# On estimating reliability function for the family of power series distribution

18 Jun 2021-Communications in Statistics-theory and Methods (Taylor & Francis)-Vol. 50, Iss: 12, pp 2801-2830

TL;DR: In this article, an explicit expression of the stress - strength reliability function, R=P(X≤Y) is derived, when the stress (X) and strength (Y) distributions are different members of the Power series family of distri...

Abstract: Explicit expression of the Stress - Strength reliability function, R=P(X≤Y) is derived, when the stress (X) and strength (Y) distributions are different members of the Power series family of distri...

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TL;DR: Alho and Spencer as discussed by the authors published a book on statistical and mathematical demography, focusing on mature population models, the particular focus of the new author (see, e.g., Caswell 2000).

Abstract: Here are two books on a topic new to Technometrics: statistical and mathematical demography. The first author of Applied Mathematical Demography wrote the first two editions of this book alone. The second edition was published in 1985. Professor Keyfritz noted in the Preface (p. vii) that at age 90 he had no interest in doing another edition; however, the publisher encouraged him to find a coauthor. The result is an additional focus for the book in the world of biology that makes it much more relevant for the sciences. The book is now part of the publisher’s series on Statistics for Biology and Health. Much of it, of course, focuses on the many aspects of human populations. The new material focuses on mature population models, the particular focus of the new author (see, e.g., Caswell 2000). As one might expect from a book that was originally written in the 1970s, it does not include a lot of information on statistical computing. The new book by Alho and Spencer is focused on putting a better emphasis on statistics in the discipline of demography (Preface, p. vii). It is part of the publisher’s Series in Statistics. The authors are both statisticians, so the focus is on statistics as used for demographic problems. The authors are targeting human applications, so their perspective on science does not extend any further than epidemiology. The book actually strikes a good balance between statistical tools and demographic applications. The authors use the first two chapters to teach statisticians about the concepts of demography. The next four chapters are very similar to the statistics content found in introductory books on survival analysis, such as the recent book by Kleinbaum and Klein (2005), reported by Ziegel (2006). The next three chapters are focused on various aspects of forecasting demographic rates. The book concludes with chapters focusing on three areas of applications: errors in census numbers, financial applications, and small-area estimates.

710 citations

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TL;DR: In this article , the authors compared NHPP and α-series to obtain a better process for using monotone trend data and prediction, meanwhile, the other studies in this field focused on comparing methods of estimation parameters of NHPP.

Abstract: This study aims to compare the stochastic process model designed as a nonhomogeneous Poisson process and α-series process, to obtain a better process for using monotonous trend data. The α-series process is a stochastic process with a monotone trend, while the NHPP is a general process of the ordinary Poisson process and it is used as a model for a series of events that occur randomly over a variable period of time. Data on the daily fault time of machines in Bahrri Thermal Station in Sudan was analyzed during the interval from first January 2021, to July 31, 2021, to acquire the best stochastic process model used to analyze monotone trend data. The results revealed that the NHPP model could be the most suitable process model for the description of the daily fault time of machines in Bahrri Thermal Station according to lowest MSE, RMSE, Bias, MPE, and highest. The current study concluded through the NHPP the fault time of machines and repair rate occurs in an inconsistent way. The further value of this study is that it compared NHPP and α-series to obtain a better process for using monotone trend data and prediction, meanwhile, the other studies in this field focused on comparing methods of estimation parameters of NHPP and α-series process. The distinctive scientific addition of this study stems from displaying the precision of the NHPP better than the α-series process in the case of monotone trend data.

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TL;DR: In this article, the relation between the variances of unbiased estimators is established, and a theorem letting one construct unbiased estimation for functions of an unknown vector parameter λ is proved.

Abstract: k-dimensional Poisson walks are considered. Giving stopping boundaries of the process of the walk defines a family of probability measures. A theorem letting one construct unbiased estimators for functions of an unknown vector parameter λ is proved. The relation between the variances of unbiased estimators is established.

15 citations

### "On estimating reliability function ..." refers background in this paper

...Belyaev and Lumelskii (1988) and Barbiero (2013) used Poisson distribution to explore different...

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TL;DR: In this article, the maximum likelihood and uniformly minimum variance unbiased estimator (UMVUE) of P [X ⩽ Y ] are derived, where both X and Y have negative binomial distribution whose exact variance is calculated in both the methods.

Abstract: The maximum likelihood and uniformly minimum variance unbiased estimator (UMVUE) of P [ X ⩽ Y ] are derived, where both X and Y have negative binomial distribution whose exact variance is calculated in both the methods. It has been shown that UMVUE is better than the maximum likelihood estimator (MLE) in the case of geometric distribution.

13 citations

### "On estimating reliability function ..." refers background in this paper

...The Negative Binomial distribution was considered by Ivshin, Lumelskii, and Ya (1995) and Sathe and Dixit (2001) in the context of system reliability estimator....

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05 Jun 2013TL;DR: In this article, the authors compared the performance of the maximum likelihood estimator and the uniformly minimum variance unbiased estimator in terms of their mean square error for the reliability of stress-strength models when stress and strength are independent Poisson random variables.

Abstract: Researchers in reliability engineering regularly encounter variables that are discrete in nature, such as the number of events (e.g., failures) occurring in a certain spatial or temporal interval. The methods for analyzing and interpreting such data are often based on asymptotic theory, so that when the sample size is not large, their accuracy is suspect. This paper discusses statistical inference for the reliability of stress-strength models when stress and strength are independent Poisson random variables. The maximum likelihood estimator and the uniformly minimum variance unbiased estimator are here presented and empirically compared in terms of their mean square error; recalling the delta method, confidence intervals based on these point estimators are proposed, and their reliance is investigated through a simulation study, which assesses their performance in terms of coverage rate and average length under several scenarios and for various sample sizes. The study indicates that the two estimators possess similar properties, and the accuracy of these estimators is still satisfactory even when the sample size is small. An application to an engineering experiment is also provided to elucidate the use of the proposed methods.

10 citations

### "On estimating reliability function ..." refers methods in this paper

...Belyaev and Lumelskii (1988) and Barbiero (2013) used Poisson distribution to explore different CONTACT Rahul Bhattacharya rahul_bhattya@yahoo.com Department of Statistics, University of Calcutta, 35, B. C. Road, Kolkata 700019, India....

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TL;DR: In this article, the maximum likelihood estimator of R and its asymptotic distribution are derived and a procedure for deriving bootstrap confidence intervals is presented. And also the Bayes estimator for conjugate prior distributions is obtained.

Abstract: In this paper we estimate R = P{X ≤ Y } when X and Y are independent random variables from geometric and Poisson distribution respectively. We find maximum likelihood estimator of R and its asymptotic distribution. This asymptotic distribution is used to construct asymptotic confidence intervals. A procedure for deriving bootstrap confidence intervals is presented. UMVUE of R and UMVUE of its variance are derived and also the Bayes estimator of R for conjugate prior distributions is obtained. Finally, we perform a simulation study in order to compare these estimators. keywords: stress-strength, geometric distribution, Poisson distribution, maximum likelihood estimator, Bayes estimator, UMVUE, bootstrap confidence intervals. MSC(2010): 62F10, 62F12, 62F15, 62F25, 62F40.

6 citations

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TL;DR: In this paper, the Bayesian estimation of R = P(Y > X) when Y and X are two independent but not identically distributed geometric random variables is obtained by using Lindley's approximation form.

Abstract: In this paper, Bayesian estimation of R = P(Y > X) when Y and X are two independent but not identically distributed geometric random variables is obtained by using Lindley's approximation form. Based on a Monte Carlo simulation, the Bayes estimate of R is compared with their corresponding maximum likelihood estimate.

4 citations