Leda D. Minkova

The Pólya-Aeppli process and ruin problems

The Polya-Aeppli process as a generalization of the homogeneous Poisson process is defined. We consider the risk model in which the counting process is the Polya-Aeppli process. It is called a Polya-Aeppli risk model. The problem of finding the ruin probability and the Cramer-Lundberg approximation is studied. The Cramer condition and the Lundberg exponent are defined. Finally, the comparison between the Pelya-Aeppli risk model and the corresponding classical risk model is given.

A GENERALIZATION OF THE CLASSICAL DISCRETE DISTRIBUTIONS

ABSTRACT Some results related to the family of Inflated-parameter generalized power series distributions (IGPSD) are presented. The probability mass functions, useful properties, and recursion formulas are given. A characterization of the Inflated-parameter geometric distribution in terms of the ρ-type lack of memory property is given. The distributions are discussed as counting distributions in risk theory.

On a Bivariate Pólya-Aeppli Distribution

In this article, we use the bivariate Poisson distribution obtained by the trivariate reduction method and compound it with a geometric distribution to derive a bivariate Pólya-Aeppli distribution. We then discuss a number of properties of this distribution including the probability generating function, correlation structure, probability mass function, recursive relations, and conditional distributions. The generating function of the tail probabilities is also obtained. Moment estimation of the parameters is then discussed and illustrated with a numerical example.

Discrete distributions related to success runs of lengthk in aulti-state markov chain

We consider the time-homogeneous multi-state Markov chain {X , n ≥0} with states labeled as “0” (success) and “f” (failure), f = 1,2,… Let k be a fixed positive integer and SK be the event of a success run of length k in the sequence X0: X ,… In this article, joint probability generating functions for various statistics, related to the event SK0 are derived. In particular cases the exact distribution of the total number of successes S, the total number of failures F and the total number of trials N are deduced. The distribution of N is called the geometric distribution of order k for time-homogeneous {0,1,2,…}-valued Markov chain.

A new Markov Binomial distribution

In this article, we introduce a two-state homogeneous Markov chain and define a geometric distribution related to this Markov chain. We define also the negative binomial distribution similar to the classical case and call it NB related to interrupted Markov chain. The new binomial distribution is related to the interrupted Markov chain. Some characterization properties of the geometric distributions are given. Recursion formulas and probability mass functions for the NB distribution and the new binomial distribution are derived.

Characterization of the Pólya-Aeppli Process

In this article, we study the Pólya-Aeppli process (PAP). We define PAP from three different points of view: as a compound Poisson process, as a delayed renewal process, and as a pure birth process. We show that these definitions are equivalent. Also, using these definitions we identify several interesting characterizations of PAP.

I-Pólya Process and Applications

The Inflated-parameter negative binomial process (I-Pólya process) as a mixed Pólya–Aeppli process is defined. Some basic properties are given. We consider the risk model in which the counting process is the I-Pólya process. It is called I-Pólya risk model. The joint probability distribution of the time to ruin and the deficit after ruin occurs is studied. The particular case of exponentially distributed claims is given.

Pólya–Aeppli Distribution of Order k

In this article, Pólya–Aeppli distribution of order k as a compound Poisson distribution is defined. The probability mass function and recursion formulas are given. We consider the interpretation of the distribution and a definition of Pólya–Aeppli process of order k.

Pólya–Aeppli of Order k Risk Model

In this study, we define the Pólya–Aeppli process of order k as a compound Poisson process with truncated geometric compounding distribution with success probability 1 − ρ > 0 and investigate some of its basic properties. Using simulation, we provide a comparison between the sample paths of the Pólya–Aeppli process of order k and the Poisson process. Also, we consider a risk model in which the claim counting process {N(t)} is a Pólya-Aeppli process of order k, and call it a Pólya—Aeppli of order k risk model. For the Pólya–Aeppli of order k risk model, we derive the ruin probability and the distribution of the deficit at the time of ruin. We discuss in detail the particular case of exponentially distributed claims and provide simulation results for more general cases.

Type II Family of Bivariate Inflated-Parameter Generalized Power Series Distributions

The family of Inated-parameter Generalized Power Series distributions (IGPSD) was introduced by Minkova in 2002 as a compound Generalized Power Series distributions (GPSD) with geometric compounding distribution. In these notes we introduce a family of compound GPSDs with bivariate geometric compounding distribution. The probability mass function, recursion formulas, conditional distributions and some properties are given. A member of this family is a Type II bivariate Polya-Aeppli distribution, introduced by Minkova and Balakrishanan (2014). In this notes the particular cases of bivariate compound binomial, negative binomial and logarithmic series distributions are analyzed in detail.