Edward Omey

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.

Note on the article: Maximum entropy analysis of the M[x]/M/1 queueing system with multiple vacations and server breakdowns

Wang et al. [Wang, K. H., Chan, M. C., & Ke, J. C. (2007). Maximum entropy analysis of the M^[^x^]/M/1 queueing system with multiple vacations and server breakdowns. Computers &Industrial Engineering, 52, 192-202] elaborate on an interesting approach to estimate the equilibrium distribution for the number of customers in the M^[^x^]/M/1 queueing model with multiple vacations and server breakdowns. Their approach consists of maximizing an entropy function subject to constraints, where the constraints are formed by some known exact results. By a comparison between the exact expression for the expected delay time and an approximate expected delay time based on the maximum entropy estimate, they argue that their maximum entropy estimate is sufficiently accurate for practical purposes. In this note, we show that their maximum entropy estimate is easily rejected by simulation. We propose a minor modification of their maximum entropy method that significantly improves the quality of the estimate.

A Branching Process with Immigration in Varying Environments

Bienaymé–Galton–Watson branching processes with varying offspring variance and an immigration component are studied in the critical case. The asymptotic formulas for the probability for non extinction are derived, in dependence of immigration component. A limit theorem is proved too.

Domains of attraction of the real random vector (X,X²) and applications

Many statistics are based on functions of sample moments. Important examples are the sample variance , the sample coefficient of variation SV(n), the sample dispersion SD(n) and the non-central t-statistic t(n). The definition of these quantities makes clear that the vector defined by plays an important role. In studying the asymptotic behaviour of this vector we start by formulating best possible conditions under which the vector (X, X 2) belongs to a bivariate domain of attraction of a stable law. This approach is new, uniform and simple. Our main results include a full discussion of the asymptotic behaviour of SV(n), SD(n) and t 2(n). For simplicity, in restrict ourselves to positive random variables X.

Asymptotics of convolution with the semi-regular-variation tail and its application to risk

In this paper, according to a certain criterion, we divide the exponential distribution class into some subclasses. One of them is closely related to the regular-variation-tailed distribution class, and is called the semi-regular-variation-tailed distribution class. The new class possesses several nice properties, although distributions in it are not convolution equivalent. We give the precise tail asymptotic expression of convolutions of these distributions, and prove that the class is closed under convolution. In addition, we do not need to require the corresponding random variables to be identically distributed. Finally, we apply these results to a discrete time risk model with stochastic returns, and obtain the precise asymptotic estimation of the finite time ruin probability.

New results on the order of functions at infinity

Recently, new classes of positive and measurable functions, M(ρ) and M(±∞), have been defined in terms of their asymptotic behaviour at infinity, when normalized by a logarithm (Cadena et al., 2015, 2016, 2017). Looking for other suitable normalizing functions than logarithm seems quite natural. It is what is developed in this paper, studying new classes of functions of the type lim x→∞ log U (x)/H(x) = ρ < ∞ for a large class of normalizing functions H. It provides subclasses of M(0) and M(±∞).

A Main Class of Integral Inequalities with Applications

AbstractIn this paper, we define some integral transforms and obtain suitable bounds for them in order to introduce a main class of integral inequalities including Ostrowski and Ostrowski-Gruss inequalities and various kinds of new integral inequalities. In this sense, we also introduce a three point quadrature formula and obtain its error bounds.

Local limit theorems for shock models

In this paper we study the local behaviour of a characteristic of two types of shock models. In many physical systems, a failure occurs when the stress or the fatigue, represented by

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Extensions and Applications

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