Thomas L. Toulias

New Information Measures for the Generalized Normal Distribution

We introduce a three-parameter generalized normal distribution, which belongs to the Kotz type distribution family, to study the generalized entropy type measures of information. For this generalized normal, the Kullback-Leibler information is evaluated, which extends the well known result for the normal distribution, and plays an important role for the introduced generalized information measure. These generalized entropy type measures of information are also evaluated and presented.

Fitting the Michaelis–Menten model

Abstract The target of this paper is to introduce and investigate different methods for the solution of the Michaelis–Menten (M–M) parameters. One of the main results is that the estimation provides no unique estimators. Two main approaches for parameter estimation of the M–M model are discussed: The analytic one, and the iterative one. The former regards the linearization or Linear Least Squares (LSS), as well as the actual Non-Linear Least Squares (NLLS) evaluation, while the latter regards certain iterative methods for the NLLS estimation. The iterative methods are: An optimized Gauss–Newton (GN) approach, a quadratic and linear expansion approaches for the M–M model, as well as a Batch Sequential approach. All these methods are investigated, evaluated and compared through examples using certain datasets, in which the M–M is the assumed model.

Generalizations of Entropy and Information Measures

This paper presents and discusses two generalized forms of the Shannon entropy, as well as a generalized information measure. These measures are applied on a exponential-power generalization of the usual Normal distribution, emerged from a generalized form of the Fisher’s entropy type information measure, essential to Cryptology. Information divergences between these random variables are also discussed. Moreover, a complexity measure, related to the generalized Shannon entropy, is also presented, extending the known SDL complexity measure.

Generalized Information for the -Order Normal Distribution

This paper investigates a generalization of Fisher’s entropy type information measure under the multivariate -order normal distribution, related to his measure, as well as its corresponding Shannon entropy. Certain boundaries of this information measure are also proved and discussed.

On the Generalized Lognormal Distribution

This paper introduces, investigates, and discusses the -order generalized lognormal distribution (-GLD). Under certain values of the extra shape parameter , the usual lognormal, log-Laplace, and log-uniform distribution, are obtained, as well as the degenerate Dirac distribution. The shape of all the members of the -GLD family is extensively discussed. The cumulative distribution function is evaluated through the generalized error function, while series expansion forms are derived. Moreover, the moments for the -GLD are also studied.

Discrete Mathematics for Statistical and Probability Problems

This paper offers a compact presentation of the solid involvement of Discrete Mathematics in various fields of Statistics and Probability Theory. As far as the discrete methodologies in Statistics are concerned, our interest is focused on the foundations and applications of the Experimental Design Theory. The set-theoretic approach of the foundations of Probability Theory is also presented, while the notions of concepts and fuzzy logic are formulated and discussed.

Hazard Rate and Future Lifetime for the Generalized Normal Distribution

The target of this paper is to discuss a generalized form of the well-known Law of Frequency Error. This particular Law of Frequency of Errors is what is known as “Gaussian” or “Normal” distribution and appeared to have an aesthetic appeal to all the branches of Science. The Generalized Normal Distribution is presented as a basis to our study. We derive also the corresponding hazard function as well as the future lifetime of the Generalized Normal Distribution (GND), while new results are also presented. Moreover, due to some of the important distribution the GND family includes, specific results can also be extracted for some other distributions.

Plane-Geometric Investigation of a Proof of the Pohlke’s Fundamental Theorem of Axonometry

Consider a bundle of three given coplanar line segments (radii) where only two of them are permitted to coincide. Each pair of these radii can be considered as a pair of two conjugate semidiameters of an ellipse. Thus, three concentric ellipses Ei, i = 1, 2, 3, are then formed. In a proof by G.A. Peschka of Karl Pohlke’s fundamental theorem of axonometry, a parallel projection of a sphere onto a plane, say, \(\mathbb E\), is adopted to show that a new concentric (to Ei) ellipse E exists, “circumscribing” all Ei, i.e., E is simultaneously tangent to all \(E_i\subset \mathbb E\), i = 1, 2, 3. Motivated by the above statement, this paper investigates the problem of determining the form and properties of the circumscribing ellipse E of Ei, i = 1, 2, 3, exclusively from the analytic plane geometry’s point of view (unlike the sphere’s parallel projection that requires the adoption of a three-dimensional space). All the results are demonstrated by the actual corresponding figures as well as with the calculations given in various examples.

Generalized Information Criteria for the Best Logit Model

In this paper the \(\gamma \)–order Generalized Fisher’s entropy type Information measure (\(\gamma \)–GFI) is adopted as a criterion for the selection of the best Logit model. Thus the appropriate Relative Risk model can be evaluated through an algorithm. The case of the entropy power is also discussed as such a criterion. Analysis of a real breast cancer data set is conducted to demonstrate the proposed algorithm, while algorithm’s realizations, through MATLAB scripts, are cited in Appendix.

Inequalities for the Fisher’s Information Measures

The objective of this chapter is to provide a thorough discussion on inequalities related to the entropy measures in connection to the γ-order generalized normal distribution (γ–GND). This three-term (position, scale and shape) family of distributions plays the role of the usual multivariate normal distribution in information theory. Moreover, the γ–GND is the appropriate family of distributions to support a generalized version of the entropy type Fisher’s information measure. This generalized (entropy type) Fisher’s information is also discussed as well as the generalized entropy power, while the γ-GND heavily contributes to these generalizations. The appropriate bounds and inequalities of these measures are also provided.