scholarly journals Shape measures of generalized beta distributions

2021 ◽  
Vol 2094 (2) ◽  
pp. 022009
Author(s):  
V G Polosin
Keyword(s):  
Complex Systems ◽  
Information Entropy ◽  
Beta Distribution ◽  
Random Variable ◽  
Uncertainty Interval ◽  
Information Measures ◽  
Beta Distributions ◽  
Independent Information ◽  
Probabilistic Measures ◽  
Generalized Beta Distribution

Abstract This paper presents shape measures for generalized beta distributions that unit many subfamilies of distributions. For the study of complex systems, the information entropy of the whole family of the generalized beta distribution is obtained. The paper uses the interval of entropy uncertainty as an estimate of the entropy uncertainty for probable models, which are given in units of an observable random variable. The entropy uncertainty interval was used to construct the entropy coefficient of unbiased subfamilies of the generalized beta distribution. Particular entropy coefficients are given for frequently used subfamilies of beta distribution, that greatly facilitates the use of coefficients as independent information measures in determining the shape of models. The paper contains the most general formulas for probabilistic measures of the distributions shape also.

Mapping of Beta Distribution for the Study of Dispersed Materials

2022 ◽  
Vol 1049 ◽  
pp. 295-304
Author(s):  
Vitaly Polosin
Keyword(s):  
Computer Technology ◽  
Beta Distribution ◽  
Powder Materials ◽  
Size Distributions ◽  
Particle Size Distributions ◽  
Beta Distributions ◽  
Distribution Shape ◽  
Negative Skewness ◽  
Different Shapes ◽  
Probabilistic Measures

In the study of polydisperse materials, most of the experimental particle size distributions were obtained on bounded intervals. In these cases, it is also desirable to use bounded models with different shapes to simulate the results of studying polydisperse and powder materials. The beta distribution is often used to approximate results due to the fact that this distribution contains many forms for displaying realizations on a limited interval. With the development of computer technology, there has been an increased interest in the use of beta distribution in the modern practice of analyzing results. Meanwhile, there remains a limitation in the use of the beta distribution that is associated with the choice of distribution shape. The possibilities of using known shape measures for mapping beta distribution in this paper is discusses. On the example of the space of shape measure of kurtosis and skewness, the limited use of only probabilistic measures of shapes is illustrated. It is proposed to use the entropy coefficients as an additional informational parameter of the beta distribution shape. On the base of a features comparison of the entropy coefficients for biased and unbiased beta distributions, recommendations for their application are given. By using the example of beta distributions mapping in the space of asymmetry and the entropy coefficient, it is shown that the synergistic combination of probabilistic and informational measures of the shape allows expanding the possibilities of estimating the shape parameters beta distributions. Two methods to display the positions of realizations of beta distributions is proposed. There are trajectories on a constant ratio of shape and realizations position curve on equal values of one parameter. In particular, the features of the choice of beta distributions with negative skewness are discussed.

scholarly journals New Inequalities between Information Measures of Network Information Content

2013 ◽  
Vol 2013 ◽  
pp. 1-3 ◽  
Author(s):  
Pantelimon-George Popescu ◽  
Florin Pop ◽  
Alexandru Herişanu ◽  
Nicolae Ţăpuş
Keyword(s):  
Information Processing ◽  
Complex Systems ◽  
Complex Networks ◽  
Information Content ◽  
Discrete Case ◽  
Network Information ◽  
Information Inequalities ◽  
Information Measures ◽  
New Inequalities

We refine a classical logarithmic inequality using a discrete case of Bernoulli inequality, and then we refine furthermore two information inequalities between information measures for graphs, based on information functionals, presented by Dehmer and Mowshowitz in (2010) as Theorems 4.7 and 4.8. The inequalities refer to entropy-based measures of network information content and have a great impact for information processing in complex networks (a subarea of research in modeling of complex systems).

scholarly journals Dual Lukacs regressions of negative orders for noncommutative variables

2014 ◽  
Vol 17 (03) ◽  
pp. 1450021 ◽  
Author(s):  
Kamil Szpojankowski
Keyword(s):  
Free Probability ◽  
Random Variable ◽  
Probability Measures ◽  
Classical Probability ◽  
Binomial Distributions ◽  
Beta Distributions

In the paper we study characterizations of probability measures in free probability. By constancy of regressions for random variable 𝕍1/2(𝕀 - 𝕌)𝕍1/2 given by 𝕍1/2𝕌𝕍1/2, where 𝕌 and 𝕍 are free, we characterize free Poisson and free binomial distributions. Our paper is a free probability analogue of results known in classical probability,3 where gamma and beta distributions are characterized by constancy of 𝔼((V(1 - U))i|UV), for i ∈ {-2, -1, 1, 2}. This paper together with previous results18 exhaust all cases of characterizations from Ref. 3.

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