On Statistical Properties of Jizba-Arimitsu Hybrid Entropy

Jizba-Arimitsu entropy (also called hybrid entropy) combines axiomatics of R\'enyi and Tsallis entropy. It has many common properties with them, on the other hand, some aspects as e.g., MaxEnt distributions, are completely different from the former two entropies. In this paper, we demonstrate the statistical properties of hybrid entropy, including the definition of hybrid entropy for continuous distributions, its relation to discrete entropy and calculation of hybrid entropy for some well-known distributions. Additionally, definition of hybrid divergence and its connection to Fisher metric is also discussed. Interestingly, the main properties of continuous hybrid entropy and hybrid divergence are completely different from measures based on R\'enyi and Tsallis entropy. This motivates us to introduce average hybrid entropy, which can be understood as an average between Tsallis and R\'enyi entropy

A Bimodal Extension of the Generalized Gamma Distribution

A bimodal extension of the generalized gamma distribution is proposed by using a mixing approach. Some distributional properties of the new distribution are investigated. The maximum likelihood (ML) estimators for the parameters of the new distribution are obtained. Real data examples are given to show the strength of the new distribution for modeling data.Una extensión bimodal de la distribución gamma generalizada es propuesta a través de un enfoque de mixturas. Algunas propiedades de la nueva distribución son investigadas. Los estimadores máximo verosímiles (ML por sus siglas en inglés) de los parámetros de la nueva distribución son obtenidos. Algunos ejemplos con datos reales son utilizados con el fin de mostrar las fortalezas de la nueva distribución en la modelación de datos

Propositions for confidence interval in systematic sampling on real line

Systematic sampling is used as a method to get the quantitative results from tissues and radiological images. Systematic sampling on a real line (?) is a very attractive method within which biomedical imaging is consulted by practitioners. For the systematic sampling on ?, the measurement function (MF) occurs by slicing the three-dimensional object equidistant systematically. The currently-used covariogram model in variance approximation is tested for the different measurement functions in a class to see the performance on the variance estimation of systematically-sampled ?. An exact calculation method is proposed to calculate the constant ?(q, N) of the confidence interval in the systematic sampling. The exact value of constant ?(q, N) is examined for the different measurement functions, as well. As a result, it is observed from the simulation that the proposed MF should be used to check the performances of the variance approximation and the constant ?(q, N). Synthetic data can support the results of real data. © 2016 by the authors

Asymmetric bimodal exponential power distribution on the real line

The asymmetric bimodal exponential power (ABEP) distribution is an extension of the generalized gamma distribution to the real line via adding two parameters that fit the shape of peakedness in bimodality on the real line. The special values of peakedness parameters of the distribution are a combination of half Laplace and half normal distributions on the real line. The distribution has two parameters fitting the height of bimodality, so capacity of bimodality is enhanced by using these parameters. Adding a skewness parameter is considered to model asymmetry in data. The location-scale form of this distribution is proposed. The Fisher information matrix of these parameters in ABEP is obtained explicitly. Properties of ABEP are examined. Real data examples are given to illustrate the modelling capacity of ABEP. The replicated artificial data from maximum likelihood estimates of parameters of ABEP and other distributions having an algorithm for artificial data generation procedure are provided to test the similarity with real data. A brief simulation study is presented. © 2018 by the authors.Acknowledgments: I would like to sincerely thank referees for their comments. I would like to thank the partial financial support from the Turkish government and also the English language editing from the University of Us¸ak, School of Foreign Language. I also wish to thank my great parents, and the memory of my kind brother, whose name is Sabri Çankaya

Least informative distributions in maximum q-log-likelihood estimation

We use the maximum q-log-likelihood estimation for Least informative distributions (LIDs) in order to estimate the parameters in probability density functions (PDFs) efficiently and robustly when data include outlier(s). LIDs are derived by using convex combinations of two PDFs. A convex combination of two PDFs is composed of an underlying distribution and a contamination. The optimal criterion is obtained by minimizing the change of maximum q-log-likelihood function when the data contain small amount of contamination. In this paper, we make a comparison between ordinary maximum likelihood estimation, maximum q-log-likelihood estimation (MqLE) and LIDs based on MqLE for parameter estimation from data with outliers. We derive a new Fisher information matrix based on the score function for LID from M-function and use it for choice of optimal estimator in the class of MqLE. The model selection is done by the robust information criteria. We test the methods on the real data with outliers and estimate shape and scale parameters of probability distributions. As a result, we show that the LIDs based on MqLE provide the most robust and efficient estimation of the model parameters. © 2018 Elsevier B.V.Austrian Science Fund: I 3073 Grantová Agentura ?eské Republiky: 17-33812LWe are indebted to Prof. Dr. James F. Peters from the University of Manitoba, Winnipeg, Canada, for critical reading. We also thank to Foreign Language School of Uşak University for editing language and partial support from Turkish government as an added salary for M.N.Ç. J. K. was supported by the Austrian Science Fund , grant No. I 3073 and by the Czech Science Foundation , grant No. 17-33812L . Appendix

On the robustness properties for maximum likelihood estimators of parameters in exponential power and generalized T distributions

Examining the robustness properties of maximum likelihood (ML) estimators of parameters in exponential power and generalized t distributions has been considered together. The well-known asymptotic properties of ML estimators of location, scale and added skewness parameters in these distributions are studied. The ML estimators for location, scale and scale variant (skewness) parameters are represented as an iterative reweighting algorithm (IRA) to compute the estimates of these parameters simultaneously. The artificial data are generated to examine performance of IRA for ML estimators of parameters simultaneously. We make a comparison between these two distributions to test the fitting performance on real data sets. The goodness of fit test and information criteria approve that robustness and fitting performance should be considered together as a key for modeling issue to have the best information from real data sets. © 2018, © 2018 Taylor & Francis Group, LLC.We would like to thank sincerely Editor in Chief and Associate Editor and finally referees for their supports and efforts in the process. We also thank to Foreign Language School of U?ak University for editing language

The informational entropy endowed in cortical oscillations

A two-dimensional shadow may encompass more information than its corresponding three-dimensional object. Indeed, if we rotate the object, we achieve a pool of observed shadows from different angulations, gradients, shapes and variable length contours that make it possible for us to increase our available information. Starting from this simple observation, we show how informational entropies might turn out to be useful in the evaluation of scale-free dynamics in the brain. Indeed, brain activity exhibits a scale-free distribution that leads to the variations in the power law exponent typical of different functional neurophysiological states. Here we show that modifications in scaling slope are associated with variations in Rényi entropy, a generalization of Shannon informational entropy. From a three-dimensional object’s perspective, by changing its orientation (standing for the cortical scale-free exponent), we detect different two-dimensional shadows from different perception angles (standing for Rényi entropy in different brain areas). We show how, starting from known values of Rényi entropy (easily detectable in brain fMRIs or EEG traces), it is feasible to calculate the scaling slope in a given moment and in a given brain area. Because changes in scale-free cortical dynamics modify brain activity, this issue points towards novel approaches to mind reading and description of the forces required for transcranial stimulation. © 2018, Springer Nature B.V