Mapping of Beta Distribution for the Study of Dispersed Materials

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.

Shape measures of generalized beta distributions

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.

Doubly Non-Central Generalized Beta Distributions

The doubly non-central generalized beta type 1 and type 2 distributions have been derived by using two independent non-central gamma variables with different scale parameters. These distributions generalize several well known central and non-central beta distributions. The doubly non-central generalized beta densities are much more flexible than many exiting beta models and can assume a large variety of shapes. Several properties of these distributions have been studied.

MANAGEMENT OF TESTS OF COMPLEX TECHNICAL SYSTEMS BASED ON A CONSISTENT RELIABILITY ASSESSMENT AND TAKING INTO ACCOUNT A PRIORI DATA ON THEIR ELEMENTS

The article deals with the problem of reducing the volume of tests of complex systems by using a priori data on the reliability of their elements. At the preliminary stage, the a priori distribution of the probability of failure of the system as a whole is determined. To do this, the results of element tests are processed and the parameters of the a posteriori probability distribution of element failure are determined based on the Bayesian procedure. The type of distribution law (beta distribution) is chosen from the conjugacy condition. Statistical modeling of the system failure probability of a known structural-logical reliability scheme is performed for random values of the failure probabilities of each element, set in accordance with the obtained distribution law. The system failure probability distribution law is formed as a mixture of beta distributions; the advantage of this distribution law is a fairly high accuracy of the simulation data description and conjugacy to the binomial distribution. The parameters of a mixture of beta distributions are determined using the EM (Expectation-Maximization) algorithm. The quality of selection of the desired distribution density is checked using the nonparametric Kolmogorov criterion. When testing the system, after each experiment, the a posteriori density of the probability distribution is recalculated; it is represented as a mixture of beta distributions with a constant proportion of components. The parameters of each element of the mixture are easily determined by the results of the experiment. As a point Bayesian estimate, the average value calculated from the a posteriori distribution is taken, the confidence interval for a given confidence probability is found as the central interval. An example is given and the possibility of minimizing the number of tests is shown.