed that the distorted log-likelihood for the estimations of parameters of exponential power distribution has superior performance than other score and objective functions for the contaminated data sets.