MultinomialN‐mixture models for removal sampling

If X 1, X 2, … is a sequence of independent Bernoulli random variables, the number of successes in the first n trials has a binomial distribution and the number of failures before the rth success has a negative binomial distribution. From both the binomial and the negative binomial distributions, the Poisson distribution is obtainable as a limit. Moreover, gamma distributions (integer shape parameters) are limits of negative binomial distributions, and the normal distribution is a limit of negative binomial, Poisson, and gamma distributions. These basic facts from elementary probability have natural extensions to two dimensions because there is a unique natural bivariate Bernoulli distribution. In this article, such extensions yielding a family of bivariate distributions are obtained and studied.

The focus of this article is on the nature of the likelihood associated with N-mixture models for repeated count data. It is shown that the infinite sum embedded in the likelihood associated with the Poisson mixing distribution can be expressed in terms of a hypergeometric function and, thence, in closed form. The resultant expression for the likelihood can be readily computed to a high degree of accuracy and is algebraically tractable. Specifically, the likelihood equations can be simplified to some advantage, the concentrated likelihood in the probability of detection formulated and problematic cases identified. The results are illustrated by means of a simulation study and a real world example. The study is extended to N-mixture models with a negative binomial mixing distribution and results similar to those for the Poisson case obtained. N-mixture models with mixing distributions which accommodate excess zeros and, separately, with a beta-binomial distribution rather than a binomial used to model the intra-site counts are also investigated. However the results for these settings, while computationally attractive, do not provide insight into the nature of the maximum likelihood estimates.

This paper introduced a new mixed distribution, namely the negative binomial-generalized exponential (NB-GE) distribution, which it obtained by mixing the negative binomial (NB) distribution with a generalized exponential (GE) distribution. The NB-GE distribution offers the advantage of being able to handle this kind of data sets, while still maintaining similar characteristics as the traditional negative binomial. The closed form and the factorial moment of the NB-GE distribution are derived. The NB-GE distribution appropriate when we find the one related to count data which have a large number of zeros and overdispersion in Poisson distribution. In addition, we present the basic properties of the new distribution such as mean, variance, skewness and kurtosis. Including, the negative binomial-exponential distributions are presented as special cases of this NB-GE distribution. Parameters estimation is also implemented using maximum likelihood method and the usefulness of the NB-GE distribution is illustrated by real data set. When using a goodness of fit test, the results show that the NB-GE distribution can provide a better fit than the NB distribution and Poisson distribution for count data that contain a large number of zeros.

Negative binomial group distribution was proposed in the literature which was motivated by inverse sampling when considering group inspection: products are inspected group by group, and the number of non-conforming items of a group is recorded only until the inspection of the whole group is finished. The non-conforming probability p of the population is thus the parameter of interest. In this paper, the confidence interval construction for this parameter is investigated. The common normal approximation and exact method are applied. To overcome the drawbacks of these commonly used methods, a composite method that is based on the confidence intervals of the negative binomial distribution is proposed, which benefits from the relationship between negative binomial distribution and negative binomial group distribution. Simulation studies are carried out to examine the performances of our methods. A real data example is also presented to illustrate the application of our method.

The question of how to characterize the bacterial density in a body of water when data are available as counts from a number of small-volume samples was examined for cases where either the Poisson or negative binomial probability distributions could be used to describe the bacteriological data. The suitability of the Poisson distribution when replicate analyses were performed under carefully controlled conditions and of the negative binomial distribution for samples collected from different locations and over time were illustrated by two examples. In cases where the negative binomial distribution was appropriate, a procedure was given for characterizing the variability by dividing the bacterial counts into homogeneous groups. The usefulness of this procedure was illustrated for the second example based on survey data for Lake Erie. A further illustration of the difference between results based on the Poisson and negative binomial distributions was given by calculating the probability of obtaining all samples sterile, assuming various bacterial densities and sample sizes.

There are several parametric models to describe the charged particle multiplicity distributions in literature. Among them, log-normal distribution and negative binomial distribution are the most well-known parametric models. In this study, negative binomial and log-normal distributions are compared and tested in neutrino interactions produced in DONUT experiment. This analysis was carried out using the real experimental results of the DONUT Collaboration. The results of the analysis show that the neutrino data, reported by DONUT collaboration, is well described by negative binomial and log-normal distribution. In terms of fit parameters, the χ/ndf value of the both distribution NBD and LND are close to unity but log-normal distribution data slightly better than negative binomial distribution.

Methodologic Challenges in the Analysis of Count Data in Radiology Health Services Research

Estimating Retail Demand with Poisson Mixtures and Out-of-Sample Likelihood

We model the interaction between a qubit and optical radiation field (ORF) using an excited binomial distribution (EBD) and excited negative binomial distribution (ENBD). The explicit form of the density matrix of the qubit–ORF system is given in terms of the photon number distribution. The Mandel parameter is used to quantify the statistical properties of the ORF during the interaction. We derive the relation between the tomographic entropy and the quantum Fisher information of the estimators of model parameters in the EBD and ENBD. We also explore the properties and dynamics of these statistical quantities.

Measures of absolute animal abundance may be estimated by capture-recapture, removal, or distance sampling methods. We investigate the usage of binomial mixture models to estimate local group abundance of major prey species that is frequently used as a surrogate for prey abundance to study predator or prey-mediated ecological interactions such as predator-prey relationships. We evaluate mixture models using data from a camera-trapping survey intended for a tiger Panthera tigris census in Chitwan National Park, Nepal, where the entire park was surveyed in 361 4-km2 quadrats. We chose four prey species (chital Axis axis, sambar Rusa unicolor, muntjac Muntiacus muntjac, and wild boar Sus scrofa) that collectively account for > 75% of prey biomass consumed by tigers. Abundance of prey group was modeled as a random variable with a Poisson or a negative binomial distribution, with the mean abundance affected by distance from water sources, elevation, and normalized difference vegetation index (NDVI). Except for wild boar, the top models for all other species included the hypothesized covariates while the null model was the most parsimonious model for the wild boar. The most parsimonious chital model included effects of distance from water sources (−) and elevation (−). The sambar model supported all three covariates: distance from water sources (−), elevation (+), and NDVI (+). Only distance from water sources (−) was supported by the most parsimonious muntjac model. Our abundance estimates also conformed to the results obtained from recently conducted labor-intensive distance sampling procedure. We conclude that camera-trapping survey data can be effectively utilized adopting the binomial mixture model framework to understand animal abundance-habitat relationships and estimate abundance of animal that are not identifiable individually.

In this paper we examine estimation in a mixture model under various assumptions for the underlying distribution. We suggest that statistical inference should be carried out using the conditional distribution given the appropriate ancillary statistics. The conditional likelihood approach used here can be justified by Cox's (1958) criterion about ancillarity in the presence of a nuisance parameter, and in the spirit of Godambe's (1976) conditional likelihood optimum estimating equation. The analysis is known to be easier and more stable than the unconditional likelihood function approach. The probability distributions in the analysis of count data discussed here include Poisson, binomial, geometric and negative binomial distributions. The information loss incurred by using the conditional approach is discussed

Background: In multiple sclerosis (MS) studies, the most appropriate model for the distribution of the number of relapses was shown to be the negative binomial (NB) distribution. Objective: To determine whether the sample-size estimation (SSE) and the analysis of annualized relapse rates (ARRs) in randomized controlled trials (RCTs) were aligned and compare the SSE between normal and NB distributions. Methods: Systematic review of phase 3 and 4 RCTs for which the primary endpoint was ARR in relapsing remitting MS published since 2008 in pre-selected major medical journals. A PubMed search was performed on 30 November 2020. We checked whether the SSE and ARR analyses were congruent. We also performed standardized (fixed α/β, number of arms and overdispersion) SSEs using data collected from the studies. Results: Twenty articles (22 studies) were selected. NB distribution (or quasi-Poisson) was used for SSE in only 7/22 studies, whereas 21/22 used it for ARR analyses. SSE relying on NB regression necessitated a smaller sample size in 21/22 of our calculations. Conclusion: SSE was rarely performed using the most appropriate model. However, the use of an NB model is recommended to optimize the number of included patients and to be congruent with the final analysis.

The foliage in a fully-leafed deciduous forest canopy is clumped. Consequently, theory indicates that the probability of beam penetration will be estimated more accurately with a model based on the negative binomial distribution than with a model based on the Poisson distribution, incorporating an assumption of a spherical leaf inclination angle distribution. Flux densities of photosynthetically active radiation (PAR) were measured in and above a deciduous forest canopy and were computed with the canopy radiative transfer models based on the negative binomial and Poisson distributions. These radiation values were used to compute canopy photosynthesis and stomatal conductance. Canopy photosynthesis and stomatal conductance, based on the negative binomial model, overestimated values computed from measured PAR profiles by 8 and 9%. respectively. The canopy photosynthesis and stomatal conductance values computed with the spherical Poisson model under-estimated measured values by 17 and 10%, respectively. Thus, the negative binomial radiative transfer model improves estimates of canopy photosynthesis and, to a lesser extent, stomatal conductance, inside a deciduous forest.

BackgroundRNA sequencing (RNA-seq) is a powerful tool for genome-wide expression profiling of biological samples with the advantage of high-throughput and high resolution. There are many existing algorithms nowadays for quantifying expression levels and detecting differential gene expression, but none of them takes the misaligned reads that are mapped to non-exonic regions into account. We developed a novel algorithm, XBSeq, where a statistical model was established based on the assumption that observed signals are the convolution of true expression signals and sequencing noises. The mapped reads in non-exonic regions are considered as sequencing noises, which follows a Poisson distribution. Given measureable observed and noise signals from RNA-seq data, true expression signals, assuming governed by the negative binomial distribution, can be delineated and thus the accurate detection of differential expressed genes.ResultsWe implemented our novel XBSeq algorithm and evaluated it by using a set of simulated expression datasets under different conditions, using a combination of negative binomial and Poisson distributions with parameters derived from real RNA-seq data. We compared the performance of our method with other commonly used differential expression analysis algorithms. We also evaluated the changes in true and false positive rates with variations in biological replicates, differential fold changes, and expression levels in non-exonic regions. We also tested the algorithm on a set of real RNA-seq data where the common and different detection results from different algorithms were reported.ConclusionsIn this paper, we proposed a novel XBSeq, a differential expression analysis algorithm for RNA-seq data that takes non-exonic mapped reads into consideration. When background noise is at baseline level, the performance of XBSeq and DESeq are mostly equivalent. However, our method surpasses DESeq and other algorithms with the increase of non-exonic mapped reads. Only in very low read count condition XBSeq had a slightly higher false discovery rate, which may be improved by adjusting the background noise effect in this situation. Taken together, by considering non-exonic mapped reads, XBSeq can provide accurate expression measurement and thus detect differential expressed genes even in noisy conditions.

A generalized negative binomial (GNB) distribution with an additional parameter $\beta $ has been obtained by using Lagrange’s expansion. The parameter is such that both mean and variance tend to increase or decrease with an increase or decrease in its value but the variance increases or decreases faster than the mean. For $\beta = \frac{1} {2}$, the mean and variance are approximately equal and so the GNB distribution resembles the Poisson distribution. When $\beta = 0$ or 1, the GNB distribution reduces to the binomial or negative binomial distribution respectively. It has been shown that the generalized negative binomial distribution converges to a Poisson-type distribution in which the variance may be more or less than the mean, depending upon the value of a parameter. Expected frequencies have been calculated for a number of examples to show that the distribution provides a very satisfactory fit in different practical situations. Its convolution property together with other properties are quite inter...