# 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.

648 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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01 Jan 1992

TL;DR: In this paper, the authors propose a family of Discrete Distributions, which includes Hypergeometric, Mixture, and Stopped-Sum Distributions (see Section 2.1).

Abstract: Preface. 1. Preliminary Information. 2. Families of Discrete Distributions. 3. Binomial Distributions. 4. Poisson Distributions. 5. Neggative Binomial Distributions. 6. Hypergeometric Distributions. 7. Logarithmic and Lagrangian Distributions. 8. Mixture Distributions. 9. Stopped-Sum Distributions. 10. Matching, Occupancy, Runs, and q-Series Distributions. 11. Parametric Regression Models and Miscellanea. Bibliography. Abbreviations. Index.

2,105 citations

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TL;DR: Rao's Linear Statistical Inference and Its Applications as discussed by the authors is one of the earliest works in statistical inference in the literature and has been translated into six major languages of the world.

Abstract: "C. R. Rao would be found in almost any statistician's list of five outstanding workers in the world of Mathematical Statistics today. His book represents a comprehensive account of the main body of results that comprise modern statistical theory." -W. G. Cochran "[C. R. Rao is] one of the pioneers who laid the foundations of statistics which grew from ad hoc origins into a firmly grounded mathematical science." -B. Efrom Translated into six major languages of the world, C. R. Rao's Linear Statistical Inference and Its Applications is one of the foremost works in statistical inference in the literature. Incorporating the important developments in the subject that have taken place in the last three decades, this paperback reprint of his classic work on statistical inference remains highly applicable to statistical analysis. Presenting the theory and techniques of statistical inference in a logically integrated and practical form, it covers: * The algebra of vectors and matrices * Probability theory, tools, and techniques * Continuous probability models * The theory of least squares and the analysis of variance * Criteria and methods of estimation * Large sample theory and methods * The theory of statistical inference * Multivariate normal distribution Written for the student and professional with a basic knowledge of statistics, this practical paperback edition gives this industry standard new life as a key resource for practicing statisticians and statisticians-in-training.

1,579 citations

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

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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.

648 citations

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01 Mar 2003

TL;DR: A survey of applications theory and general estimation procedures for stress strength models can be found in this paper, along with examples and details on applications and their application in the context of point estimation and statistical inference.

Abstract: Stress-strength models - history, mathematical tools and survey of applications theory and general estimation procedures parametric point estimation parametric statistical inference nonparametric methods special cases and generalizations examples and details on applications

330 citations

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

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