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Journal articles on the topic 'Negative binomial distribution. Parameter estimation'
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1
Famoye, Felix. "Parameter estimation for generalized negative binomial distribution." Communications in Statistics - Simulation and Computation 26, no. 1 (January 1997): 269–79. http://dx.doi.org/10.1080/03610919708813378.
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Whitaker, Thomas, Francis Giesbrecht, and Jeremy Wu. "Suitability of Several Statistical Models to Simulate Observed Distribution of Sample Test Results in Inspections of Aflatoxin-Contaminated Peanut Lots." Journal of AOAC INTERNATIONAL 79, no. 4 (July 1, 1996): 981–88. http://dx.doi.org/10.1093/jaoac/79.4.981.
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Abstract The acceptability of 10 theoretical distributions to simulate observed distribution of sample aflatoxin test results was evaluated by using 2 parameter estimation methods and 3 goodness of fit (GOF) tests. All theoretical distributions were compared with 120 observed distributions of aflatoxin test results of farmers' stock peanuts. For a given parameter estimation method and GOF test, the negative binomial distribution had the highest percentage of statistically acceptable fits. The log normal and Poisson-gamma (gamma shape parameter = 0.5) distributions had slightly fewer but an almost equal percentage of acceptable fits. For the 3 most acceptable statistical models, the negative binomial had the greatest percentage of best or closest fits. Both the parameter estimation method and the GOF test had an influence on which theoretical distribution had the largest number of acceptable fits. All theoretical distributions, except the negative binomial distribution, had more acceptable fits when model parameters were determined by the maximum likelihood method. The negative binomial had slightly more acceptable fits when model parameters were estimated by the method of moments. The results also demonstrated the importance of using the same GOF test for comparing the acceptability of several theoretical distributions.
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Wang, Yining. "Estimation problems for the two-parameter negative binomial distribution." Statistics & Probability Letters 26, no. 2 (February 1996): 113–14. http://dx.doi.org/10.1016/0167-7152(94)00259-2.
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Aryuyuen, Sirinapa, and Issaraporn Thiamsorn. "Methods for Parameter Estimation of the Negative Binomial-Generalized Exponential Distribution." Applied Mechanics and Materials 866 (June 2017): 383–86. http://dx.doi.org/10.4028/www.scientific.net/amm.866.383.
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Abstract. We proposed several estimation methods for the parameters of the negative binomial-generalized exponential (NB-GE) distribution. In the simulation study, the maximum likelihood estimation (MLE) with nlm function seems to have the most efficiency to estimate the parameters and of the NB-GE distribution when it compares with method of the moments (MM) and MLE with optim function by using the average mean square error (AMSE) for a criteria. The AMSE values of each parameter estimation methods are decreasing when the sample size increasing. Moreover, the example dataset is illustrated. Based on the chi-square values for the fitting distribution via the MLE with nlm function is better than other estimation methods.
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Al-Saleh, Mohammad F., and Fatima K. Al-Batainah. "Estimation of the shape parameter k of the negative binomial distribution." Applied Mathematics and Computation 143, no. 2-3 (November 2003): 431–41. http://dx.doi.org/10.1016/s0096-3003(02)00374-0.
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Shanker, Rama, and Kamlesh Kumar Shukla. "A new three-parameter size-biased poisson-lindley distribution with properties and applications." Biometrics & Biostatistics International Journal 9, no. 1 (February 11, 2020): 1–4. http://dx.doi.org/10.15406/bbij.2020.09.00294.
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A new three-parameter size-biased Poisson-Lindley distribution which includes several one parameter and two-parameter size-biased distributions including size-biased geometric distribution (SBGD), size-biased negative binomial distribution (SBNBD), size-biased Poisson-Lindley distribution (SBPLD), size-biased Poisson-Shanker distribution (SBPSD), size-biased two-parameter Poisson-Lindley distribution-1 (SBTPPLD-1), size-biased two-parameter Poisson-Lindley distribution-2(SBTPPLD-2), size-biased quasi Poisson-Lindley distribution (SBQPLD) and size-biased new quasi Poisson-Lindley distribution (SBNQPLD) for particular cases of parameters has been proposed. Its various statistical properties based on moments including coefficient of variation, skewness, kurtosis and index of dispersion have been studied. Maximum likelihood estimation has been discussed for estimating the parameters of the distribution. Goodness of fit of the proposed distribution has been discussed.
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Lee, Simon CK. "Delta Boosting Implementation of Negative Binomial Regression in Actuarial Pricing." Risks 8, no. 1 (February 19, 2020): 19. http://dx.doi.org/10.3390/risks8010019.
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This study proposes an efficacious approach to analyze the over-dispersed insurance frequency data as it is imperative for the insurers to have decisive informative insights for precisely underwriting and pricing insurance products, retaining existing customer base and gaining an edge in the highly competitive retail insurance market. The delta boosting implementation of the negative binomial regression, both by one-parameter estimation and a novel two-parameter estimation, was tested on the empirical data. Accurate parameter estimation of the negative binomial regression is complicated with considerations of incomplete insurance exposures, negative convexity, and co-linearity. The issues mainly originate from the unique nature of insurance operations and the adoption of distribution outside the exponential family. We studied how the issues could significantly impact the quality of estimation. In addition to a novel approach to simultaneously estimate two parameters in regression through boosting, we further enrich the study by proposing an alteration of the base algorithm to address the problems. The algorithm was able to withstand the competition against popular regression methodologies in a real-life dataset. Common diagnostics were applied to compare the performance of the relevant candidates, leading to our conclusion to move from light-tail Poisson to negative binomial for over-dispersed data, from generalized linear model (GLM) to boosting for non-linear and interaction patterns, from one-parameter to two-parameter estimation to reflect more closely the reality.
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Elsaied, Hanan, and Roland Fried. "On robust estimation of negative binomial INARCH models." METRON 79, no. 2 (April 24, 2021): 137–58. http://dx.doi.org/10.1007/s40300-021-00207-8.
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AbstractWe discuss robust estimation of INARCH models for count time series, where each observation conditionally on its past follows a negative binomial distribution with a constant scale parameter, and the conditional mean depends linearly on previous observations. We develop several robust estimators, some of them being computationally fast modifications of methods of moments, and some rather efficient modifications of conditional maximum likelihood. These estimators are compared to related recent proposals using simulations. The usefulness of the proposed methods is illustrated by a real data example.
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Van De Ven, R. "Estimating the shape parameter for the negative binomial distribution." Journal of Statistical Computation and Simulation 46, no. 1-2 (April 1993): 111–23. http://dx.doi.org/10.1080/00949659308811497.
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Savani, V., and A. A. Zhigljavsky. "Efficient Estimation of Parameters of the Negative Binomial Distribution." Communications in Statistics - Theory and Methods 35, no. 5 (June 2006): 767–83. http://dx.doi.org/10.1080/03610920500501346.
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Hussain, Tassaddaq, Muhammad Aslam, and Munir Ahmad. "A Two Parameter Discrete Lindley Distribution." Revista Colombiana de Estadística 39, no. 1 (January 18, 2016): 45–61. http://dx.doi.org/10.15446/rce.v39n1.55138.
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<p>In this article we have proposed and discussed a two parameter discrete Lindley distribution. The derivation of this new model is based on a two step methodology i.e. mixing then discretizing, and can be viewed as a new generalization of geometric distribution. The proposed model has proved itself as the least loss of information model when applied to a number of data sets (in an over and under dispersed structure). The competing models such as Poisson, Negative binomial, Generalized Poisson and discrete gamma distributions are the well known standard discrete distributions. Its Lifetime classification, kurtosis, skewness, ascending and descending factorial moments as well as its recurrence relations, negative moments, parameters estimation via maximum likelihood method, characterization and discretized bi-variate case are presented.</p>
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Willson, L. J., J. L. Folks, and J. H. Young. "Complete sufficiency and maximum likelihood estimation for the two-parameter negative binomial distribution." Metrika 33, no. 1 (December 1986): 349–62. http://dx.doi.org/10.1007/bf01894768.
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Zhao, Zhiwen. "Parameter Estimation and Hypothesis Testing of Two Negative Binomial Distribution Population with Missing Data." Physics Procedia 33 (2012): 1475–80. http://dx.doi.org/10.1016/j.phpro.2012.05.241.
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Amato, Timothy W. "On Difference Equations, Probability Models and the “Generalized Event Count” Distribution." Political Analysis 6 (1996): 175–212. http://dx.doi.org/10.1093/pan/6.1.175.
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In this article, the mathematical and probabilistic foundations of Gary King's “generalized event count” (GEC) model for dealing with unequally dispersed event count data are explored. It is shown that the GEC model is a probability model that joins together the binomial, negative binomial, and Poisson distributions. Some aspects of the GEC's reparameterization are described and extended and it is shown how different reparameterizations lead to different interpretations of the dispersion parameter. The common mathematical and statistical structure of “unequally dispersed” event count models as models that require estimation of the “number of trials” parameter along with the “probability” component is derived. Some questions pertaining to estimation of this class of models are raised for future discussion.
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Prasongporn, Pralongpol, and Winai Bodhisuwan. "Bayesian parameters estimation of the negative binomial - two parameter weighted exponential distribution with application in biological data." Journal of Applied Science 18, no. 1 (June 19, 2019): 39–48. http://dx.doi.org/10.14416/j.appsci.2019.03.001.
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Power, James H., and E. Barry Moser. "Linear model analysis of net catch data using the negative binomial distribution." Canadian Journal of Fisheries and Aquatic Sciences 56, no. 2 (February 1, 1999): 191–200. http://dx.doi.org/10.1139/f98-150.
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Sampling with nets or trawls remains a common technique for determining the comparative abundances of aquatic organisms, and the objective of such studies is frequently to evaluate relationships among the counts of individuals caught and exogenous variables. Analysis of such data is often done with a general linear model (e.g., ANOVA, ANCOVA, regression), assuming an underlying normal probability distribution. Such analyses are not fully satisfactory because of the symmetry and continuous nature of the assumed normal probability distribution and the high variance to low mean value relationships common to counts of biological populations. The negative binomial is a discrete probability distribution that is recognized as a suitable descriptor of organism count data. We present an approach for undertaking linear model analyses of net catch data that permits estimation of model parameters (including the negative binomial k parameter) and hypothesis testing of both continuous and discrete model effects and their interactions using bootstrap replication. The analysis incorporates adjustment for varying element sizes, such as differences in the amounts of water filtered during sampling.
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Mansoor, Muhammad, Muhammad Hussain Tahir, Gauss M. Cordeiro, Sajid Ali, and Ayman Alzaatreh. "The Lindley negative-binomial distribution: Properties, estimation and applications to lifetime data." Mathematica Slovaca 70, no. 4 (August 26, 2020): 917–34. http://dx.doi.org/10.1515/ms-2017-0404.
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AbstractA generalization of the Lindley distribution namely, Lindley negative-binomial distribution, is introduced. The Lindley and the exponentiated Lindley distributions are considered as sub-models of the proposed distribution. The proposed model has flexible density and hazard rate functions. The density function can be decreasing, right-skewed, left-skewed and approximately symmetric. The hazard rate function possesses various shapes including increasing, decreasing and bathtub. Furthermore, the survival and hazard rate functions have closed form representations which make this model tractable for censored data analysis. Some general properties of the proposed model are studied such as ordinary and incomplete moments, moment generating function, mean deviations, Lorenz and Bonferroni curve. The maximum likelihood and the Bayesian estimation methods are utilized to estimate the model parameters. In addition, a small simulation study is conducted in order to evaluate the performance of the estimation methods. Two real data sets are used to illustrate the applicability of the proposed model.
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Al-Babtain, Abdulhakim A., Abdul Hadi N. Ahmed, and Ahmed Z. Afify. "A New Discrete Analog of the Continuous Lindley Distribution, with Reliability Applications." Entropy 22, no. 6 (May 28, 2020): 603. http://dx.doi.org/10.3390/e22060603.
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In this paper, we propose and study a new probability mass function by creating a natural discrete analog to the continuous Lindley distribution as a mixture of geometric and negative binomial distributions. The new distribution has many interesting properties that make it superior to many other discrete distributions, particularly in analyzing over-dispersed count data. Several statistical properties of the introduced distribution have been established including moments and moment generating function, residual moments, characterization, entropy, estimation of the parameter by the maximum likelihood method. A bias reduction method is applied to the derived estimator; its existence and uniqueness are discussed. Applications of the goodness of fit of the proposed distribution have been examined and compared with other discrete distributions using three real data sets from biological sciences.
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Zhang, Yunlong, Zhirui Ye, and Dominique Lord. "Estimating Dispersion Parameter of Negative Binomial Distribution for Analysis of Crash Data." Transportation Research Record: Journal of the Transportation Research Board 2019, no. 1 (January 2007): 15–21. http://dx.doi.org/10.3141/2019-03.
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León-Novelo, Luis, Claudio Fuentes, and Sarah Emerson. "Marginal likelihood estimation of negative binomial parameters with applications to RNA-seq data." Biostatistics 18, no. 4 (March 19, 2017): 637–50. http://dx.doi.org/10.1093/biostatistics/kxx006.
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SUMMARY RNA-Seq data characteristically exhibits large variances, which need to be appropriately accounted for in any proposed model. We first explore the effects of this variability on the maximum likelihood estimator (MLE) of the dispersion parameter of the negative binomial distribution, and propose instead to use an estimator obtained via maximization of the marginal likelihood in a conjugate Bayesian framework. We show, via simulation studies, that the marginal MLE can better control this variation and produce a more stable and reliable estimator. We then formulate a conjugate Bayesian hierarchical model, and use this new estimator to propose a Bayesian hypothesis test to detect differentially expressed genes in RNA-Seq data. We use numerical studies to show that our much simpler approach is competitive with other negative binomial based procedures, and we use a real data set to illustrate the implementation and flexibility of the procedure.
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Whitaker, Thomas B., Francis G. Giesbrecht, Jeremy Wu, Winston M. Hagler, and Floyd E. Dowell. "Predicting the Distribution of Aflatoxin Test Results from Farmers’ Stock Peanuts." Journal of AOAC INTERNATIONAL 77, no. 3 (May 1, 1994): 659–66. http://dx.doi.org/10.1093/jaoac/77.3.659.
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Abstract Suitability of the negative binomial function for use in estimating the distribution of sample aflatoxin test results associated with testing farmers1 stock peanuts for aflatoxin was studied. A 900 kg portion of peanut pods was removed from each of 40 contaminated farmers1 stock lots. The lots averaged about 4100 kg. Each 900 kg portion was divided into fifty 2.26 kg samples, fifty 4.21 kg samples, and fifty 6.91 kg samples. The aflatoxin in each sample was quantified by liquid chromatography. An observed distribution of sample aflatoxin test results consisted of 50 aflatoxin test results for each lot and each sample size. The mean aflatoxin concentration, m; the variance, s2xamong the 50 sample aflatoxin test results; and the shape parameter, k, for the negative binomial function were determined for each of the 120 observed distributions (40 lots times 3 sample sizes). Regression analysis indicated the functional relationship between k and m to be k = 0.000006425m0.8047. The 120 observed distributions of sample aflatoxin test results were compared to the negative binomial function by using the Kolmogorov–Smirnov (KS) test. The null hypothesis that the true unknown distribution function was negative binomial was not rejected at the 5% significance level for 114 of the 120 distributions. The negative binomial function failed the KS test at a sample concentration of 0 ng/g in all 6 of the distributions where the negative binomial function was rejected. The negative binomial function always predicted a smaller percentage of samples testing 0 ng/g than was actually observed. However, the negative binomial function did fit the observed distribution for sample test results at a concentration greater than 0 in 4 of the 6 cases. As a result, the negative binomial function provides an accurate estimate of the acceptance probabilities associated with accepting contaminated lots of farmers' stock peanuts for various sample sizes and various sample acceptance levels greater than 0 ng/g.
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Thongteeraparp, Ampai, and A. Volodin. "Parameter Estimation of the Negative Binomial—New Weighted Lindley Distribution by the Method of Maximum Likelihood." Lobachevskii Journal of Mathematics 41, no. 3 (March 2020): 430–34. http://dx.doi.org/10.1134/s1995080220030178.
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Morgan, CJ, IB Aban, CR Katholi, and GR Cutter. "Modeling lesion counts in multiple sclerosis when patients have been selected for baseline activity." Multiple Sclerosis Journal 16, no. 8 (June 18, 2010): 926–34. http://dx.doi.org/10.1177/1352458510373110.
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The number of new gadolinium-enhancing lesions discovered via magnetic resonance imaging is a well-established outcome for multiple sclerosis studies, especially Phase II Studies. Due to the high cost of magnetic resonance imaging scans, many investigators select participants for the presence of lesions. While this selection procedure is thought to improve the power of inferences, the effect of screening for baseline activity on parameter estimation and interval coverage has not yet been examined. The objective of this study was to investigate the performance of the negative binomial distribution for modeling lesion count data in multiple sclerosis when patients have been selected for activity on a baseline scan. We performed computer simulations to investigate the influence of the screening process on inferences made using a negative binomial model about treatment effects in two independent samples. We also demonstrate how the statistical properties of screening can be incorporated into trial design. We demonstrate that when the negative binomial distribution is used to model lesion counts, while screening for baseline activity improves point estimation, this practice also has the potential to decrease interval coverage and inflate the Type I error rate. For data that is to be modeled using a negative binomial distribution, screening for baseline activity can create a trade-off between cost effectiveness and a higher than desired false positive rate that must be carefully considered in planning Phase II trials.
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Liu, Ya Shu, and Han Bing Yan. "The Development of Topic Model Based on Beta-Negative Binomial Process." Applied Mechanics and Materials 427-429 (September 2013): 1597–600. http://dx.doi.org/10.4028/www.scientific.net/amm.427-429.1597.
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. Topic Model is one of the important subfields in Data Mining, which has been developed very quickly and has been applicated in many fields in recent years. Many researchers have been engaged in this field. In this paper, we introduce the BNB process based on Beta and Negative Binomial distribution, using the hierarchical distribution instead of Dirichlet in LDA. And we give the expression of parameter estimation used by Gibbs sampling. Then, BNB process is applicated in the text topic classification. We design experiments to decide the numbers of topics and compare the BNB process with LDA. Experiment results show that the BNB process has better performance over LDA in English Dataset, but they have almost the same result in Chinese micro-blog topic classification. Finally we analyze the problem and give the idea in further research.
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GABA, S., V. GINOT, and J. CABARET. "Modelling macroparasite aggregation using a nematode-sheep system: the Weibull distribution as an alternative to the Negative Binomial distribution?" Parasitology 131, no. 3 (April 25, 2005): 393–401. http://dx.doi.org/10.1017/s003118200500764x.
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Macroparasites are almost always aggregated across their host populations, hence the Negative Binomial Distribution (NBD) with its exponent parameter k is widely used for modelling, quantifying or analysing parasite distributions. However, many studies have pointed out some drawbacks in the use of the NBD, with respect to the sensitivity of k to the mean number of parasites per host or the under-representation of the heavily infected hosts in the estimate of k. In this study, we compare the fit of the NBD with 4 other widely used distributions on observed parasitic gastrointestinal nematode distributions in their sheep host populations (11 datasets). Distributions were fitted to observed data using maximum likelihood estimator and the best fits were selected using the Akaike's Information Criterion (AIC). A simulation study was also conducted in order to assess the possible bias in parameter estimations especially in the case of small sample sizes. We found that the NBD is seldom the best fit for gastrointestinal nematode distributions. The Weibull distribution was clearly more appropriate over a very wide range of degrees of aggregation, mainly because it was more flexible in fitting the heavily infected hosts. Moreover, the Weibull distribution estimates are less sensitive to sample size. Thus, when possible, we suggest to carefully check on observed data if the NBD is appropriate before conducting any further analysis on parasite distributions.
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Hahn, Jinyong. "A Note on the Efficient Semiparametric Estimation of Some Exponential Panel Models." Econometric Theory 13, no. 4 (February 1997): 583–88. http://dx.doi.org/10.1017/s0266466600006010.
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This paper investigates the semiparametric efficiency of the conditional maximum likelihood estimation in some panel models. The nonparametric component of the model is the unknown distribution of the fixed effect. For the exponential panel model, there exists a complete sufficient statistic for the fixed effect. When the complete sufficient statistic does not depend on the parameter of interest, the conditional maximum likelihood estimator (CMLE) achieves the semiparametric efficiency bound. In particular, the CMLE is semiparametrically efficient for the panel Poisson regression model and the panel negative binomial model.
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Chen, Cathy WS, and K. Khamthong. "Bayesian modelling of nonlinear negative binomial integer-valued GARCHX models." Statistical Modelling 20, no. 6 (July 8, 2019): 537–61. http://dx.doi.org/10.1177/1471082x19845541.
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This study focuses on modelling dengue cases in northeastern Thailand through two meteorological covariates: cumulative rainfall and average maximum temperature. We propose two nonlinear integer-valued GARCHX models (Markov switching and threshold specification) with a negative binomial distribution, as they take into account the stylized features of weekly dengue haemorrhagic fever cases, which contain nonlinear dynamics, lagged dependence, overdispersion, consecutive zeros and asymmetric effects of meteorological covariates. We conduct parameter estimation and one-step-ahead forecasting for two proposed models based on Bayesian Markov chain Monte Carlo (MCMC) methods. A simulation study illustrates that the adaptive MCMC sampling scheme performs well. The empirical results offer strong support for the Markov switching integer-valued GARCHX model over its competitors via Bayes factor and deviance information criterion. We also provide one-step-ahead forecasting based on the prediction interval that offers a useful early warning signal of outbreak detection.
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Hung, Lai-Fa. "A Negative Binomial Regression Model for Accuracy Tests." Applied Psychological Measurement 36, no. 2 (January 24, 2012): 88–103. http://dx.doi.org/10.1177/0146621611429548.
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Rasch used a Poisson model to analyze errors and speed in reading tests. An important property of the Poisson distribution is that the mean and variance are equal. However, in social science research, it is very common for the variance to be greater than the mean (i.e., the data are overdispersed). This study embeds the Rasch model within an overdispersion framework and proposes new estimation methods. The parameters in the proposed model can be estimated using the Markov chain Monte Carlo method implemented in WinBUGS and the marginal maximum likelihood method implemented in SAS. An empirical example based on models generated by the results of empirical data, which are fitted and discussed, is examined.
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Su, Zhangwen, Haiqing Hu, Mulualem Tigabu, Guangyu Wang, Aicong Zeng, and Futao Guo. "Geographically Weighted Negative Binomial Regression Model Predicts Wildfire Occurrence in the Great Xing’an Mountains Better Than Negative Binomial Model." Forests 10, no. 5 (April 30, 2019): 377. http://dx.doi.org/10.3390/f10050377.
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Wildfire is a major disturbance that affects large area globally every year. Thus, a better prediction of the likelihood of wildfire occurrence is essential to develop appropriate fire prevention measures. We applied a global negative Binomial (NB) and a geographically weighted negative Binomial regression (GWNBR) models to determine the relationship between wildfire occurrence and its drivers factors in the boreal forests of the Great Xing’an Mountains, northeast China. Using geo-weighted techniques to consider the geospatial information of meteorological, topographic, vegetation type and human factors, we aimed to verify whether the performance of the NB model can be improved. Our results confirmed that the model fitting and predictions of GWNBR model were better than the global NB model, produced more precise and stable model parameter estimation, yielded a more realistic spatial distribution of model predictions, and provided the detection of the impact hotpots of these predictor variables. We found slope, vegetation cover, average precipitation, average temperature, and average relative humidity as important predictors of wildfire occurrence in the Great Xing’an Mountains. Thus, spatially differing relations improves the explanatory power of the global NB model, which does not explain sufficiently the relationship between wildfire occurrence and its drivers. Thus, the GWNBR model can complement the global NB model in overcoming the issue of nonstationary variables, thereby enabling a better prediction of the occurrence of wildfires in large geographical areas and improving management practices of wildfire.
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Almazah, Mohammed Mohammed Ahmed, Tenzile Erbayram, Yunus Akdoğan, Mashail M. AL Sobhi, and Ahmed Z. Afify. "A New Extended Geometric Distribution: Properties, Regression Model, and Actuarial Applications." Mathematics 9, no. 12 (June 9, 2021): 1336. http://dx.doi.org/10.3390/math9121336.
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In this paper, a new modified version of geometric distribution is proposed. The newly introduced model is called transmuted record type geometric (TRTG) distribution. TRTG distribution is a good alternative to the negative binomial, Poisson and geometric distributions in modeling real data encountered in several applied fields. The main statistical properties of the new distribution were obtained. We determined the measures of value at risk and tail value at risk for the TRTG distribution. These measures are important quantities in actuarial sciences for portfolio optimization under uncertainty. The TRTG parameters were estimated via maximum likelihood, moments, proportions, and Bayesian estimation methods, and the simulation results were determined to explore their performance. Furthermore, a new count regression model based on the TRTG distribution was proposed. Four real data applications were adopted to illustrate the applicability of the TRTG distribution and its count regression model. These applications showed empirically that the TRTG distribution outperforms some important discrete models such as the negative binomial, transmuted geometric, discrete Burr, discrete Chen, geometric, and Poisson distributions.
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Sellers, Kimberly F., Tong Li, Yixuan Wu, and Narayanaswamy Balakrishnan. "A Flexible Multivariate Distribution for Correlated Count Data." Stats 4, no. 2 (April 15, 2021): 308–26. http://dx.doi.org/10.3390/stats4020021.
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Multivariate count data are often modeled via a multivariate Poisson distribution, but it contains an underlying, constraining assumption of data equi-dispersion (where its variance equals its mean). Real data are oftentimes over-dispersed and, as such, consider various advancements of a negative binomial structure. While data over-dispersion is more prevalent than under-dispersion in real data, however, examples containing under-dispersed data are surfacing with greater frequency. Thus, there is a demonstrated need for a flexible model that can accommodate both data types. We develop a multivariate Conway–Maxwell–Poisson (MCMP) distribution to serve as a flexible alternative for correlated count data that contain data dispersion. This structure contains the multivariate Poisson, multivariate geometric, and the multivariate Bernoulli distributions as special cases, and serves as a bridge distribution across these three classical models to address other levels of over- or under-dispersion. In this work, we not only derive the distributional form and statistical properties of this model, but we further address parameter estimation, establish informative hypothesis tests to detect statistically significant data dispersion and aid in model parsimony, and illustrate the distribution’s flexibility through several simulated and real-world data examples. These examples demonstrate that the MCMP distribution performs on par with the multivariate negative binomial distribution for over-dispersed data, and proves particularly beneficial in effectively representing under-dispersed data. Thus, the MCMP distribution offers an effective, unifying framework for modeling over- or under-dispersed multivariate correlated count data that do not necessarily adhere to Poisson assumptions.
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Shilane, David, and Derek Bean. "Growth Estimators and Confidence Intervals for the Mean of Negative Binomial Random Variables with Unknown Dispersion." Journal of Probability and Statistics 2013 (2013): 1–9. http://dx.doi.org/10.1155/2013/602940.
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The negative binomial distribution becomes highly skewed under extreme dispersion. Even at moderately large sample sizes, the sample mean exhibits a heavy right tail. The standard normal approximation often does not provide adequate inferences about the data's expected value in this setting. In previous work, we have examined alternative methods of generating confidence intervals for the expected value. These methods were based upon Gamma and Chi Square approximations or tail probability bounds such as Bernstein's inequality. We now propose growth estimators of the negative binomial mean. Under high dispersion, zero values are likely to be overrepresented in the data. A growth estimator constructs a normal-style confidence interval by effectively removing a small, predetermined number of zeros from the data. We propose growth estimators based upon multiplicative adjustments of the sample mean and direct removal of zeros from the sample. These methods do not require estimating the nuisance dispersion parameter. We will demonstrate that the growth estimators' confidence intervals provide improved coverage over a wide range of parameter values and asymptotically converge to the sample mean. Interestingly, the proposed methods succeed despite adding both bias and variance to the normal approximation.
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Sah, Binod Kumar, and A. Mishra. "A Generalised Exponential-Lindley Mixture of Poisson Distribution." Nepalese Journal of Statistics 3 (September 11, 2019): 11–20. http://dx.doi.org/10.3126/njs.v3i0.25575.
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Background: The exponential and the Lindley (1958) distributions occupy central places among the class of continuous probability distributions and play important roles in statistical theory. A Generalised Exponential-Lindley Distribution (GELD) was given by Mishra and Sah (2015) of which, both the exponential and the Lindley distributions are the particular cases. Mixtures of distributions form an important class of distributions in the domain of probability distributions. A mixture distribution arises when some or all the parameters in a probability function vary according to certain probability law. In this paper, a Generalised Exponential- Lindley Mixture of Poisson Distribution (GELMPD) has been obtained by mixing Poisson distribution with the GELD.
Materials and Methods: It is based on the concept of the generalisations of some continuous mixtures of Poisson distribution.
Results: The Probability mass of function of generalized exponential-Lindley mixture of Poisson distribution has been obtained by mixing Poisson distribution with GELD. The first four moments about origin of this distribution have been obtained. The estimation of its parameters has been discussed using method of moments and also as maximum likelihood method. This distribution has been fitted to a number of discrete data-sets which are negative binomial in nature and it has been observed that the distribution gives a better fit than the Poisson–Lindley Distribution (PLD) of Sankaran (1970).
Conclusion: P-value of the GELMPD is found greater than that in case of PLD. Hence, it is expected to be a better alternative to the PLD of Sankaran for similar type of discrete data-set which is negative binomial in nature.
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Mishra, A. "A new generalization of the logarithmic series distribution." Studia Scientiarum Mathematicarum Hungarica 51, no. 1 (March 1, 2014): 41–49. http://dx.doi.org/10.1556/sscmath.51.2014.1.1257.
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A new generalization of the logarithmic series distribution has been obtained as a limiting case of the zero-truncated Mishra’s [10] generalized negative binomial distribution (GNBD). This distribution has an advantage over the Mishra’s [9] quasi logarithmic series distribution (QLSD) as its moments appear in compact forms unlike the QLSD. This makes the estimation of parameters easier by the method of moments. The first four moments of this distribution have been obtained and the distribution has been fitted to some well known data-sets to test its goodness of fit.
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Shanker, R., and K. K. Shukla. "On Poisson-Weighted Lindley Distribution and Its Applications." Journal of Scientific Research 11, no. 1 (January 1, 2019): 1–13. http://dx.doi.org/10.3329/jsr.v11i1.35745.
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In this paper the nature and behavior of its coefficient of variation, skewness, kurtosis and index of dispersion of Poisson- weighted Lindley distribution (P-WLD), a Poisson mixture of weighted Lindley distribution, have been proposed and the nature and behavior have been explained graphically. Maximum likelihood estimation has been discussed to estimate its parameters. Applications of the proposed distribution have been discussed and its goodness of fit has been compared with Poisson distribution (PD), Poisson-Lindley distribution (PLD), negative binomial distribution (NBD) and generalized Poisson-Lindley distribution (GPLD).
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Bonat, Wagner H., Bent Jørgensen, Célestin C. Kokonendji, John Hinde, and Clarice G. B. Demétrio. "Extended Poisson–Tweedie: Properties and regression models for count data." Statistical Modelling 18, no. 1 (August 30, 2017): 24–49. http://dx.doi.org/10.1177/1471082x17715718.
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We propose a new class of discrete generalized linear models based on the class of Poisson–Tweedie factorial dispersion models with variance of the form [Formula: see text], where [Formula: see text] is the mean and [Formula: see text] and [Formula: see text] are the dispersion and Tweedie power parameters, respectively. The models are fitted by using an estimating function approach obtained by combining the quasi-score and Pearson estimating functions for the estimation of the regression and dispersion parameters, respectively. This provides a flexible and efficient regression methodology for a comprehensive family of count models including Hermite, Neyman Type A, Pólya–Aeppli, negative binomial and Poisson-inverse Gaussian. The estimating function approach allows us to extend the Poisson–Tweedie distributions to deal with underdispersed count data by allowing negative values for the dispersion parameter [Formula: see text]. Furthermore, the Poisson–Tweedie family can automatically adapt to highly skewed count data with excessive zeros, without the need to introduce zero-inflated or hurdle components, by the simple estimation of the power parameter. Thus, the proposed models offer a unified framework to deal with under-, equi-, overdispersed, zero-inflated and heavy-tailed count data. The computational implementation of the proposed models is fast, relying only on a simple Newton scoring algorithm. Simulation studies showed that the estimating function approach provides unbiased and consistent estimators for both regression and dispersion parameters. We highlight the ability of the Poisson–Tweedie distributions to deal with count data through a consideration of dispersion, zero-inflated and heavy tail indices, and illustrate its application with four data analyses. We provide an R implementation and the datasets as supplementary materials.
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Huang, Jie, and Fukang Zhu. "A New First-Order Integer-Valued Autoregressive Model with Bell Innovations." Entropy 23, no. 6 (June 4, 2021): 713. http://dx.doi.org/10.3390/e23060713.
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A Poisson distribution is commonly used as the innovation distribution for integer-valued autoregressive models, but its mean is equal to its variance, which limits flexibility, so a flexible, one-parameter, infinitely divisible Bell distribution may be a good alternative. In addition, for a parameter with a small value, the Bell distribution approaches the Poisson distribution. In this paper, we introduce a new first-order, non-negative, integer-valued autoregressive model with Bell innovations based on the binomial thinning operator. Compared with other models, the new model is not only simple but also particularly suitable for time series of counts exhibiting overdispersion. Some properties of the model are established here, such as the mean, variance, joint distribution functions, and multi-step-ahead conditional measures. Conditional least squares, Yule–Walker, and conditional maximum likelihood are used for estimating the parameters. Some simulation results are presented to access these estimates’ performances. Real data examples are provided.
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Pepin, P., K. A. Curtis, P. V. R. Snelgrove, B. de Young, and J. A. Helbig. "Optimal estimation of catch by the continuous underway fish egg sampler based on a model of the vertical distribution of American plaice (Hippoglossoides platessoides) eggs." ICES Journal of Marine Science 64, no. 1 (November 6, 2006): 18–30. http://dx.doi.org/10.1093/icesjms/fsl016.
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Abstract Pepin, P., Curtis, K.A., Snelgrove, P.V.R., de Young, B., and Helbig, J.A. 2007. Optimal estimation of catch by the continous underway fish egg sampler based on a model of the vertical distribution of American plaice (Hippoglossoides platessoides) eggs – ICES Journal of Marine Science, 64, 18–30. We investigate how the vertical stratification of the water column (specifically density) affects predictions of the catch of American plaice eggs (Hipploglossoides platessoides) from a fixed-depth sampler [the continuous underway fish egg sampler (CUFES)] relative to the integrated abundance in the water column measured in bongo tows. A steady-state model of the vertical distribution of fish eggs coupled with a simple model of the vertical profile of eddy diffusivity (i.e. mixing) is applied. Key model parameters are estimated through optimization of a one-to-one relationship between predicted and observed catches fit, using a generalized linear model with a Poisson, negative binomial, or gamma error structure. The incorporation of data on the vertical structure of the water column significantly improved the ability to forecast CUFES catches when using Poisson or negative binomial error structure, but not using a gamma distribution. Optimal maximum likelihood parameter estimates for eddy diffusivity and egg buoyancy fell within the range of expected values. The degree of uncertainty in the parameterization of eddy diffusivity suggests, however, that greater understanding of the forces that determine the vertical profile of mixing is critical to achieving strong predictive capabilities. The inverse problem of predicting integrated abundance from CUFES catches did not benefit from the environmental-driven model because of the high uncertainty in the catches from the CUFES.
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Sah, Binod Kumar. "A Generalised Poisson Mishra Distribution." Nepalese Journal of Statistics 2 (September 26, 2018): 27–36. http://dx.doi.org/10.3126/njs.v2i0.21153.
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Background: “Mishra distribution" of B. K. Sah (2015) has been obtained in honor of Professor A. Mishra, Department of Statistics, Patna University, Patna (Sah, 2015). A one parameter Poisson-Mishra distribution (PMD) of B. K. Sah (2017) has been obtained by compounding Poisson distribution with Mishra distribution. It has been found that this distribution gives better fit to all the discrete data sets which are negative binomial in nature used by Sankarn (1970) and others. A generalisation of PMD has been obtained by mixing the generalised Poisson distribution of Consul and Jain (1973) with the Mishra distribution.Materials and Methods: It is based on the concept of the generalisations of some continuous mixtures of Poisson distribution.Results: Probability density function and the first four moments about origin of the proposed distribution have been obtained. The estimation of parameters of this distribution has been discussed by using the first moment about origin and the probability mass function at x = 0 . This distribution has been fitted to a number of discrete data-sets to which earlier Poisson-Lindley distribution (PLD) and PMD have been fitted.Conclusion: P-value of generalised Poisson-Mishra distribution is greater than PLD and PMD. Hence, it provides a better alternative to the PLD of Sankarn and PMD of B. K. Sah.Nepalese Journal of Statistics, Vol. 2, 27-36
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Dzidzornu, S. K. B., and R. Minkah. "Assessing the Performance of the Discrete Generalised Pareto Distribution in Modelling Non-Life Insurance Claims." Journal of Probability and Statistics 2021 (June 11, 2021): 1–8. http://dx.doi.org/10.1155/2021/5518583.
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The generalised Pareto distribution (GPD) offers a family of probability spaces which support threshold exceedances and is thus suitable for modelling high-end actuarial risks. Nonetheless, its distributional continuity presents a critical limitation in characterising data of discrete forms. Discretising the GPD, therefore, yields a derived distribution which accommodates the count data while maintaining the essential tail modelling properties of the GPD. In this paper, we model non-life insurance claims under the three-parameter discrete generalised Pareto (DGP) distribution. Data for the study on reported and settled claims, spanning the period 2012–2016, were obtained from the National Insurance Commission, Ghana. The maximum likelihood estimation (MLE) principle was adopted in fitting the DGP to yearly and aggregated data. The estimation involved two steps. First, we propose a modification to the μ and μ + 1 frequency method in the literature. The proposal provides an alternative routine for generating initial estimators for MLE, in cases of varied count intervals, as is a characteristic of the claim data under study. Second, a bootstrap algorithm is implemented to obtain standard errors of estimators of the DGP parameters. The performance of the DGP is compared to the negative binomial distribution in modelling the claim data using the Akaike and Bayesian information criteria. The results show that the DGP is appropriate for modelling the count of non-life insurance claims and provides a better fit to the regulatory claim data considered.
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Lan, Chang-Jen, and Patricia S. Hu. "Mixed Generalized Linear Model for Estimating Household Trip Production." Transportation Research Record: Journal of the Transportation Research Board 1718, no. 1 (January 2000): 61–67. http://dx.doi.org/10.3141/1718-08.
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An innovative modeling framework to estimate household trip rates using 1995 Nationwide Personal Transportation Survey data is presented. A generalized linear model with a mixture of negative binomial probability distribution functions was developed on the basis of characteristics observed from the empirical distribution of household daily trips. This model provides a more flexible framework and a better model specification for analyzing household-specific trip production behavior. Compared with traditional least squares-based regression models, the parameter estimates from the proposed model are more efficient. Although the mean accuracies from the two modeling approaches are comparable, the mixed generalized linear model is more robust in identifying outliers due to its unsymmetric prediction bounds derived from more correct model specification.
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Endo, Akira, Sam Abbott, Adam J. Kucharski, and Sebastian Funk. "Estimating the overdispersion in COVID-19 transmission using outbreak sizes outside China." Wellcome Open Research 5 (April 9, 2020): 67. http://dx.doi.org/10.12688/wellcomeopenres.15842.1.
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Background: A novel coronavirus disease (COVID-19) outbreak has now spread to a number of countries worldwide. While sustained transmission chains of human-to-human transmission suggest high basic reproduction number R0, variation in the number of secondary transmissions (often characterised by so-called superspreading events) may be large as some countries have observed fewer local transmissions than others. Methods: We quantified individual-level variation in COVID-19 transmission by applying a mathematical model to observed outbreak sizes in affected countries. We extracted the number of imported and local cases in the affected countries from the World Health Organization situation report and applied a branching process model where the number of secondary transmissions was assumed to follow a negative-binomial distribution. Results: Our model suggested a high degree of individual-level variation in the transmission of COVID-19. Within the current consensus range of R0 (2-3), the overdispersion parameter k of a negative-binomial distribution was estimated to be around 0.1 (median estimate 0.1; 95% CrI: 0.05-0.2 for R0 = 2.5), suggesting that 80% of secondary transmissions may have been caused by a small fraction of infectious individuals (~10%). A joint estimation yielded likely ranges for R0 and k (95% CrIs: R0 1.4-12; k 0.04-0.2); however, the upper bound of R0 was not well informed by the model and data, which did not notably differ from that of the prior distribution. Conclusions: Our finding of a highly-overdispersed offspring distribution highlights a potential benefit to focusing intervention efforts on superspreading. As most infected individuals do not contribute to the expansion of an epidemic, the effective reproduction number could be drastically reduced by preventing relatively rare superspreading events.
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Endo, Akira, Sam Abbott, Adam J. Kucharski, and Sebastian Funk. "Estimating the overdispersion in COVID-19 transmission using outbreak sizes outside China." Wellcome Open Research 5 (July 3, 2020): 67. http://dx.doi.org/10.12688/wellcomeopenres.15842.2.
Full textAbstract:
Background: A novel coronavirus disease (COVID-19) outbreak has now spread to a number of countries worldwide. While sustained transmission chains of human-to-human transmission suggest high basic reproduction number R0, variation in the number of secondary transmissions (often characterised by so-called superspreading events) may be large as some countries have observed fewer local transmissions than others. Methods: We quantified individual-level variation in COVID-19 transmission by applying a mathematical model to observed outbreak sizes in affected countries. We extracted the number of imported and local cases in the affected countries from the World Health Organization situation report and applied a branching process model where the number of secondary transmissions was assumed to follow a negative-binomial distribution. Results: Our model suggested a high degree of individual-level variation in the transmission of COVID-19. Within the current consensus range of R0 (2-3), the overdispersion parameter k of a negative-binomial distribution was estimated to be around 0.1 (median estimate 0.1; 95% CrI: 0.05-0.2 for R0 = 2.5), suggesting that 80% of secondary transmissions may have been caused by a small fraction of infectious individuals (~10%). A joint estimation yielded likely ranges for R0 and k (95% CrIs: R0 1.4-12; k 0.04-0.2); however, the upper bound of R0 was not well informed by the model and data, which did not notably differ from that of the prior distribution. Conclusions: Our finding of a highly-overdispersed offspring distribution highlights a potential benefit to focusing intervention efforts on superspreading. As most infected individuals do not contribute to the expansion of an epidemic, the effective reproduction number could be drastically reduced by preventing relatively rare superspreading events.
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Endo, Akira, Sam Abbott, Adam J. Kucharski, and Sebastian Funk. "Estimating the overdispersion in COVID-19 transmission using outbreak sizes outside China." Wellcome Open Research 5 (July 10, 2020): 67. http://dx.doi.org/10.12688/wellcomeopenres.15842.3.
Full textAbstract:
Background: A novel coronavirus disease (COVID-19) outbreak has now spread to a number of countries worldwide. While sustained transmission chains of human-to-human transmission suggest high basic reproduction number R0, variation in the number of secondary transmissions (often characterised by so-called superspreading events) may be large as some countries have observed fewer local transmissions than others. Methods: We quantified individual-level variation in COVID-19 transmission by applying a mathematical model to observed outbreak sizes in affected countries. We extracted the number of imported and local cases in the affected countries from the World Health Organization situation report and applied a branching process model where the number of secondary transmissions was assumed to follow a negative-binomial distribution. Results: Our model suggested a high degree of individual-level variation in the transmission of COVID-19. Within the current consensus range of R0 (2-3), the overdispersion parameter k of a negative-binomial distribution was estimated to be around 0.1 (median estimate 0.1; 95% CrI: 0.05-0.2 for R0 = 2.5), suggesting that 80% of secondary transmissions may have been caused by a small fraction of infectious individuals (~10%). A joint estimation yielded likely ranges for R0 and k (95% CrIs: R0 1.4-12; k 0.04-0.2); however, the upper bound of R0 was not well informed by the model and data, which did not notably differ from that of the prior distribution. Conclusions: Our finding of a highly-overdispersed offspring distribution highlights a potential benefit to focusing intervention efforts on superspreading. As most infected individuals do not contribute to the expansion of an epidemic, the effective reproduction number could be drastically reduced by preventing relatively rare superspreading events.
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Albers, Willem. "Risk-Adjusted Control Charts for Health Care Monitoring." International Journal of Mathematics and Mathematical Sciences 2011 (2011): 1–16. http://dx.doi.org/10.1155/2011/895273.
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Attribute data from high-quality processes can be monitored effectively by deciding on whether or not to stop at each time where failures have occurred. The smaller the degree of change in failure rate during out of control one wants to be optimally protected against, the larger thershould be. Under homogeneity, the distribution involved is negative binomial. However, in health care monitoring, (groups of) patients will often belong to different risk categories. In the present paper, we will show how information about category membership can be used to adjust the basic negative binomial charts to the actual risk incurred. Attention is also devoted to comparing such conditional charts to their unconditional counterparts. The latter do take possible heterogeneity into account but refrain from risk-adjustment. Note that in the risk adjusted case several parameters are involved, which will all be typically unknown. Hence, the potentially considerable estimation effects of the new charts will be investigated as well.
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Rezapour, Mahdi, and Khaled Ksaibati. "Comprehensive Evaluation of a Sparse Dataset, Assessment and Selection of Competing Models." Signals 1, no. 2 (November 3, 2020): 157–69. http://dx.doi.org/10.3390/signals1020009.
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With tremendous associated economic and social costs of crashes, researchers have been trying not only to identify the factors affecting crashes, but also to estimate those coefficients in the most accurate ways. Estimating model coefficients without accounting for a correct distribution would result in biased and erroneous results. This risk especially holds true when modeling skewed equivalent property damage only (EPDO) crashes with a preponderance of zeroes. The distribution of EPDO is known for not being modeled with known distributions such as Poisson or negative binomial distributions. This issue is highlighted in particular for a mountainous state like Wyoming that has very low traffic levels and a severely high crash rate. In addition, we included barriers in the model that did not experience any crashes but did suffer from being under-designed by geometric architects, thereby adding to the number of zero count observations. Various models with different distributional characteristics were considered and compared in this study. Comparisons were not just made across models in terms of their goodness of fit, but the estimated coefficients were also compared to see the impact of considering the wrong distributional assumptions on model parameter estimates. As the objectives of this study are to implement the identified results for optimization purposes and locate hazardous locations that could host future crashes, the results highlight accurate model estimations and the consequences of a failure to account for the right distributions. After conducting different goodness-of-fit measures, a hurdle model was proposed in this study to accommodate observations with zero crashes, and to account for a sparse distribution of EPDO crashes in the state of Wyoming. For the hurdle model, binary logistic regression was used to account for observations with zero crashes, while the negative binomial method was considered for non-zero observations. The findings of this study have direct implications on the allocation of limited funds for policymakers in Wyoming, as optimization could be conducted on the geometric characteristics of traffic barriers in future studies.
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Gertsbakh, Ilya B. "Mixed Vehicle Flow At Signalized Intersection: Markov Chain Analysis." Transport and Telecommunication Journal 16, no. 3 (September 1, 2015): 190–96. http://dx.doi.org/10.1515/ttj-2015-0017.
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Abstract We assume that a Poisson flow of vehicles arrives at isolated signalized intersection, and each vehicle, independently of others, represents a random number X of passenger car units (PCU’s). We analyze numerically the stationary distribution of the queue process {Zn}, where Zn is the number of PCU’s in a queue at the beginning of the n-th red phase, n → ∞. We approximate the number Yn of PCU’s arriving during one red-green cycle by a two-parameter Negative Binomial Distribution (NBD). The well-known fact is that {Zn} follow an infinite-state Markov chain. We approximate its stationary distribution using a finite-state Markov chain. We show numerically that there is a strong dependence of the mean queue length E[Zn] in equilibrium on the input distribution of Yn and, in particular, on the ”over dispersion” parameter γ= Var[Yn]/E[Yn]. For Poisson input, γ = 1. γ > 1 indicates presence of heavy-tailed input. In reality it means that a relatively large ”portion” of PCU’s, considerably exceeding the average, may arrive with high probability during one red-green cycle. Empirical formulas are presented for an accurate estimation of mean queue length as a function of load and g of the input flow. Using the Markov chain technique, we analyze the mean ”virtual” delay time for a car which always arrives at the beginning of the red phase.
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Almazah, Mohammed Mohammed Ahmed, Badr Alnssyan, Abdul Hadi N. Ahmed, and Ahmed Z. Afify. "Reliability Properties of the NDL Family of Discrete Distributions with Its Inference." Mathematics 9, no. 10 (May 18, 2021): 1139. http://dx.doi.org/10.3390/math9101139.
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The natural discrete Lindley (NDL) distribution is an intuitive idea that uses discrete analogs to well-known continuous distributions rather than using any of the published discretization techniques. The NDL is a flexible extension of both the geometric and the negative binomial distributions. In the present article, we further investigate new results of value in the areas of both theoretical and applied reliability. To be specific, several closure properties of the NDL are proved. Among the results, sufficient conditions that maintain the preservation properties under useful partial orderings, convolution, and random sum of random variables are introduced. Eight different methods of estimation, including the maximum likelihood, least squares, weighted least squares, Cramér–von Mises, the maximum product of spacing, Anderson–Darling, right-tail Anderson–Darling, and percentiles, have been used to estimate the parameter of interest. The performance of these estimators has been evaluated through extensive simulation. We have also demonstrated two applications of NDL in modeling real-life problems, including count data. It is worth noting that almost all the methods have resulted in very satisfactory estimates on both simulated and real-world data.
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Geedipally, Srinivas Reddy, and Dominique Lord. "Effects of Varying Dispersion Parameter of Poisson–Gamma Models on Estimation of Confidence Intervals of Crash Prediction Models." Transportation Research Record: Journal of the Transportation Research Board 2061, no. 1 (January 2008): 46–54. http://dx.doi.org/10.3141/2061-06.
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In estimating safety performance, the most common probabilistic structures of the popular statistical models used by transportation safety analysts for modeling motor vehicle crashes are the traditional Poisson and Poisson–gamma (or negative binomial) distributions. Because crash data often exhibit overdispersion, Poisson–gamma models are usually the preferred model. The dispersion parameter of Poisson–gamma models had been assumed to be fixed, but recent research in highway safety has shown that the parameter can potentially be dependent on the covari-ates, especially for flow-only models. Given that the dispersion parameter is a key variable for computing confidence intervals, there is reason to believe that a varying dispersion parameter could affect the computation of confidence intervals compared with confidence intervals produced from Poisson–gamma models with a fixed dispersion parameter. This study evaluates whether the varying dispersion parameter affects the computation of the confidence intervals for the gamma mean (m) and predicted response (y) on sites that have not been used for estimating the predictive model. To accomplish that objective, predictive models with fixed and varying dispersion parameters were estimated by using data collected in California at 537 three-leg rural unsignalized intersections. The study shows that models developed with a varying dispersion parameter greatly influence the confidence intervals of the gamma mean and predictive response. More specifically, models with a varying dispersion parameter usually produce smaller confidence intervals, and hence more precise estimates, than models with a fixed dispersion parameter, both for the gamma mean and for the predicted response. Therefore, it is recommended to develop models with a varying dispersion whenever possible, especially if they are used for screening purposes.
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Caicedo R., Luis Sigifredo, Edgar Herney Varón D., and Helena Luisa Brochero. "Binomial sampling of Paraleyrodes Quaintance pos. bondari (Hemiptera: Aleyrodidae) in Persea americana Mill." Agronomía Colombiana 34, no. 2 (May 1, 2016): 209–16. http://dx.doi.org/10.15446/agron.colomb.v34n2.54084.
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Fresno (Tolima), in Colombia, is a notable avocado producer, with 36% of the national production. In this paper, two sampling methods are presented to assess natural populations of Paraleyrodes Quaintance pos. bondari attacking avocado trees of Hass and Lorena cultivars under field conditions. The presence/absence of whitefly nymph colonies on 30 leaves located at the high, medium and low strata per host plant from both cultivars was evaluated. Visual estimations were performed to count the number ofwhitefly nymphs on 1.25 cm2 of five leaves/ bud in low and medium strata per tree to evaluate the spatial distribution of whitefly population in accordance to Poisson distribution, Negative Binomial distribution and b parameter of Law of Taylor. Significant differences in percentages of infestation (P≤0.03) from leaves that belonged to the low avocado tree strata were found between the Lorena (31.88±1.2%) and Hass (15.64±1.8%) cultivars. Natural populations of P. pos. bondari were located on the abaxial leaf side, showing an aggregate distribution in avocado tree from orchards located at different altitudes. Our findings recommend entomological surveillance for Paraleyrodes sp. pos. bondari in Fresno (Tolima), sampling four branches from the medium and low avocado tree strata through inspection of five buds/branches/tree throughout each branch with the presence/absence method to count whitefly nymph colonies on the abaxial side of pre-basal leaves. In total, the sampling involved five leaves/branch (20 leaves/strata or 40 leaves/tree) on 13 avocado trees per hectare.