Lifetime distributions motivated by series and parallel structures

According to Ross, any system can be represented either as a series arrangement of parallel structures or as a parallel arrangement of series structures. Motivated by this, we propose new three-parameter lifetime distributions by compounding geometric, power series, and exponential distributions. The distributions can allow for decreasing, increasing, bathtub-shaped, and upside down bathtub-shaped hazard rates. A mathematical treatment of the new distributions is provided including expressions for their density functions, Shannon and Rényi entropies, mean residual life functions, hazard rate functions, quantiles, and moments. The method of maximum likelihood is used for estimating parameters. Five of the new distributions are studied in detail. Finally, two illustrative data examples and a sensitivity analysis are presented.     

The Geometric ArcTan distribution with applications to model demand for health services

In this paper , a new discrete two–parameter distribution α ∈ ? ? {0} and 0 < θ < 1, the Geometric ArcTan (GAT) distribution is introduced. The geometric distribution is a limiting case of this model when α tends to zero. Similarly to the the latter distribution, this probabilistic family is unimodal but the mode can be located at zero or in other point of the support. Then, after deriving some of its more relevant properties , the issue of parameter investigation is investigated. Next, the GAT distribution is used to explain the demand for health services by means of a regression model. Numerical results show that this new model outperforms the negative binomial distribution.     

An optimal age-replacement policy for a simple repairable system with delayed repair

In this article, a simple repairable system (i.e., a repairable system consisting of one component and one repairman) with delayed repair is studied. Assume that the system after repair is not “as good as new”, and the degeneration of the system is stochastic. Under these assumptions, using the geometric process repair model, we consider a replacement policy T based on system age under which the system is replaced when the system age reaches T. Our problem is to determine an optimal replacement policy T*, such that the average cost rate (i.e., the long-run average cost per unit time) of the system is minimized. The explicit expression of the average cost rate is derived, the corresponding optimal replacement policy T* can be determined by minimizing the average cost rate of the system. Finally, a numerical example is given to illustrate some theoretical results and the model's applicability.     

A geometric process repair model for a cold standby repairable system with imperfect delay repair and priority in use

In this article, a two-dissimilar-component cold standby repairable system with one repairman is studied. Assume that the repair after failure for each component is delayed or undelayed. Component 2 after repair is “as good as new” while Component 1 after repair is not, but Component 1 has priority in use. Under these assumptions, using a geometric process, we consider a replacement policy N based on the failure number of Component 1. An optimal replacement policy N* is determined by minimizing the average cost rate C(N) of the system. Finally, a numerical example is given to illustrate some theoretical results and the model applicability.     

Bounds for the expected value of the stochastic Divisia's price index

In this article, we estimate bounds for the expected value of the stochastic Divisia's price index, that is, we assume that prices and quantities of the given commodities are stochastic processes with continuous time. We consider some special case of the stochastic model in which prices and quantities are described by the geometric Brownian motion. It is shown that the precision of this estimation depends rather on the volatility of prices than quantities volatilities.     

A non uniform bound on geometric approximation with w-functions

The aim of this paper is a use of Stein’s method and w-functions to determine a non uniform bound on the geometric approximation for a non negative integer-valued random variable. Some applications of the obtained results are provided to approximate the negative hypergeometric, Pólya and negative Pólya distributions.     

Some improved tail bounds for the sum of variables with geometric distribution

Abstract

In this paper, compared with the results in Janson (2018 Janson, S. 2018. Tail bounds for sums of geometric and exponential variables. Statistics & Probability Letters 135 (C):1–6. doi:10.1016/j.spl.2017.11.017.[Crossref] , [Google Scholar]), we provide some improved explicit bounds for the tail probabilities of the sum of independent geometric variables with their expectations and variances. Particularly, in some cases, we demonstrate that our bounds are sharper than the ones in Janson (2018 Janson, S. 2018. Tail bounds for sums of geometric and exponential variables. Statistics & Probability Letters 135 (C):1–6. doi:10.1016/j.spl.2017.11.017.[Crossref] , [Google Scholar]).     

Projection Properties of Certain Three-Level Main Effect Plans with Quantitative Factors

ABSTRACT

Orthogonal arrays are used as screening designs to identify active main effects, after which the properties of the subdesign for estimating these effects and possibly their interactions become important. Such a subdesign is known as a “projection design”. In this article, we have identified all the geometric non isomorphic projection designs of an OA(27,13,3,2), an OA(18,7,3,2) and an OA(36,13,3,2) into k = 3,4, and 5 factors when they are used for screening out active quantitative experimental factors, with regard to the prior selection of the middle level of factors. We use the popular D-efficiency criterion to evaluate the ability of each design found in estimating the parameters of a second order model.     

A Mixed Bivariate Distribution Connected with Geometric Maxima of Exponential Variables

We study the joint distribution of X and N, where N has a geometric distribution and X is the maximum of N i.i.d. exponential variables, independent of N. We present basic properties of these mixed bivariate distributions and discuss parameter estimation for this model. An example from finance, where N represents the number of consecutive positive daily log-returns of currency exchange rates, illustrates stochastic modeling potential of these laws.     

Asymptotic Distributions of the Generalized Range,Midrange, Extremal Quotient,and Extremal Product,with a Comparison Study

Necessary and sufficient conditions for the weak convergence of the generalized range, midrange, extremal quotient, and extremal product are obtained. The classes of possible non degenerate limit distribution functions of these simple statistics are characterized. Comparison study between these statistics with some examples for the most important distribution functions are given.