Two-Stage Estimation of Mean in a Negative Binomial Distribution with Applications to Mexican Bean Beetle Data

Mukhopadhyay, Nitis; Silva, B. M. de

doi:10.1081/sqa-200046838

Cited by 15 publications

(4 citation statements)

References 25 publications

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“…Incidentally, both Willson and Folks (1983) and Mukhopadhyay and Diaz (1985) proved asymptotic first-order properties. Recently, Mukhopadhyay and de Silva (2005) proved the asymptotic second-order efficiency property for the Mukhopadhyay-Diaz two-stage negative binomial estimation problem.…”

Section: Discussionmentioning

confidence: 99%

Distributions of Sequential and Two-Stage Stopping Times for Fixed-Width Confidence Intervals in Bernoulli Trials: Application in Reliability

Zacks

Mukhopadhyay

2007

Sequential Analysis

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We develop the exact distribution of the stopping variable of a sequential procedure that was originally given by Robbins and Siegmund (1974). The stopping variable was designed for estimating the log-odds in a sequence of Bernoulli trials. Using our exact distribution of the stopping variable, we give explicit formulas for the expected value and mean squared error for the estimator of the odds at stopping. An alternative two-stage procedure is then given and some of its important characteristics are exactly evaluated. It is shown that if the probability of success p is not too small or too large, the two-stage procedure is nearly as efficient as the purely sequential procedure. The results of this paper are then applied for designing an appropriate stopping time in a reliability experiment for estimating the ratio of the mean time between failures of two independent systems with exponential lifetimes.

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Section: Discussionmentioning

confidence: 99%

Distributions of Sequential and Two-Stage Stopping Times for Fixed-Width Confidence Intervals in Bernoulli Trials: Application in Reliability

Zacks

Mukhopadhyay

2007

Sequential Analysis

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“…These facts make our results valid in those situations where the Poisson process is inappropriate, namely, when the events occur in clusters and are characterized by overdispersion. Our study is motivated by the application of the negative binomial process and the related negative binomial distribution in several fields, such as distribution theory (Anscombe (1950), Barndorff-Nielsen and Yeo (1969), Vaillant (1991), Zhou and Carin (2013)), agriculture and pest management (Mukhopadhyay (2014) and Mukhopadhyay and de Silva (2005)), cosmology (Carruthers and Minh (1983)), entomology (Nedelman (1983) and Wilson and Room (1983)), and hydrology (Kozubowski and Podgórski (2009)). Problems of sequential testing for such a process were faced in Buonaguidi and Muliere (2013a) and Buonaguidi and Muliere (2013b).…”

Section: Introductionmentioning

confidence: 99%

On the Disorder Problem for a Negative Binomial Process

Buonaguidi

Muliere

2015

Journal of Applied Probability

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We study the Bayesian disorder problem for a negative binomial process. The aim is to determine a stopping time which is as close as possible to the random and unknown moment at which a sequentially observed negative binomial process changes the value of its characterizing parameter p ∈ (0, 1). The solution to this problem is explicitly derived through the reduction of the original optimal stopping problem to an integro-differential free-boundary problem. A careful analysis of the free-boundary equation and of the probabilistic nature of the boundary point allows us to specify when the smooth fit principle holds and when it breaks down in favour of the continuous fit principle.

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“…The exponential distribution of the change point and the linear penalty for the correct identification of the disorder that we assume throughout this paper ensure an explicit and analytical tractability of the problem, which, instead, would require numerical solution methods if these settings were removed (see, e.g. Our study is motivated by the application of the negative binomial process and the related negative binomial distribution in several fields, such as distribution theory (Anscombe (1950), Barndorff-Nielsen and Yeo (1969), Vaillant (1991), Zhou and Carin (2013)), agriculture and pest management (Mukhopadhyay (2014) and Mukhopadhyay and de Silva (2005)), cosmology (Carruthers and Minh (1983)), entomology (Nedelman (1983) and Wilson and Room (1983)), and hydrology (Kozubowski and Podgórski (2009)). Furthermore, from (1.1) we observe that the height of the jumps of a negative binomial process ranges in the positive integer numbers and E[X t ] = qt/p < qt/p 2 = var[X t ], t > 0.…”

Section: Introductionmentioning

confidence: 99%

On the Disorder Problem for a Negative Binomial Process

Buonaguidi¹,

Muliere²

2015

J. Appl. Probab.

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We study the Bayesian disorder problem for a negative binomial process. The aim is to determine a stopping time which is as close as possible to the random and unknown moment at which a sequentially observed negative binomial process changes the value of its characterizing parameter p ∈ (0, 1). The solution to this problem is explicitly derived through the reduction of the original optimal stopping problem to an integrodifferential free-boundary problem. A careful analysis of the free-boundary equation and of the probabilistic nature of the boundary point allows us to specify when the smooth fit principle holds and when it breaks down in favour of the continuous fit principle.

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Two-Stage Estimation of Mean in a Negative Binomial Distribution with Applications to Mexican Bean Beetle Data

Cited by 15 publications

References 25 publications

Distributions of Sequential and Two-Stage Stopping Times for Fixed-Width Confidence Intervals in Bernoulli Trials: Application in Reliability

Distributions of Sequential and Two-Stage Stopping Times for Fixed-Width Confidence Intervals in Bernoulli Trials: Application in Reliability

On the Disorder Problem for a Negative Binomial Process

On the Disorder Problem for a Negative Binomial Process

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