## “Mathematics Days in Sofia”

Section “Numerical Analysis, Operations Research, Probability and Statistics”

### Participants

#### Invited Speakers

**Geno Nikolov**, Sofia University “St. Kliment Ohridski”, Bulgaria**George Yanev**, University of Texas Rio Grande Valley, USA

#### Contributors

**Alessandro Barbiero**, Università degli Studi di Milano, Italy**Detelina Kamburova**, Institute of Mathematics and Informatics, Bulgaria**Hristo Sariev**, Institute of Mathematics and Informatics, Bulgaria**Martin Minchev**, Sofia University “St. Kliment Ohridski”, Bulgaria**Mohammed Beggas**, University of El-Oued, Algeria**Pando Georgiev**, Institute of Mathematics and Informatics and Sofia University “St. Kliment Ohridski”, Bulgaria

### Program and Abstracts

The problem of characterization of probability distributions can be described as follows. It is known that a family of distributions F possesses certain property P. Is it true, conversely, that a distribution has the property P only if it is a member of F? If so, then P characterizes the family F.

In this talk, we will discuss characterizations of the exponential distribution. More specifically, we will present recent results regarding characterization properties of the exponential distribution involving order statistics, record values, and sums of random variables. Let us point out that the contributions of Bulgarian mathematicians in the area of probability characterizations can be traced back to a 1961 paper by Apostol (a.k.a. Shefa) Obretenov.

A count distribution obtained as a discrete version of the continuous half-logistic distribution is introduced. It is derived by assigning to each non-negative integer value a probability proportional to the corresponding value of the density function of the parent model. Statistical properties of this new distribution, in particular related to its shape, moments, and reliability concepts, are described. Parameter estimation, which can be carried out resorting to different methods including maximum likelihood, is discussed and a numerical comparison between the methods, based on Monte Carlo simulations, is presented. The applicability of the proposed distribution is proved on a real dataset, which has been already fitted by other well-established count distributions. In order to increase the flexibility of this counting model, a generalization is finally suggested, which is obtained by adding a shape parameter to the continuous one-parameter half-logistic and then applying the same discretization technique, based on the mimicking of the density function.

This is a joint work with Rumen Marinov and Nadia Zlateva.