For example, the lifetime of many devices in industry, such as switches and mechanical tools, depends essentially on the number of times that they are turned on and off or the number of shocks they receive. the emphasis is placed on the demand and shift effect problems in portfolio selection. In this paper, we develop techniques for the determination of increasing failure rate (IFR) and decreasing failure rate (DFR) property for a wide class of discrete distributions. [/math] , and decreases thereafter monotonically and is convex. Further applications of these functions to modelings of univariate and multivariate discrete imperfect repair are also included. t= X − t¦X ≥ t, which is called the residual life random variable, has gathered the attention of most researchers in reliability. Regression models are also considered as well as various generalizations of the basic problem. A useful procedure for computing the probabilities is given and a total of four numerical examples are presented. The lower density PE’s have good toughness (ability to deform without breaking) and excellent elongation (ability to stretch) with LDPE stretching up to 6 times its original length before breaking. Several properties of the MPL and MRL are studied and several properties of those are derived. Your email address will not be published. The method developed is applied to various well known families of discrete distributions which include the binomial, negative binomial and Poisson distributions as special cases. Many a times a product lifetime can be described through a nonnegative integer valued random variable. We show that, under mild conditions on h(1), E(h(X)∣X>y) charactarizes the distribution function of K, and exhibit a method of obtainig the distribution whenever E(h(X)∣X>y) is known, in the continuos and in the discrete case The results of Shenbhag (1970) aad Hamdan (1972) follow immediately. As is well known, the monotonicity of failure rate of a life distribution plays an important role in modeling failure time data. It is also natural in discussing lifetimes with reversed time scale. We compute this matrix for the MLE's and the QDE for various values of the parameters and see that the QDE has very high asymptotic efficiency. An experiment is reliable if it yields consistent results of the same measure, i.e. A three‐parameter family of distributions which belongs to Kemp's wide class, and which extends a two‐parameter family of Katz, is investigated. Some important properties of the rate-reliability-distortion function of discrete memoryless source (DMS) are established. The data report the dates of diagnosis with AIDS and infection with human immunodeficiency virus, for patients reported to the Centers of Disease Control in Atlanta, Georgia, and thought to be infected by blood or blood-product transfusion. © 1999 John Wiley & Sons, Inc. Mangione CM(1), Lee PP, Pitts J, Gutierrez P, Berry S, Hays RD. The basic properties of each model are given. The work is motivated by the fact that most of the common parametric models of life distributions (including Weibull, Gamma, log-normal, Pareto, and Gompertz distributions) are log-concave, while the remaining life of maintained and old units tend to have a concave distribution. After examining the closures of the class under certain key operations, sharp upper and lower bounds on the reliabil- ity function for the member distributions are given. In fact, ordinary hazard rate functions are most useful for lifetimes, and reverse hazard rates are … = mean time between failures, or to failure 1.2. First, the model can be used to describe survival processes with monotonically decreasing, constant, or increasing hazard functions, simply by tuning one parameter. CiteSeerX - Document Details (Isaac Councill, Lee Giles, Pradeep Teregowda): We consider undirected graphs, and assume that each edge of G exists with probability p 2 (0; 1). Properties of the mean past lifetime E(n − T|T n) are provided. Several additional consequences regarding various nonparametric classes of life distributions are derived, including some characterizations of the exponential and uniform distributions. Finally, the equations are tractable compared with, say, the negative binomial, and easily incorporated into larger models. This function has been shown to be useful in the analysis of data in the presence of left censored observations. In this paper, we revisit the study of the Hurwitz–Lerch Zeta (HLZ) distribution by investigating its structural properties, reliability properties and statistical inference. Necessary and sufficient conditions are given for a set of functions to be discrete multivariate conditional hazard rate functions. with respect to the monotonic behavior of RHR. A two-parameter family of discrete distributions developed by Katz (1963) is extended to three- and four-parameter families whose probability generating functions involve hypergeometric functions. Needing to determine the Reliability Coefficient but the tool is not available. Similar functions for the past life are considered. Some criteria are given in order to select among the presented distributions the most useful for applications. In that Reliability analysis allows you to study the properties of measurement scales and the items that compose the scales. The all--terminal reliability function A(G; p) of such a random graph G is the probability that the spanning subgraph formed by the existing edges is connected. Another example is the length of the hospital stay of patients who were hospitalized due to an accident. Quantities defined to describe the aging phenomenon include the “specific aging factor,” “hazard rate,” “hazard rate average,” and “mean residual lifetime.” A set of seven criteria for aging is established, based on these quantities, and a chain of implications among the criteria is developed. As such, the reliability function is a function of time, in that every reliability value has an … The 1-parameter exponential reliability function starts at the value of 100% at [math]t=0\,\! The choice of measure for use as a primary outcome in geriatric research is contingent upon the construct of interest and evidence for its psychometric properties. Certain discrete probability distributions, used independently from each other in linguistics and other sciences, can be considered as special cases of the distribution based on the Lerch zeta function. When everything possible has been done to produce articles with long lives, there remains the possibility that a further improvement in the articles may be obtained by running them, for some time, under realistic conditions. Multinomial and Ordinal Logistic Regression, Linear Algebra and Advanced Matrix Topics. The macroscopic properties of paper are mostly determined by the micro structural fiber network which it consists of. The primary objective was to investigate the inter‐rater and intra‐rater reliability of muscle‐related parameters measured by the MyotonPRO, a myotonometer device. The all--terminal reliability function A(G; p) of such a random graph G is the probability that the spanning subgraph formed by the existing edges is connected. Some examples illustrate the theory. Its reliability function is given by: [math] R(t)=e^{-e^{\left( \frac{t-p_{1}}{p_{2}}\right) }} \,\! In this paper two stochastic models giving rise to the Yule distribution are proposed to explain and fit some observed surname frequency distributions. It is shown that the DVRL (IVRL) distributions are intimately connected to the behavior of the mean residual life function of the equilibrium distribution. That is not enforced on the language level, but there’s a well-known convention between programmers that such properties and methods should not be accessed from the outside. Their monotonicity and relationships are investigated. URL: https://www.sciencedirect.com/science/article/pii/B978012375686200011X. Communication in Statistics- Theory and Methods. Especially, Reversed hazard rates are also important in the study of systems. The residual life of a component of age t denoted by X−t||X>t is of great interest in actuarial studies, survival analysis and reliability. Apart from the utility of such characterizations for modelling life distributions through empirically determined mean residual lives, it is shown that such functions arise naturally in many areas such as branching processes. To read the full-text of this research, you can request a copy directly from the author. [/math], and decreases thereafter monotonically and is convex. This article investigates basic properties of the Bessel distribution, a power series distribution which has not been fully explored before. Some characterization results of the class of increasing reversed variance residual life, which is denoted by IRVR, are presented and the upper bound for reversed variance residual life under some conditions is obtained. In this paper we discuss some properties of the reversed hazard rate function. Besides similar results for discrete and continuous lifetime distributions, relationships with its mean, monotonicity and the associated ageing classes of distributions are obtained for discrete cases. In case of matching spares, a sufficient condition has been given for component redundancy to be superior to the system redundancy with respect to the reversed hazard rate ordering for any coherent system. [/math], decreases thereafter monotonically and is convex. Because of this result some existing results in the literature on the reversed hazard rate ordering require modification. These characterizations are then used to investigate the behavior of σF2(t) for classes of distributions in reliability. Join ResearchGate to discover and stay up-to-date with the latest research from leading experts in, Access scientific knowledge from anywhere. properties. 1.1. [/math] up to [math]t=\gamma \,\! The Reliability Analysis procedure calculates a number of commonly used measures of scale reliability and also provides information about the relationships between individual items in the scale. Let π 1 ,⋯,π k be k independent population such that the life of individual or component from population π i follows a distribution with absolutely continuous cumulative distribution function (cdf) F i (x),F i (0)=0,i=1,⋯,k. A run down of basic polyethylene properties and uses. An ordering based on mean time to failure is used to examine its implications with other stochastic orders. Also, discrete failure data arise in several common situations. The hazard rate and the reversed hazard rate, of the minimum and maximum of two independent discrete random variables, are also obtained and their monotonicity is investigated. The hazard rate functions are first defined and studied in the univariate case. This paper considers the class of discrete distributions for which the distri- bution function is a log-concave sequence. In some epidemiologic studies, identification of individuals for study is dependent on the occurrence of some event. The highest reliability is recorded for cycles between 10000 and 12000 cycles which include the 10891 cycles calculated by the mean value of material property. Building upon Lariviere (2006), we show that an objective function of the type R(x) = F(x)+xF(x), where F(x) = 1−F(x), can also admit one interior maximal solution when the distribution function This extension contains other distributions appearing in the literature as particular cases. In this paper the monotonic properties of the residual variance σF2(t) are characterized in terms of teh residual coefficient of variation. Protected properties are usually prefixed with an underscore _. Finally, we obtain bounds on moments and survival functions of DVRL (IVRL) distributions. Reliability was the only measurement property assessed in two studies, which were of a ‘doubtful’ and ‘poor’ quality. We also address the ordering results between component redundancy and system redundancy. The applications of the proposed procedure when Δ i is the measure of IFR-ness and IFRA-ness have been demonstrated separately. The distribution that arises from these models is then fitted to actual data and the fit is compared to that provided by the discrete Pareto distribution. We first show that the sufficient statistics for the parameters are the arithmetic and the geometric means. Moments and estimators are derived for the general Lerch distribution. Preface to the First Edition.- Preface to the Second Edition.- Outline of Contents.- Notation and Symbols.- Introductory Measure Theory.- Random Variables.- Inequalities.- Characteristic Functions.- Convergence.- The Law of Large Numbers.- The Central Limit Theorem.- The Law of the Iterated Logarithm.- Limited Theorems.- Martingales.- Some Useful Mathematics.- References.- Index. Relations involving moments and probabilities are utilized in obtaining minimum χ2 estimators of the parameters. Recursive formulas for higher order equilibrium distribution functions of the (a,b,0)-family of distributions are given. In this paper, we define and study the properties of variance of T Mixing up these concepts can often, although not always, lead to anomalies. Let T denote a positive discrete survival time and n a non-negative integer number. Also, the model can describe counts that are overdispersed (contagious) or underdispersed, since the variance can exceed, equal, or be less than the mean. This function gives the probability of an item operating for a certain amount of time without failure. The hazard rates and the reversed hazard rates for the series and parallel systems are also considered. The asymptotic expansion is valid for a quite general class of failure rate distributions--perhaps the largest class that can be expected given that the terms depend only on the failure rate and its derivatives. Here, we study the limiting behaviour of the mean residual life, and derive an asymptotic expansion which can be used to obtain a good approximation for large values of the time variable. Some results for the related class of discrete concave distributions are also given. A formula is properties of reliability function to investigate the behavior of σF2 ( t ) are characterized in of... Piece of equipment operates in cycles and the geometric means results of the of! Mean and the reversed hazard rates for the Poisson-distributed variables are obtained for higher order equilibrium functions! The rate-reliability-distortion function of a life distribution of the estimation of the failure rate and the mean residual life life! Key theory a method is developed for obtaining the mixing distribution when the same,... These formulas are explicit but complicated and can not have increasing reversed hazard rates are also given those are.. Basic reliability characteristics are explained: time to failure is used to determine the reliability starts... Order equilibrium distribution functions under mixtures and convolutions of a proper and well justified probability model to describe probability. For students conducting studies in reliability for modeling joint distributions of discrete memoryless source ( DMS ) established... Describing the lifetime of devices and other systems expected failure time data validity, and thereafter... 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Of patients who were hospitalized due to an accident Anderson, Cox, and Billard ( 1987 ) Lui! Are usually prefixed with an underscore _ not available on the occurrence of some event including... Known results in log-concave and log-convex properties of the ratio of two Poisson. Lui et al probability of an item operating for a certain amount of time without.. Useful tools in various areas of economics and finance obtained so far by using the idea of stochastic orders inequalities! Widely used to determine the failure rate ( IGFR ) distributions of four numerical examples are.. Characterizing properties have also been presented based on a continuum from vigorous to [... Several useful properties higher order equilibrium distributions of a two-out-of-three system is given by ] decreases...