The Negative Binomial Distribution

Ross, G. J.; Preece, D. A.

doi:10.2307/2987659

Cited by 62 publications

(39 citation statements)

References 16 publications

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“…In fact, in cases of under-dispersion, φ cannot be calculated directly (5,10). Second, the confidence intervals for α are also continuous and more symmetric than for φ .…”

Section: Probability Mass Functionmentioning

confidence: 98%

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Adjustment for Maximum Likelihood Estimate of Negative Binomial Dispersion Parameter

Park

Lord

2008

Transportation Research Record

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Negative Binomial (or Poisson-gamma) model has been used extensively by highway safety analysts because it can accommodate the over-dispersion, often exhibited in crash data. However, it has been reported in the literature that the maximum likelihood estimate of the dispersion parameter of NB models can be significantly affected when the data are characterized by small sample size and low sample mean. Given the important roles of the dispersion parameter in various types of highway safety analyses, there is a need to determine whether the bias could be potentially corrected or minimized. The objectives of this paper are to explore whether a systematic relationship exists between the estimated and true dispersion parameters; determine the bias as a function of the sample size and sample mean; and, develop a procedure for correcting the bias caused by these two conditions. For this purpose, simulated data were used to derive the relationship under the various combinations of sample mean, dispersion parameter and sample size, which encompass all simulation conditions performed in previous researches. The dispersion parameter was estimated using the maximum likelihood method. The results confirmed previous studies, and developed a reasonable relationship between the estimated and true dispersion parameter for reducing the bias. Details about the application of the correction procedure were also provided using the crash data collected at 458 three-legged unsignalized intersections in California. Finally, the study provided several discussion points for further work.Park& Lord 2

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“…In fact, in cases of under-dispersion, φ cannot be calculated directly (5,10). Second, the confidence intervals for α are also continuous and more symmetric than for φ .…”

Section: Probability Mass Functionmentioning

confidence: 98%

“…Second, the confidence intervals for α are also continuous and more symmetric than for φ . As the dataset becomes more homogeneous (φ ∞ ), the confidence intervals for φ can become erratic or discontinuous (10). For these reasons, it is recommended to estimate α directly.…”

Section: Probability Mass Functionmentioning

confidence: 99%

Adjustment for Maximum Likelihood Estimate of Negative Binomial Dispersion Parameter

Park

Lord

2008

Transportation Research Record

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“…Hence, taking k 5 1 (Bernoulli distribution) we get a 5 À1 and S pop ðtÞ in (4) becomes S pop ðtÞ ¼ 1 À yFðtÞ, corresponding to the mixture cure model (Boag, 1949;Berkson and Gage, 1952). Ross and Preece (1985) showed that even if k 5 À1/a (ao0) is not an integer, the negative binomial distribution still furnishes positive values of P(N 5 m), m 5 0,1,y,k Ã , where k Ã is the largest integer less than k. From (3) it follows that the variance of the number of competing causes under the negative binomial model is flexible. If À1/yrao0, there is under-dispersion from the Poisson model.…”

Section: Modelmentioning

confidence: 99%

“…In fact, the range of models in Yin and Ibrahim (2005) is enlarged in our paper. Second, by adopting the so-called Fisher's parametrization for the negative binomial distribution (Ross and Preece, 1985), the likelihood function of our model includes the cured fraction. Furthermore, irrespective of the model, a unique expression relates the cured fraction to covariates.…”

Section: Introductionmentioning

confidence: 99%

A Bayesian Long‐term Survival Model Parametrized in the Cured Fraction

2009

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The main goal of this paper is to investigate a cure rate model that comprehends some well-known proposals found in the literature. In our work the number of competing causes of the event of interest follows the negative binomial distribution. The model is conveniently reparametrized through the cured fraction, which is then linked to covariates by means of the logistic link. We explore the use of Markov chain Monte Carlo methods to develop a Bayesian analysis in the proposed model. The procedure is illustrated with a numerical example.

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“…Entretanto, como não é intuito desta dissertação descrever o método da máxima verossimilhança para a estimação de parâmetros, são informadas as seguintes referências de (Crowley, 2012) e (Ross & Preece, 1985) De forma ratificar, que a função binomial negativa é a função que mais se aproxima da amostra analisada, mostra-se na tabela 7 os valores de logverossimilhança, AIC (Aikake Information Criterion) e BIC (Bayesian Information Criterion), ratificando que para função de log-verossimilhança quanto mais próximo de zero o número estiver mais próxima da amostra estará a distribuição estimada. Para os demais indicadores (i.e.…”

Section: Mge (Maximum Goodness Fit Estimation)unclassified

Aplicação Multiperíodo Do Modelo De Localização De Hubs Não-Capacitados Na Distribuição Física De Equipamentos De Uma Empresa De Telecomunicações via Satélite: Um Estudo De Caso

BRITTO¹

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Britto, Marcos Lopes; Oliveira, Fabrício (Advisor) Application of multiperiod uncapacitated hub location model for equipment physical distribution of a satellite telecommunications company -a case study. Rio de Janeiro, 2015, 77p. MSc. Dissertation -Departamento de Engenharia Industrial, Pontifícia Universidade Católica do Rio de Janeiro.The relationship between logistics activities performed on telecommunications companies and their service delivery seems, to the public, is disassociated. However, the need to service large areas associated with reducing costs, puts these activities nonessential into to the group of strategic activities.Through the introduction of the Brazilian telecommunications environment, the importance of logistics for this service and the study location problems, this master thesis develops a dynamic MIP model -Mix Integer Programming -for the hub location problem known as ULP -Uncapacitated Hub Location Problem, and this model is used in the analysis of a real case study of an satellite telecommunications operator. which were obtained insights into the level of reducing cost by redesigning of distribution network and the choice of new warehouse points, being demonstrated by a stochastic study of 500 random scenarios.

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The Negative Binomial Distribution

Cited by 62 publications

References 16 publications

Adjustment for Maximum Likelihood Estimate of Negative Binomial Dispersion Parameter

Adjustment for Maximum Likelihood Estimate of Negative Binomial Dispersion Parameter

A Bayesian Long‐term Survival Model Parametrized in the Cured Fraction

Aplicação Multiperíodo Do Modelo De Localização De Hubs Não-Capacitados Na Distribuição Física De Equipamentos De Uma Empresa De Telecomunicações via Satélite: Um Estudo De Caso

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