Robust Estimates of the Negative Binomial Model

We consider robust parametric procedures for univariate discrete distributions, focusing on the negative binomial model. The procedures are based on three steps: ?First, a very robust, but possibly inefficient, estimate of the model parameters is computed. ?Second, this initial model is used to identify outliers, which are then removed from the sample. ?Third, a corrected maximum likelihood estimator is computed with the remaining observations. The final estimate inherits the breakdown point (bdp) of the initial one and its efficiency can be significantly higher. Analogous procedures were proposed in [1], [2], [5] for the continuous case. A comparison of the asymptotic bias of various estimates under point contamination points out the minimum Neyman's chi-squared disparity estimate as a good choice for the initial step. Various minimum disparity estimators were explored by Lindsay [4], who showed that the minimum Neyman's chi-squared estimate has a 50% bdp under point contamination; in addition, it is asymptotically fully efficient at the model. However, the finite sample efficiency of this estimate under the uncontaminated negative binomial model is usually much lower than 100% and the bias can be strong. We show that its performance can then be greatly improved using the three step procedure outlined above. In addition, we compare the final estimate with the procedure described in

A Note on Bayesian Estimation for the Negative-Binomial Model

2000 Mathematics Subject Classification: 62F15.

On estimation and influence diagnostics for zero-inflated negative binomial regression models

The zero-inflated negative binomial model is used to account for overdispersion detected in data that are initially analyzed under the zero-Inflated Poisson model A frequentist analysis a jackknife estimator and a non-parametric bootstrap for parameter estimation of zero-inflated negative binomial regression models are considered In addition an EM-type algorithm is developed for performing maximum likelihood estimation Then the appropriate matrices for assessing local influence on the parameter estimates under different perturbation schemes and some ways to perform global influence analysis are derived In order to study departures from the error assumption as well as the presence of outliers residual analysis based on the standardized Pearson residuals is discussed The relevance of the approach is illustrated with a real data set where It is shown that zero-inflated negative binomial regression models seems to fit the data better than the Poisson counterpart (C) 2010 Elsevier B V All rights reserved

A Nonstationary Negative Binomial Time Series with Time-Dependent Covariates: Enterococcus Counts in Boston Harbor

Boston Harbor has had a history of poor water quality, including contamination by enteric pathogens. We conduct a statistical analysis of data collected by the Massachusetts Water Resources Authority (MWRA) between 1996 and 2002 to evaluate the effects of court-mandated improvements in sewage treatment. Motivated by the ineffectiveness of standard Poisson mixture models and their zero-inflated counterparts, we propose a new negative binomial model for time series of Enterococcus counts in Boston Harbor, where nonstationarity and autocorrelation are modeled using a nonparametric smooth function of time in the predictor. Without further restrictions, this function is not identifiable in the presence of time-dependent covariates; consequently we use a basis orthogonal to the space spanned by the covariates and use penalized quasi-likelihood (PQL) for estimation. We conclude that Enterococcus counts were greatly reduced near the Nut Island Treatment Plant (NITP) outfalls following the transfer of wastewaters from NITP to the Deer Island Treatment Plant (DITP) and that the transfer of wastewaters from Boston Harbor to the offshore diffusers in Massachusetts Bay reduced the Enterococcus counts near the DITP outfalls.

Elección de la duración del viaje vacacional: una aproximación con modelos de recuento

La duración del viaje vacacional es una decisión del turista con unas implicaciones fundamentales para las organizaciones turísticas, pero que ha recibido una escasa atención por la literatura. Además, los escasos estudios se han centrado en los destinos costeros, cuando el turismo de interior se está erigiendo como una alternativa importante en algunos países. El presente trabajo analiza los factores determinantes de la elección temporal del viaje turístico, distinguiendo el tipo de destino elegido -costa e interior-, y proponiendo varias hipótesis acerca de la influencia de las características de los individuos relacionadas con el destino, de las restricciones personales y de las características sociodemográficas. La metodología aplicada estima, como novedad en este tipo de decisiones, un Modelo Binomial Negativo Truncado que evita los sesgos de estimación de los modelos de regresión y el supuesto restrictivo de igualdad media-varianza del Modelo de Poisson. La aplicación empírica realizada en España sobre una muestra de 1.600 individuos permite concluir, por un lado, que el Modelo Binomial Negativo es más adecuado que el de Poisson para realizar este tipo de análisis. Por otro lado, las dimensiones determinantes de la duración del viaje vacacional son, para ambos destinos, el alojamiento en hotel y apartamento propio, las restricciones temporales, la edad del turista y la forma de organizar el viaje; mientras que el tamaño de la ciudad de residencia y el atributo “precios baratos” es un aspecto diferencial de la costa; y el alojamiento en apartamentos alquilados lo es de los destinos de interior.

Extended truncated Tweedie-Poisson model

It has been argued that by truncating the sample space of the negative binomial and of the inverse Gaussian-Poisson mixture models at zero, one is allowed to extend the parameter space of the model. Here that is proved to be the case for the more general three parameter Tweedie-Poisson mixture model. It is also proved that the distributions in the extended part of the parameter space are not the zero truncation of mixed poisson distributions and that, other than for the negative binomial, they are not mixtures of zero truncated Poisson distributions either. By extending the parameter space one can improve the fit when the frequency of one is larger and the right tail is heavier than is allowed by the unextended model. Considering the extended model also allows one to use the basic maximum likelihood based inference tools when parameter estimates fall in the extended part of the parameter space, and hence when the m.l.e. does not exist under the unextended model. This extended truncated Tweedie-Poisson model is proved to be useful in the analysis of words and species frequency count data.

The negative binomial-beta Weibull regression model to predict the cure of prostate cancer

In this article, for the first time, we propose the negative binomial-beta Weibull (BW) regression model for studying the recurrence of prostate cancer and to predict the cure fraction for patients with clinically localized prostate cancer treated by open radical prostatectomy. The cure model considers that a fraction of the survivors are cured of the disease. The survival function for the population of patients can be modeled by a cure parametric model using the BW distribution. We derive an explicit expansion for the moments of the recurrence time distribution for the uncured individuals. The proposed distribution can be used to model survival data when the hazard rate function is increasing, decreasing, unimodal and bathtub shaped. Another advantage is that the proposed model includes as special sub-models some of the well-known cure rate models discussed in the literature. We derive the appropriate matrices for assessing local influence on the parameter estimates under different perturbation schemes. We analyze a real data set for localized prostate cancer patients after open radical prostatectomy.

A Poisson mixed model with nonnormal random effect distribution

In this paper, we propose a random intercept Poisson model in which the random effect is assumed to follow a generalized log-gamma (GLG) distribution. This random effect accommodates (or captures) the overdispersion in the counts and induces within-cluster correlation. We derive the first two moments for the marginal distribution as well as the intraclass correlation. Even though numerical integration methods are, in general, required for deriving the marginal models, we obtain the multivariate negative binomial model from a particular parameter setting of the hierarchical model. An iterative process is derived for obtaining the maximum likelihood estimates for the parameters in the multivariate negative binomial model. Residual analysis is proposed and two applications with real data are given for illustration. (C) 2011 Elsevier B.V. All rights reserved.

Statnote 37 : the negative binomial distribution

An organism living in water, and present at low density, may be distributed at random and therefore, samples taken from the water are likely to be distributed according to the Poisson distribution. The distribution of many organisms, however, is not random, individuals being either aggregated into clusters or more uniformly distributed. By fitting a Poisson distribution to data, it is only possible to test the hypothesis that an observed set of frequencies does not deviate significantly from an expected random pattern. Significant deviations from random, either as a result of increasing uniformity or aggregation, may be recognized by either rejection of the random hypothesis or by examining the variance/mean (V/M) ratio of the data. Hence, a V/M ratio not significantly different from unity indicates a random distribution, greater than unity a clustered distribution, and less then unity a regular or uniform distribution . If individual cells are clustered, however, the negative binomial distribution should provide a better description of the data. In addition, a parameter of this distribution, viz., the binomial exponent (k), may be used as a measure of the ‘intensity’ of aggregation present. Hence, this Statnote describes how to fit the negative binomial distribution to counts of a microorganism in samples taken from a freshwater environment.

Mixed Negative Binomial Distribution by Weighted Gamma Mixing Distribution

Павел Т. Стойнов - В тази работа се разглежда отрицателно биномното разпределение, известно още като разпределение на Пойа. Предполагаме, че смесващото разпределение е претеглено гама разпределение. Изведени са вероятностите в някои частни случаи. Дадени са рекурентните формули на Панжер.

A Characterization of the Negative Binomial Distribution

Only a few characterizations have been obtained in literatute for the negative binomial distribution (see Johnson et al., Chap. 5, 1992). In this article a characterization of the negative binomial distribution related to random sums is obtained which is motivated by the geometric distribution characterization given by Khalil et al. (1991). An interpretation in terms of an unreliable system is given.

Institutions and firm formation: An empirical analysis of Portuguese municipalities

A Work Project, presented as part of the requirements for the Award of a Masters Degree in Economics from the NOVA – School of Business and Economics

Under the veil of citations

Academics are often ranked on citation counts’, which is considered an adequate proxy for author's quality and reputation. This paper seeks to find what is behind a cited academic / a cited article. We constructed a rich dataset from Portuguese affiliated economists and use zero inflated negative binomial model. This procedure is appropriate for count outcomes, correcting for overdispersion and excess zeros. We also use a fixed effect poisson model to accomodate authors' unobserved heterogeneity. We analyze results in detail comparing with existing literature and making some theoretical considerations around.

Canakinumab Reduces The Risk Of Acute Gouty Arthritis Flares During Initiation Of Allopurinol Therapy: Results Of A Double-blind, Randomised Study

RATIONALE: This study assessed the efficacy and safety of canakinumab, a fully human anti-interleukin-1beta monoclonal antibody, for prophylaxis against acute gouty arthritis flares in patients initiating uratelowering therapy.METHODS: In this double-blind, double-dummy, dose-ranging study, 432 patients with gouty arthritis initiating allopurinol therapy were randomised 1:1:1:1:1:1:2 to receive: a single dose of canakinumab, 25, 50, 100, 200, or 300 mg subcutaneously (sc); four 4-weekly doses of canakinumab (50150125125 mg sc); or daily colchicine 0.5 mg orally for 16 weeks. Patients recorded details of flares in diaries. The study aimed to determine the canakinumab dose having equivalent efficacy to colchicine 0.5 mg at 16 weeks.RESULTS: A dose-response for canakinumab was not apparent with any of the four pre-defined dose-responsemodels. The estimated canakinumab dose with equivalent efficacy to colchicinewas belowthe range of doses tested.At 16 weeks, therewas a 62-72% reduction in themean number of flares per patient for canakinumab doses >50 mg vs colchicine based on a negative binomial model (rate ratio: 0.28-0.38, p50.0083), and the percentage of patients experiencing >1 flarewas significantly lower for all canakinumab doses (15- 27%) vs colchicine (44%, p<0.05). Therewas a 64-72%reduction in the risk of experiencing >1 flare for canakinumab doses >50 mg vs colchicine at 16 weeks (hazard ratio: 0.28-0.36, p50.05). The incidence of adverse events was similar across treatment groups.CONCLUSIONS: Single canakinumab doses >50 mg or four 4-weekly doses provided superior prophylaxis against flares compared with daily colchicine 0.5 mg.