Small Sample Properties of the Pareto/Negative Binomial Distribution Model

Marketing ZFP ◽  
2010 ◽  
Vol 32 (JRM 1) ◽  
pp. 39-50
Author(s):  
Daniel Hoppe ◽  
Udo Wagner
Keyword(s):  
Negative Binomial Distribution ◽  
Binomial Distribution ◽  
Negative Binomial ◽  
Small Sample ◽  
Distribution Model ◽  
Small Sample Properties

Modelling macroparasite aggregation using a nematode-sheep system: the Weibull distribution as an alternative to the Negative Binomial distribution?

Parasitology ◽  
2005 ◽  
Vol 131 (3) ◽  
pp. 393-401 ◽  
Author(s):  
S. GABA ◽  
V. GINOT ◽  
J. CABARET
Keyword(s):  
Weibull Distribution ◽  
Negative Binomial Distribution ◽  
Binomial Distribution ◽  
Negative Binomial ◽  
Information Criterion ◽  
Gastrointestinal Nematode ◽  
Small Sample ◽  
Parameter Estimations ◽  
Wide Range ◽  
Small Sample Sizes

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.

The Stability of Weed Seedling Population Models and Parameters in Eastern Nebraska Corn (Zea mays) and Soybean (Glycine max) Fields

Weed Science ◽  
1995 ◽  
Vol 43 (4) ◽  
pp. 604-611 ◽  
Author(s):  
Gregg A. Johnson ◽  
David A. Mortensen ◽  
Linda J. Young ◽  
Alex R. Martin
Keyword(s):  
Negative Binomial Distribution ◽  
Binomial Distribution ◽  
Negative Binomial ◽  
Sequential Sampling ◽  
Mean Value ◽  
Distribution Model ◽  
Ratio Test ◽  
Weed Species ◽  
Sampling Procedures ◽  
Grass Weed

Intensive field surveys were conducted in eastern Nebraska to determine the frequency distribution model and associated parameters of broadleaf and grass weed seedling populations. The negative binomial distribution consistently fit the data over time (1992 to 1993) and space (fields) for both the inter and intrarow broadleaf and grass weed seedling populations. The other distributions tested (Poisson with zeros, Neyman type A, logarithmic with zeros, and Poisson-binomial) did not fit the data as consistently as the negative binomial distribution. Associated with the negative binomial distribution is akparameter.kis a nonspatial aggregation parameter related to the variance at a given mean value. Thekparameter of the negative binomial distribution was consistent across weed density for individual weed species in a given field except for foxtail spp. populations. Stability of thekparameter across field sites was assessed using the likelihood ratio test There was no stable or commonkvalue across field sites and years for all weed species populations. The lack of stability inkacross field sites is of concern, because this parameter is used extensively in the development of parametric sequential sampling procedures. Becausekis not stable across field sites,kmust be estimated at the time of sampling. Understanding the variability in it is critical to the development of parametric sequential sampling strategies and understanding the dynamics of weed species in the field.

scholarly journals Spatial Distribution of Eggs of Alabama argillacea Hübner and Heliothis virescens Fabricius (Lepidoptera: Noctuidae) on Bt and non-BtCotton

2015 ◽  
Vol 87 (4) ◽  
pp. 2243-2253
Author(s):  
TATIANA R. RODRIGUES ◽  
MARCOS G. FERNANDES ◽  
PAULO E. DEGRANDE ◽  
THIAGO A. MOTA
Keyword(s):  
Spatial Distribution ◽  
Negative Binomial Distribution ◽  
Binomial Distribution ◽  
Heliothis Virescens ◽  
Negative Binomial ◽  
Model Fitting ◽  
Spatial Arrangement ◽  
Distribution Model ◽  
Aggregation Index ◽  
Alabama Argillacea

ABSTRACT Among the options to control Alabama argillacea (Hübner, 1818) and Heliothis virescens (Fabricius, 1781) on cotton, insecticide spraying and biological control have been extensively used. The GM'Bt' cotton has been introduced as an extremely viable alternative, but it is yet not known how transgenic plants affect populations of organisms that are interrelated in an agroecosystem. For this reason, it is important to know how the spatial arrangement of pests and beneficial insect are affected, which may call for changes in the methods used for sampling these species. This study was conducted with the goal to investigate the pattern of spatial distribution of eggs of A. argillacea and H. virescens in DeltaOpalTM (non-Bt) and DP90BTMBt cotton cultivars. Data were collected during the agricultural year 2006/2007 in two areas of 5,000 m2, located in in the district of Nova América, Caarapó municipality. In each sampling area, comprising 100 plots of 50 m2, 15 evaluations were performed on two plants per plot. The sampling consisted in counting the eggs. The aggregation index (variance/mean ratio, Morisita index and exponent k of the negative binomial distribution) and chi-square fit of the observed and expected values to the theoretical frequency distribution (Poisson, Binomial and Negative Binomial Positive), showed that in both cultivars, the eggs of these species are distributed according to the aggregate distribution model, fitting the pattern of negative binomial distribution.

scholarly journals Use of a negative binomial distribution to describe the presence of Sphyrion laevigatum in Genypterus blacodes

2013 ◽  
Vol 22 (4) ◽  
pp. 602-604 ◽  
Author(s):  
Patricio Peña-Rehbein ◽  
Patricio De los Ríos-Escalante ◽  
Raúl Castro ◽  
Carolina Navarrete
Keyword(s):  
Spatial Distribution ◽  
Negative Binomial Distribution ◽  
Binomial Distribution ◽  
Negative Binomial ◽  
Distribution Model ◽  
Economic Resource ◽  
Spatial Distribution Model

This paper describes the frequency and number of Sphyrion laevigatum in the skin of Genypterus blacodes, an important economic resource in Chile. The analysis of a spatial distribution model indicated that the parasites tended to cluster. Variations in the number of parasites per host could be described by a negative binomial distribution. The maximum number of parasites observed per host was two.

scholarly journals Testing the odds of inherent vs. observed overdispersion in neural spike counts

2016 ◽  
Vol 115 (1) ◽  
pp. 434-444 ◽  
Author(s):  
Wahiba Taouali ◽  
Giacomo Benvenuti ◽  
Pascal Wallisch ◽  
Frédéric Chavane ◽  
Laurent U. Perrinet
Keyword(s):  
Poisson Process ◽  
Negative Binomial Distribution ◽  
Binomial Distribution ◽  
Negative Binomial ◽  
Poisson Model ◽  
Probabilistic Methods ◽  
Distribution Model ◽  
Data Sets ◽  
Electrophysiological Recordings ◽  
Neurophysiological Data

The repeated presentation of an identical visual stimulus in the receptive field of a neuron may evoke different spiking patterns at each trial. Probabilistic methods are essential to understand the functional role of this variance within the neural activity. In that case, a Poisson process is the most common model of trial-to-trial variability. For a Poisson process, the variance of the spike count is constrained to be equal to the mean, irrespective of the duration of measurements. Numerous studies have shown that this relationship does not generally hold. Specifically, a majority of electrophysiological recordings show an “overdispersion” effect: responses that exhibit more intertrial variability than expected from a Poisson process alone. A model that is particularly well suited to quantify overdispersion is the Negative-Binomial distribution model. This model is well-studied and widely used but has only recently been applied to neuroscience. In this article, we address three main issues. First, we describe how the Negative-Binomial distribution provides a model apt to account for overdispersed spike counts. Second, we quantify the significance of this model for any neurophysiological data by proposing a statistical test, which quantifies the odds that overdispersion could be due to the limited number of repetitions (trials). We apply this test to three neurophysiological data sets along the visual pathway. Finally, we compare the performance of this model to the Poisson model on a population decoding task. We show that the decoding accuracy is improved when accounting for overdispersion, especially under the hypothesis of tuned overdispersion.

scholarly journals Objective Bayesian analysis of the 2 x 2 contingency table and the negative binomial distribution

2018 ◽  
Author(s):  
◽  
John Christian Snyder
Keyword(s):  
Bayesian Analysis ◽  
Negative Binomial Distribution ◽  
Binomial Distribution ◽  
Odds Ratio ◽  
Negative Binomial ◽  
Real Data ◽  
R Package ◽  
Small Sample ◽  
Reference Prior ◽  
Categorical Data Analysis

In Bayesian analysis, the “objective” Bayesian approach seeks to select a prior distribution not by using (often subjective) scientific belief or by mathematical convenience, but rather by deriving it under a pre-specified criteria. This approach takes the decision of prior selection out of the hands of the researcher. Ideally, for a given data model, we would like to have a prior which represents a "neutral" prior belief in the phenomenon we are studying. In categorical data analysis, the odds ratio is one of several approaches to quantify how strongly the presence or absence of one property is associated with the presence or absence of another property. In this project, we present a Reference prior for the odds ratio of an unrestricted 2 x 2 table. Posterior simulation can be conducted without MCMC and is implemented on a GPU via the CUDA extensions for C. Simulation results indicate that the proposed approach to this problem is far superior to the widely used Frequentist approaches that dominate this area. Real data examples also typically yield much more sensible results, especially for small sample sizes or for tables that contain zeros. An R package is also presented to allow for easy implementation of this methodology. Next, we develop an approximate reference prior for the negative binomial distribution, applying this methodology to a continuous parameterization often used for modeling over-dispersed count data as well as the typical discrete case. Results indicate that the developed prior equals the performance of the MLE in estimating the mean of the distribution but is far superior when estimating the dispersion parameter.

Niche Sharing in Intertidal Mollusks and Decapods in Rocky Shore of Easter Island

2019 ◽  
Vol 53 (5) ◽  
pp. 417-422
Author(s):  
P. De los Ríos ◽  
E. Ibáñez Arancibia
Keyword(s):  
Spatial Distribution ◽  
Negative Binomial Distribution ◽  
Binomial Distribution ◽  
Rocky Shore ◽  
Negative Binomial ◽  
Null Model ◽  
Easter Island ◽  
Published Data ◽  
Distribution Analysis ◽  
Field Works

Abstract The coastal marine ecosystems in Easter Island have been poorly studied, and the main studies were isolated species records based on scientific expeditions. The aim of the present study is to apply a spatial distribution analysis and niche sharing null model in published data on intertidal marine gastropods and decapods in rocky shore in Easter Island based in field works in 2010, and published information from CIMAR cruiser in 2004. The field data revealed the presence of decapods Planes minutus (Linnaeus, 1758) and Leptograpsus variegatus (Fabricius, 1793), whereas it was observed the gastropods Nodilittorina pyramidalis pascua Rosewater, 1970 and Nerita morio (G. B. Sowerby I., 1833). The available information revealed the presence of more species in data collected in 2004 in comparison to data collected in 2010, with one species markedly dominant in comparison to the other species. The spatial distribution of species reported in field works revealed that P. minutus and N. morio have aggregated pattern and negative binomial distribution, L. variegatus had uniform pattern with binomial distribution, and finally N. pyramidalis pascua, in spite of aggregated distribution pattern, had not negative binomial distribution. Finally, the results of null model revealed that the species reported did not share ecological niche due to competition absence. The results would agree with other similar information about littoral and sub-littoral fauna for Easter Island.

Sign in / Sign up

Export Citation Format

Share Document