Author

Bio: N. S. Upadhye is an academic researcher from Indian Institute of Technology Madras. The author has contributed to research in topic(s): Negative binomial distribution & Poisson distribution. The author has an hindex of 8, co-authored 39 publication(s) receiving 175 citation(s). Previous affiliations of N. S. Upadhye include Indian Institute of Technology Bombay.

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TL;DR: In this paper, a random parameter representation for the convolution of independent and weighted gamma variables with arbitrary parameters has been obtained and applications to the reliability of m-out-of-n:G systems and to the shortest path problem in graph theory are discussed.
Abstract: We study the convolution of compound negative binomial distributions with arbitrary parameters. The exact expression and also a random parameter representation are obtained. These results generalize some recent results in the literature. An application of these results to insurance mathematics is discussed. The sums of certain dependent compound Poisson variables are also studied. Using the connection between negative binomial and gamma distributions, we obtain a simple random parameter representation for the convolution of independent and weighted gamma variables with arbitrary parameters. Applications to the reliability of m-out-of-n:G systems and to the shortest path problem in graph theory are also discussed.

29 citations

Journal ArticleDOI
TL;DR: In this paper, negative binomial approximation to sums of independent Z +$-valued random variables using Stein's method is employed to obtain the error bounds Convolution of negative Binomial and Poisson distribution is used as a three-parametric approximation. Abstract: This paper deals with negative binomial approximation to sums of independent${\bf Z}_+\$-valued random variables Stein's method is employed to obtain the error bounds Convolution of negative binomial and Poisson distribution is used as a three-parametric approximation

19 citations

Journal ArticleDOI
TL;DR: In this article, a new method based on probability generating functions is used to obtain multiple Stein operators for various random variables closely related to Poisson, binomial and negative binomial distributions.
Abstract: In this paper, a new method based on probability generating functions is used to obtain multiple Stein operators for various random variables closely related to Poisson, binomial and negative binomial distributions. Also, the Stein operators for certain compound distributions, where the random summand satisfies Panjer’s recurrence relation, are derived. A well-known perturbation approach for Stein’s method is used to obtain total variation bounds for the distributions mentioned above. The importance of such approximations is illustrated, for example, by the binomial convoluted with Poisson approximation to sums of independent and dependent indicator random variables.

16 citations

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TL;DR: In this article, the Poisson process of order k (PPoK) time-changed with an independent Levy subordinator and its inverse was studied, which they called TCPPoK-I and TCPPoK-II.
Abstract: In this article, we study the Poisson process of order k (PPoK) time-changed with an independent Levy subordinator and its inverse, which we call, respectively, as TCPPoK-I and TCPPoK-II, t...

13 citations

Journal ArticleDOI
TL;DR: In this article, the problem of compound Poisson approximation to the convolution of compound negative binomial distributions, under total variation distance, is considered, and an interesting application to risk theory is discussed.
Abstract: In this paper, the problem of compound Poisson approximation to the convolution of compound negative binomial distributions, under total variation distance, is considered. First, we obtain an error bound using the method of exponents and it is compared with existing ones. It is known that Kerstan’s method is more powerful in compound approximation problems. We employ Kerstan’s method to obtain better estimates, using higher-order approximations. These bounds are of higher-order accuracy and improve upon some of the known results in the literature. Finally, an interesting application to risk theory is discussed.

10 citations

##### Cited by
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2,309 citations

Book
01 Jan 2013
TL;DR: In this paper, the authors consider the distributional properties of Levy processes and propose a potential theory for Levy processes, which is based on the Wiener-Hopf factorization.
Abstract: Preface to the revised edition Remarks on notation 1. Basic examples 2. Characterization and existence 3. Stable processes and their extensions 4. The Levy-Ito decomposition of sample functions 5. Distributional properties of Levy processes 6. Subordination and density transformation 7. Recurrence and transience 8. Potential theory for Levy processes 9. Wiener-Hopf factorizations 10. More distributional properties Supplement Solutions to exercises References and author index Subject index.

1,794 citations

Journal ArticleDOI

713 citations

Journal Article
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

Journal ArticleDOI
TL;DR: This text is a revision of the book by Arnold, Costillo, and Sarabia (1992), but with much more depth than the original, and comprises a lively overview of conditionally speci ed models of the conditional distribution.
Abstract: of the conditional distribution speci cations. Chapters 8 and 10 extend these methods from two to more dimensions. Chapter 9 investigates estimation in conditionally speci ed models. Chapter 11 considers models speci ed by conditioning on events speci ed by one variable exceeding a value rather than equaling a value, and Chapter 12 considers models for extreme-value data. Chapter 13 extends conditional speci cation to Bayesian analysis. Chapter 14 describes the related simultaneous-equation models, and Chapter 15 ties in some additional topics. An appendix describes methods of simulation from conditionally speci ed models. Chapters 1–4, plus Chapters 9 and 13, comprise a lively overview of conditionally speci ed models. The remainder of the text constitutes a detailed catalog of results speci c to different conditional distributions. Although this catalog is certainly of value, the reader desiring a briefer and less detailed introduction to the subject might skip the remainder at  rst reading. This text is a revision of the book by Arnold, Costillo, and Sarabia (1992). The current version is of similar breadth, but with much more depth than the original. The text is clearly written and accessible with relatively few mathematical prerequisites. I found surprisingly few typographical errors; the authors are to be congratulated for this. In a few cases, regularity conditions for results are not given in full. Generally, this causes little confusion, although something does appear to be missing in the statement of Aczél’s key theorem (Theorem 1.3). Fortunately, most of the results in the sequel are derived from corollaries to this theorem, and the corollaries are stated more precisely. I noted few gaps in the material covered. The only area that I thought was insuf ciently represented was application to Markov chain Monte Carlo. Conditional speci cation is particularly important in Gibbs sampling. I believe that many practitioners would bene t from a discussion of the issues involved in these sampling schemes. Each chapter contains numerous exercises. These exercises appear to be at an appropriate level for a graduate course in statistics, and appear to provide appropriate reinforcement for the material in the preceding chapters.

260 citations