Negative Multinomial Distribution Research Articles

Asymptotic comparison of negative multinomial and multivariate normal experiments

This note presents a refined local approximation for the logarithm of the ratio between the negative multinomial probability mass function and a multivariate normal density, both having the same mean–covariance structure. This approximation, which is derived using Stirling's formula and a meticulous treatment of Taylor expansions, yields an upper bound on the Hellinger distance between the jittered negative multinomial distribution and the corresponding multivariate normal distribution. Upper bounds on the Le Cam distance between negative multinomial and multivariate normal experiments ensue.

Moments of the Negative Multinomial Distribution

The negative multinomial distribution appears in many areas of applications such as polarimetric image processing and the analysis of longitudinal count data. In previous studies, general formulas for the falling factorial moments and cumulants of the negative multinomial distribution were obtained. However, despite the availability of the moment generating function, no comprehensive formulas for the moments have been calculated thus far. This paper addresses this gap by presenting general formulas for both central and non-central moments of the negative multinomial distribution. These formulas are expressed in terms of binomial coefficients and Stirling numbers of the second kind. Utilizing these formulas, we provide explicit expressions for all central moments up to the fourth order and all non-central moments up to the eighth order.

Economizing comparative Poisson clinical trials of multiple experimental treatments by testing against the same control.

Comparative Poisson trials of an experimental treatment versus a control typically condition on the total number of events that occur across both arms (Design A). Inference is based on the binomial distribution. Recently, an approach termed Design C to compare K experimental treatments to the same control was introduced. Under Design C without curtailment, the trial continues until a prespecified number of events occur in the control arm, leading to inference based on the negative multinomial distribution. The question remains of how advantageous it is to conduct one Design C trial comparing K experimental treatment arms to the same control arm as opposed to conducting K separate Design A trials each comparing one experimental treatment arm to a different control arm. This paper, therefore, compares the expected number of subjects to enroll for the two designs under uncurtailed and curtailed settings. The designs are evaluated when the null hypothesis and various assumptions for the alternative hypothesis hold. We simulate a variety of combinations for the Type 1 error, power, and ratio of the incidence rate of events in the experimental treatment to control arms. Design C frequently offers significant savings in terms of sample size relative to Design A.

Domain of existence of the Laplace transform of negative multinomial distributions and simulations

This article provides the domain of existence of the Laplace transform of infinitely divisible negative multinomial distributions, defined by their probability generating functions. It makes it possible to construct all of these distributions. We give examples in dimensions 2 and 3. We give a construction and a simulation of negative multinomial distributions in dimension 2.

Comparative poisson clinical trials of multiple experimental treatments vs a single control using the negative multinomial distribution.

This paper introduces a method which conditions on the number of events that occur in the control group to determine rejection regions and power for comparative Poisson trials with multiple experimental treatment arms that are each compared to one control arm. This leads to the negative multinomial as the statistical distribution used for testing. For one experimental treatment and one control with curtailed sampling, this is equivalent to Gail's (1974) approach. We provide formulas to calculate exact one-sided overall Type I error and pointwise power for tests of treatment superiority and inferiority (vs the control). Tables of trial design parameters for combinations of one-sided overall Type I error=0.05, 0.01 and pointwise power=0.90, 0.80 are provided. Curtailment approaches are presented to stop follow-up of experimental treatment arms or to stop the study entirely once the final outcomes for each arm are known.

Bayesian shrinkage estimation of negative multinomial parameter vectors

The negative multinomial distribution is a multivariate generalization of the negative binomial distribution. In this paper, we consider the problem of estimating an unknown matrix of probabilities on the basis of observations of negative multinomial variables under the standardized squared error loss. First, a general sufficient condition for a shrinkage estimator to dominate the UMVU estimator is derived and an empirical Bayes estimator satisfying the condition is constructed. Next, a hierarchical shrinkage prior is introduced, an associated Bayes estimator is shown to dominate the UMVU estimator under some conditions, and some remarks about posterior computation are presented. Finally, shrinkage estimators and the UMVU estimator are compared by simulation.

Application of a Negative Multinomial Model Gives Insight into Rarity-Area Relationships

The distribution of individuals of different species across different sampling units is typically non-random. This distributional non-independence can be interpreted and modelled as a correlated multivariate distribution. However, this correlation cannot be modelled using a totally independent and random distribution such as the Poisson distribution. In this study, we utilized the negative multinomial distribution to overcome the problem encountered by the commonly used Poisson distribution and used it to derive insight into the implications of field sampling for rare species’ distributions. Mathematically, we derived, from the negative multinomial distribution and sampling theory, contrasting relationships between sampling area, and the proportions of locally rare and regionally rare species in ecological assemblages presenting multi-species correlated distribution. With the suggested model, we explored the cross-scale relationships between the spatial extent, the population threshold for defining the rarity of species, and the multi-species correlated distribution pattern using data from two 50-ha tropical forest plots in Barro Colorado Island (Panama) and Heishiding Provincial Reserve (Guangdong Province, China). Notably, unseen species (species with zero abundance in the studied local sample) positively contributed to the distributional non-independence of species in a local sample. We empirically confirmed these findings using the plot data. These findings can help predict rare species–area relationships at various spatial scales, potentially informing biodiversity conservation and development of optimal field sampling strategies.

Community-level species' correlated distribution can be scale-independent and related to the evenness of abundance.

The spatial distribution of species is not random; instead, individuals tend to gather, resulting in a non-random pattern. Previous studies used the independent negative binomial distribution (NBD) to model the distributional aggregation of a single species, in which the independence of the distribution of individuals of a species in different quadrats had been assumed. This way of analyzing aggregation will result in the scale-dependent estimation of the aggregation or shape parameter. However, because non-random (and therefore non-independent) distribution of individuals of a species in a finite area can be caused by either correlated or clumped distribution of individuals of a species between neighboring sites, an alternative model would assume that the distribution of individuals of a species over different sampling areas is multinomial. Here, we showed that, by assuming that regional species abundance followed a NBD while using a multinomial distribution to assign individuals of species in different non-overlapped sampling quadrats that are from a partition of the entire region (quantifying positive correlation or synchrony), the estimation of the shape parameter in this probabilistic model, which is the negative multinomial distribution (NMD), was scale-invariant (i.e., the estimated shape parameter is identical across different partitions of the study region). Accordingly, the estimation of the shape parameter was related to regional species distribution alone. This implied that, the shape parameter at the community level, using the NMD model, reflected the evenness of interspecific abundance. As a comparison, if the distribution of individuals of a single species followed independent NBDs as studied previously, the shape parameter would measure the evenness of intraspecific abundance (quantifying single-species' distributional aggregation). Moreover, our study highlighted the necessity for adjusting the model for the effects of unsampled species when studying community-level distributional patterns. Collectively, as long as a target area is partitioned into non-overlapping quadrats (no matter how their sizes vary), the proposed NMD model in this study, along with the independent NBDs model, can be jointly formulated as a framework to reconcile the scale-dependent debate on the shape parameter, unifying the relationship between inter- or intraspecific abundance and distributional patterns.

Asymptotic Outage Analysis of HARQ-IR Over Time-Correlated Nakagami- $m$ Fading Channels

In this paper, outage performance of hybrid automatic repeat request with incremental redundancy (HARQ-IR) is analyzed. Unlike prior analyses, time-correlated Nakagami- $m$ fading channel is considered. The outage analysis thus involves the probability distribution analysis of a product of multiple correlated shifted Gamma random variables and is more challenging than prior analyses. Based on the findings of the conditional independence of the received signal-to-noise ratios, the outage probability is exactly derived by using conditional Mellin transform. Specifically, the outage probability of HARQ-IR under time-correlated Nakagami- $m$ fading channels can be written as a weighted sum of outage probabilities of HARQ-IR over independent Nakagami fading channels, where the weightings are determined by a negative multinomial distribution. This result enables not only an efficient truncation approximation of the outage probability with uniform convergence but also asymptotic outage analysis to further extract clear insights, which have never been discovered for HARQ-IR even under fast fading channels. The asymptotic outage probability is then derived in a simple form, which clearly quantifies the impacts of transmit powers, channel time correlation, and information transmission rate. It is proved that the asymptotic outage probability is an inverse power function of the product of transmission powers in all HARQ rounds, an increasing function of the channel time correlation coefficients, and a monotonically increasing and convex function of information transmission rate. The simple expression of the asymptotic result enables optimal power allocation and optimal rate selection of HARQ-IR with low complexity. Finally, numerical results are provided to verify our analytical results and justify the application of the asymptotic result for optimal system design.

Enhancing Safety Risk Management with Quantitative Measures

At the airfield in hub airports, many activities occur that involve a range of participants, including various-size aircraft, ground vehicles, and workers. The safety management system is FAA's approach for systematically managing aviation safety. A major component of the safety management system is safety risk management (SRM), which entails analysis, assessment, and control of safety risks, including risks on the airfield. Current SRM has few specific safety models to estimate the likelihood or frequency of risks. This paper presents an example for development and incorporation of safety models into SRM. Specifically, it discusses safety models for runway incursion that use the following variables: total and general aviation operations, length of runway by type, number of taxiway intersections, snowfall, precipitation, number of hot spots, and construction activity. Categorization and processing of data were significant because each variable used could take on multiple forms, and some types of data involved review of airfield diagrams. The data used were from 137 U.S. hub airports for 2009 through 2014. For modeling, the negative multinomial distribution was used because it proved suitable for representing overdispersed data such as runway incursion frequency. Performance of the models was assessed through the goodness-of-fit measures of log likelihood, overdispersion, and cumulative residual plots. Models were developed for five severity categories of runway incursions and three types of surface events. The safety modeling approach presented here can serve as a foundation for development of other safety models that can be integrated into SRM to enable quantitative analysis of safety risks.

Conditioning out rare events for exponential families has negligible effect on inference

ABSTRACTSuppose that we observe X ∼Binomial(n, p). Inference on p is difficult if X = 0 or n. One way around this is to condition on these events not happening. We show that this has only an exponentially small effect on its cumulants. This is also true if we condition away other rare events. Our results are presented for exponential families, with applications to the binomial, multinomial and negative multinomial distributions.

Multinomial distributions in Shintani zeta class

Multidimensional stochastic models in mathematical finance and so on are now well-studied. As to obtain more properties of them, we focus on some multidimensional discrete distributions in relation to a class of multiple zeta functions. The class called “multidimensional Shintani zeta functions” was first introduced in Aoyama and Nakamura (Tokyo J Math 36:521–538, 2013), where a class of probability distributions called “multidimensional Shintani zeta distributions associated with the zeta functions is definable. In this paper, we show that this class includes many kinds of multidimensional discrete distributions. We pick up some cases of multidimensional Shintani zeta functions and introduce some classes of distributions which contain multinomial and negative multinomial distributions as their generalizations. More precisely, we give some necessary and sufficient conditions for the functions to generate probability distributions in view of zeta functions and consider their infinite divisibilities as well.

Missing data sensitivity analysis for recurrent event data using controlled imputation

Statistical analyses of recurrent event data have typically been based on the missing at random assumption. One implication of this is that, if data are collected only when patients are on their randomized treatment, the resulting de jure estimator of treatment effect corresponds to the situation in which the patients adhere to this regime throughout the study. For confirmatory analysis of clinical trials, sensitivity analyses are required to investigate alternative de facto estimands that depart from this assumption. Recent publications have described the use of multiple imputation methods based on pattern mixture models for continuous outcomes, where imputation for the missing data for one treatment arm (e.g. the active arm) is based on the statistical behaviour of outcomes in another arm (e.g. the placebo arm). This has been referred to as controlled imputation or reference-based imputation. In this paper, we use the negative multinomial distribution to apply this approach to analyses of recurrent events and other similar outcomes. The methods are illustrated by a trial in severe asthma where the primary endpoint was rate of exacerbations and the primary analysis was based on the negative binomial model.

A note on a variance bound for the multinomial and the negative multinomial distribution

We prove a Chernoff-type upper variance bound for the multinomial and the negative multinomial distribution. An application is also given.Copyright © 2014 Wiley Periodicals, Inc. Naval Research Logistics 61: 179–183, 2014

The multivariate Meixner polynomials as matrix elements of SO(d, 1) representations on oscillator states

The multivariate Meixner polynomials are shown to arise as matrix elements of unitary representations of the SO(d, 1) group on oscillator states. These polynomials depend on d discrete variables and are orthogonal with respect to the negative multinomial distribution. The emphasis is put on the bivariate case for which the SO(2, 1) connection is used to derive the main properties of the polynomials: orthogonality relation, raising/lowering relations, generating function, recurrence relations and difference equations as well as explicit expressions in terms of standard (univariate) Krawtchouk and Meixner polynomials. It is explained how these results generalize directly to d variables.

Cumulants of multinomial and negative multinomial distributions

Abstract For the first time, explicit expressions are given for the cumulants of the multinomial and the negative multinomial distributions. These expressions are given in terms of Stirling numbers. Computational efficiency of the expressions is illustrated.

Masses of Negative Multinomial Distributions: Application to Polarimetric Image Processing

This paper derives new closed-form expressions for the masses of negative multinomial distributions. These masses can be maximized to determine the maximum likelihood estimator of its unknown parameters. An application to polarimetric image processing is investigated. We study the maximum likelihood estimators of the polarization degree of polarimetric images using different combinations of images.

Asymptotic Normality Through Factorial Cumulants and Partition Identities

In the paper we develop an approach to asymptotic normality through factorial cumulants. Factorial cumulants arise in the same manner from factorial moments as do (ordinary) cumulants from (ordinary) moments. Another tool we exploit is a new identity for 'moments' of partitions of numbers. The general limiting result is then used to (re-)derive asymptotic normality for several models including classical discrete distributions, occupancy problems in some generalized allocation schemes and two models related to negative multinomial distribution.

Dirichlet negative multinomial regression for overdispersed correlated count data

A generic random effects formulation for the Dirichlet negative multinomial distribution is developed together with a convenient regression parameterization. A simulation study indicates that, even when somewhat misspecified, regression models based on the Dirichlet negative multinomial distribution have smaller median absolute error than generalized estimating equations, with a particularly pronounced improvement when correlation between observations in a cluster is high. Estimation of explanatory variable effects and sources of variation is illustrated for a study of clinical trial recruitment.

Estimation of Progression-Free Survival for All Treated Patients in the Randomized Discontinuation Trial Design

The randomized discontinuation trial (RDT) design is an enrichment-type design that has been used in a variety of diseases to evaluate the efficacy of new treatments. The RDT design seeks to select a more homogeneous group of patients, consisting of those who are more likely to show a treatment benefit if one exists. In oncology, the RDT design has been applied to evaluate the effects of cytostatic agents, that is, drugs that act primarily by slowing tumor growth rather than shrinking tumors. In the RDT design, all patients receive treatment during an initial, open-label run-in period of duration T. Patients with objective response (substantial tumor shrinkage) remain on therapy while those with early progressive disease are removed from the trial. Patients with stable disease (SD) are then randomized to either continue active treatment or switched to placebo. The main analysis compares outcomes, for example, progression-free survival (PFS), between the two randomized arms. As a secondary objective, investigators may seek to estimate PFS for all treated patients, measured from the time of entry into the study, by combining information from the run-in and post run-in periods. For t ⩽ T, PFS is estimated by the observed proportion of patients who are progression-free among all patients enrolled. For t > T, the estimate can be expressed as , where is the estimated probability of response during the run-in period, is the estimated probability of SD, and and are the Kaplan–Meier estimates of subsequent PFS in the responders and patients with SD randomized to continue treatment, respectively. In this article, we derive the variance of , enabling the construction of confidence intervals for both S(t) and the median survival time. Simulation results indicate that the method provides accurate coverage rates. An interesting aspect of the design is that outcomes during the run-in phase have a negative multinomial distribution, something not frequently encountered in practice.