Showing papers on "Negative binomial distribution published in 1993"
TL;DR: A simple new measure of parasite aggregation is described, the index of discrepancy (D), which quantifies the difference between the observed parasite distribution, and the hypothetical distribution that corresponds to the ideal case where all hosts harbour the same number of parasites.
288 citations
TL;DR: In this paper, the authors show that the negative binomial distribution is sufficient for the detection of rare and possibly endangered species, and that the Poisson distribution is adequate if the mean density of the rare species is very low.
Abstract: Often a sampling program has the objective of detecting the presence of one or more species. One night wish to obtain a species list for the habitat, or to detect the presence of a rare and possibly endangered species. How can the sampling effort necessary for the detection of a rare species can be determined? The Poisson and the negative binomial are two possible spatial distributions that could be assumed. The Poisson assumption leads to the simple relationship n = -(1/m)log @b, where n is the number of quadrats needed to detect the presence of a species having density m, with a chance @b (the Type 2 error probability) that the species will not be collected in any of the n quadrats. Even if the animals are not randomly distributed the Poisson distribution will be adequate if the mean density is very low (i.e., the species is rare, which we arbitrarily define as a true mean density of 0.95. Only 8 of the 273 cases represented rare species that failed this requirement. Thus we conclude that a Poisson-based estimate of necessary sample size will generally be adequate and appropriate.
157 citations
TL;DR: In this article, explicit formulas are derived for finite time ruin probabilities in the discrete time and state-space compound binomial model using the technique of generating functions, and a close connection is established with the ultimate ruin probability in the usual compound Poisson model when the claim severity distribution is a (truncated) mixed Poisson distribution.
Abstract: In this paper explicit formulas are derived for finite time ruin probabilities in the discrete time and state-space compound binomial model using the technique of generating functions. Ultimate ruin probabilities are then obtained, and a close connection is extablished with the ultimate ruin probabilities in the usual compound Poisson model when the claim severity distribution is a (truncated) mixed Poisson distribution.
101 citations
Journal Article•
TL;DR: In this article, the application of the generalized linear modeling approach to the development of a model relating unsignalized intersection traffic demands to accident frequency is described, and several techniques for assessing model fit have been described and any inherent limitations noted.
Abstract: The application of the generalized linear modeling approach to the development of a model relating unsignalized intersection traffic demands to accident frequency is described. Several techniques for assessing model fit have been described and any inherent limitations noted. The model was based on the product of the intersection traffic demands raised to a power. This model form was found to explain a large portion of the variability in accidents among intersections of similar geometry and traffic control. The analysis of accident data for 125 two-way stop-controlled intersections supports the theory that the distribution of accident counts can be described by the negative binomial distribution. Also supported is the assertion that the mean accident frequency for the group of similar intersections is gamma distributed. Knowledge of these distributions and their parametric values can be used to identify hazardous locations and the true effect of safety treatments on accident frequency.
99 citations
TL;DR: The addition of a new parameter, the block size, to the two existing parameters of the fault distribution is proposed, which allows the unification of the existing models and, at the same time, adds a whole range of medium-size clustering models.
Abstract: It has been recognized that the yield of fault-tolerant VLSI circuits depends on the size of the fault clusters. Consequently, models for yield analysis have been proposed for large-area clustering and small-area clustering, based on the two-parameter negative-binomial distribution. The addition of a new parameter, the block size, to the two existing parameters of the fault distribution is proposed. This parameter allows the unification of the existing models and, at the same time, adds a whole range of medium-size clustering models. Thus, the flexibility in choosing the appropriate yield model is increased. Methods for estimating the newly defined block size are presented and the approach is validated through simulation and empirical data. >
91 citations
TL;DR: In this article, a new generalization of the binomial distribution is introduced that allows dependence between trials, nonconstant probabilities of success from trial to trial, and which contains the usual Binomial distribution as a special case.
Abstract: A new generalization of the binomial distribution is introduced that allows dependence between trials, nonconstant probabilities of success from trial to trial, and which contains the usual binomial distribution as a special case. Along with the number of trials and an initial probability of ‘success’, an additional parameter that controls the degree of correlation between trials is introduced. The resulting class of distributions includes the binomial, unirnodal distributions, and bimodal distributions. Formulas for the moments, mean, and variance of this distribution are given along with a method for fitting the distribution to sample data.
72 citations
TL;DR: In this paper, the concept and criterion for the stability of a recurrence equation are clarified based on relative error (rather than absolute error) analysis, and a family of recursions called congruent recursions is proved to be strongly stable in evaluating its non-negative solutions.
Abstract: Based on recurrence equation theory and relative error (rather than absolute error) analysis, the concept and criterion for the stability of a recurrence equation are clarified. A family of recursions, called congruent recursions, is proved to be strongly stable in evaluating its non-negative solutions. A type of strongly unstable recursion is identified. The recursive formula discussed by PANJER (1981) is proved to be strongly stable in evaluating the compound Poisson and the compound Negative Binomial (including Geometric) distributions. For the compound Binomial distribution, the recursion is shown to be unstable. A simple method to cope with this instability is proposed. Many other recursions are reviewed. Illustrative numerical examples are given.
70 citations
TL;DR: In this paper, it was shown that personal wealth is distributed as a mixture of negative binomial probability functions (NB pf s), which is closely related to the two parameter gamma probability density function (pdf) fitted to the equilibrium distribution of the process.
Abstract: The folk saying, “the rich get richer, the poor get poorer”, implies that wealth flows from poorer to richer hands, a concentrating diffusion. Its academic statement is the Surplus Theory of Social Stratification. Angle (1986) abstracted this theory as an interacting particle system model of wealth distribution, the Inequality Process. In it, random pairs continually compete for each other's wealth. The richer party has a greater chance of winning. The loser gives up a proportion of wealth. Angle (1986) found the equilibrium distribution of the Inequality Process numerically. This paper derives this distribution algebraically, showing that the Inequality Process implies that personal wealth is distributed as a mixture of negative binomial probability functions (NB pf s). The NB pf is closely related to the two parameter gamma probability density function (pdf), what Angle (1986) fitted to the equilibrium distribution of the process. The solution is given in terms of the parameters of the Inequality Proces...
56 citations
01 Sep 1993
TL;DR: In this article, the authors relax the restriction that q be a prime power, and consider a multiset construction in which the total number of possibilities of weight n is qn.
Abstract: We consider random monic polynomials of degree n over a finite field of q elements, chosen with all qn possibilities equally likely, factored into monic irreducible factors. More generally, relaxing the restriction that q be a prime power, we consider that multiset construction in which the total number of possibilities of weight n is qn. We establish various approximations for the joint distribution of factors, by giving upper bounds on the total variation distance to simpler discrete distributions. For example, the counts for particular factors are approximately independent and geometrically distributed, and the counts for all factors of sizes 1, 2, …, b, where b = O(n/log n), are approximated by independent negative binomial random variables. As another example, the joint distribution of the large factors is close to the joint distribution of the large cycles in a random permutation. We show how these discrete approximations imply a Brownian motion functional central limit theorem and a Poisson-Dirichiet limit theorem, together with appropriate error estimates. We also give Poisson approximations, with error bounds, for the distribution of the total number of factors.
50 citations
TL;DR: In this article, the authors point out a characterization of sequences of exchangeable Bernoulli random variables which can be used to develop models which show more variability than the binomial, and give sufficient conditions which will yield such models and show how existig models can be combined to generate further models.
Abstract: In many cases where the binomial dismbution fails to apply to real world data it is because of more variability in the data than can be explained by that dismbution. Several authors have proposed models that are useful in explaining extra-binomial variation. In this paper we point out a characterization of sequences of exchangeable Bernoulli random variables which can be used to develop models which show more variability than the binomial. We give sufficient conditions which will yield such models and show how existig models can be combined to generate further models. The usefulness of some of these models is illustrated by fitting them to sets of real data.
43 citations
TL;DR: The negative binomial distribution demonstrated substantially better fits than the Poisson distribution to the numbers of hospitalizations within hospital market areas and may be useful in estimation of confidence intervals for standardized rates of recurrent events when these events do not recur randomly.
Abstract: Usual approaches for estimating the variance of a standardized rate may not be applicable to rates of recurrent events. Where individuals are prone to repeated health events, Greenwood and Yule (J R Stat Soc [A], 1920;83:255-79) advocated use of the negative binomial distribution to account for departures from the assumption of randomness of recurrent events required by the Poisson distribution. In this paper, the authors implemented the negative binomial distribution in the computation of annual hospitalization rates within certain hospital market areas. Data used were from 1,549,915 New England residents aged 65 years or more who were enrolled in Medicare between October 1, 1988, and September 30, 1989, and who had 458,593 hospital admissions during that year. New England was partitioned into 170 hospital market areas ranging in population size from 162 to 70,821 elderly Medicare enrollees. The negative binomial distribution demonstrated substantially better fits than the Poisson distribution to the numbers of hospitalizations within hospital market areas. Estimated standard errors for indirectly standardized rates based on the negative binomial distribution were 25-51 percent higher than estimated standard errors that assumed an underlying Poisson distribution. Using regression analysis to smooth overdispersion parameters across hospital market areas produced similar results. The approach described in this paper may be useful in estimation of confidence intervals for standardized rates of recurrent events when these events do not recur randomly.
28 Apr 1993
TL;DR: In this paper, the authors examined the distribution of a sum S of binomial random variables, each with different success probabilities, and gave an algorithm to calculate the exact distribution of S, and several approximations are examined.
Abstract: : In this paper we examine the distribution of a sum S of binomial random variables, each with different success probabilities. The distribution arises in reliability analysis and in survival analysis. An algorithm is given to calculate the exact distribution of S, and several approximations are examined. An approximation based on a method of Kolmogorov, and another based on fitting a distribution from the Pearson family, can be recommended.
TL;DR: In this article, the statistical distribution of fluctuations on various scales is analyzed in terms of the counts in cells of smoothed density fields, using volume-limited samples of galaxy redshift catalogs, and the distribution on large scales, with volume average of the two-point correlation function of the smoothed field less than about 0.05, is consistent with Gaussian.
Abstract: The statistical distribution of fluctuations on various scales is analyzed in terms of the counts in cells of smoothed density fields, using volume-limited samples of galaxy redshift catalogs. It is shown that the distribution on large scales, with volume average of the two-point correlation function of the smoothed field less than about 0.05, is consistent with Gaussian. Statistics are shown to agree remarkably well with the negative binomial distribution, which has hierarchial correlations and a Gaussian behavior at large scales. If these observed properties correspond to the matter distribution, they suggest that our universe started with Gaussian fluctuations and evolved keeping hierarchial form.
TL;DR: A probabilistic model of the functional response of a parasitoid at the behavioural time-scale of the eulophid Sympiesis sericeicornis Nees is built and it is found that the probability of leaving by choice is a monotone increasing function of the number of hosts parasitized.
Abstract: 1. The aim of this paper is to build a probabilistic model of the functional response of a parasitoid at the behavioural time-scale. The organism used is the eulophid Sympiesis sericeicornis Nees (Hymenoptera: Eulophidae), a polyphagous ectoparasitoid attacking the apple leaf-miner Phyllonorycter cydoniella (D. & S.) (Lepidoptera: Gracillariidae). We deal exclusively with the functional response at the level of a mined leaf and restrict our attention to the cases without superparasitism. 2. We use detailed observations of the behaviour and location of the parasitoid on a leaf to define four behavioural states: searching, hunting on a mine with an unparasitized host, ovipositing, and leaving the leaf. We describe the sequence of behavioural states visited by an embedded markov chain, which is fully determined by the one-step transition probability matrix and the initial probability distribution. 3. The key element of our model is the process of leaving the leaf. We postulate that this can occur only from the searching state, and happens either by necessity, because either available hosts or useable eggs are exhausted, or by choice. We find that the probability of leaving by choice is a monotone increasing function of the number of hosts parasitized. 4. We consider a cohort of females with initial eggload distributed according to a negative binomial distribution whose parameters we obtain from independent data. An exact multinomial test of our model on the data set from which the one-step transition probabilities are obtained gives excellent results. The domain of applicability of the model is extended by relaxing two important assumptions. We also apply the exact multinomial test to an independent data set and again obtain very good results. Sensitivity analysis demonstrates that the model is sensitive to changes in the value of only one parameter.
TL;DR: Applications of the negative binomial distribution are considered in situations where either the host or parasite population can be divided into subpopulations of different types (eg. by age, sex or genotype), and they describe the relationships between the frequency distributions relevant to the different sub Populations and thoserelevant to the total population.
TL;DR: In this paper, the authors extend CTA to the nonstationary setting and compare the stationary and non-stationary models, showing that under-prediction is a mathematical artifact due to the skewness of the negative binomial distribution.
Abstract: Conditional trend analysis (CTA) predicts the number of purchases in a test period by all households that. purchase a given number of items in a base period. The underlying model assumes that households' purchases follow stationary Poisson processes with rate parameters that vary across the households in a market. However, stationarity is often an unrealistic assumption because of marketing variables and seasonal effects. This paper extends CTA to the nonstationary setting and compares the stationary and nonstationary models. Falsely assuming stationarity systematically biases forecasts. Although modeling nonstationarity reduces bias, under–prediction, especially of the zero class, persists. We show that this under–prediction is, in part, a mathematical artifact due to the skewness of the negative binomial distribution. The methodology is applied to scanner panel data.
TL;DR: In this article, the sharp bound for expected values of order statistics derived by Moriguti (1953) based on the principle of greatest convex minorants was obtained using an application of binomial-negative binomial relationship.
TL;DR: In this paper, it is shown that other recursions can be derived starting with claim number distributions from the class of modified power series distributions which also incorporates generalized Poisson, generalized negative binomial and the ones mentioned above.
Abstract: Distributions like Poisson, binomial, negative binomial or logarithmic behave well as claim number distribution in collective models. Besides, using these claim number distributions, the total claim probabilities can easily be evaluated recursively with the so-called Panjer recursion. This note shows that other recursions can be derived starting with claim number distributions from the class of modified power series distributions which also incorporates generalized Poisson, generalized negative binomial and the ones mentioned above. In addition it is shown that Panjer's recursion can be derived as a special case.
TL;DR: The transformed rejection method, a combination of inversion and rejection, is well suited to generate binomial random variates as well and for the case that the parameters of the binomial distribution vary from call to call BTRD is faster than the current state of the art algorithms.
Abstract: The transformed rejection method, a combination of inversion and rejection, which can be applied to various continuous distributions, is well suited to generate binomial random variates as well. The resulting algorithms are simple and fast, and need only a short set-up. Among the many possible variants two algorithms are described and tested: BTRS a short but nevertheless fast rejection algorithm and BTRD which is more complicated as the idea of decomposition is utilized. For BTRD the average number of uniforms required to return one binomial deviate less between 2.5 and 1.4 which is considerably lower than for any of the known uniformly fast algorithms. Timings for a C-implementation show that for the case that the parameters of the binomial distribution vary from call to call BTRD is faster than the current state of the art algorithms. Depending on the computer, the speed of the uniform generator used and the binomial parameters the savings are between 5 and 40 percent.
TL;DR: Analysis of the sampling statistics of Pacific salmon caught in a series of experimental gillnet sets on the high seas indicates that the replicate catches are adequately described by the negative binomial distribution.
Abstract: We examined the sampling statistics of Pacific salmon (Oncorhynchus spp.) caught in a series of experimental gillnet sets on the high seas and demonstrate how the reliability of the catch statistics varies with the amount of sampling effort. Our analysis indicates that the replicate catches, which were made under essentially identical conditions, are adequately described by the negative binomial distribution. We also extend the utility of this distribution for describing catch statistics by showing that (1) the shape parameter of the distribution can be directly interpreted as the number of degrees of freedom (df) associated with each observation of catch and (2) the df are related to the amount of sampling effort used and the size of the biological aggregation being encountered by the gear. For salmon caught on the high seas, approximately 1 df is obtained per 15 m of gill net used. Identification of the statistical model describing the uncertainty in high-seas gillnet catches should help in the design o...
TL;DR: In this paper, it was shown that the Stancu-Muhlbach operators preserve Lipschitz constants and also gave quantitative estimates for the approximation of Bernstein, Szasz, and Baskakov operators.
TL;DR: In this paper, an EM-algorithm is proposed to study a mixture model for the analysis of count data and iterative procedures for estimating the parameters are given for various discrete distributions including Binomial, Negative Binomial and Poisson.
TL;DR: In this article, the authors considered the binomial, hypergeometric, and negative binomial distributions and provided prediction limits of the following type for each of the three types of distributions.
Abstract: Prediction limits of the following type are considered for the binomial, hypergeometric, and negative-binomial distributions. For the binomial distribution, suppose X/sub r/ successes have occurred in the first r trials, and based on this partial information it is desired to predict the total number of successes X/sub s/ (r >
TL;DR: In this paper, it is shown that it is better to use a large number of small volume samples than vice versa, providing that the negative binomial dispersion parameter remains unaffected by volume.
TL;DR: In this article, bounds for the median of the negative binomial distribution are obtained for all possible parameter values of the distribution when the median is defined as inf {x: P(X≤x)≥1/2}.
Abstract: Bounds are obtained for the median of the negative binomial distribution which are valid for all possible parameter values of the distribution when the median is defined as inf {x: P(X≤x)≥1/2}.
TL;DR: A special urn-model distribution is presented which has an analogous behaviour to that of the Waring distribution in connection with conditional arithmetic means and combines specific properties of the two scientometric favourites, theWaring and the negative binomial distribution.
Abstract: The traditional stochastic approach to scientometric and bibliometric phenomena is based on measuring the absolute number of objects (e.g., publications, topics, citations). These measures reflect underlying rules such as the cumulative advantage principle and lead to classical statistical functions such as arithmetic mean and standard deviation. An alternative measure based on the contribution share of an individual object in the entirety of related objects reveals more about the coherence in the analyzed structure. This approach is connected with (conditional) harmonic means. The analysis of the properties of these statistical functions leads to a special urn-model distribution which has an analogous behaviour to that of the Waring distribution in connection with conditional arithmetic means. The new distribution combines specific properties (long tail, flexibility of the distribution shape) of the two scientometric favourites, the Waring and the negative binomial distribution. Five methods of parameter estimation are presented. The fit and the properties of this special urn-model distribution are illustrated by three scientometric examples, particularly, by two citation rate distributions with different shapes and one publication activity distribution with lacking zero frequencies.
TL;DR: In this article, an algorithm for maximum likelihood estimation for two-parameter generalized binomial distributions is proposed, based on the Polya urn model and the latent variable model.
Abstract: The generalized binomial distribution is defined as the distribution of a sum of symmetrically distributed Bernoulli random variates. Several two-parameter families of generalized binomial distributions have received attention in the literature, including the Polya urn model, the correlated binomial model and the latent variable model. Some properties and limitations of the three distributions are described. An algorithm for maximum likelihood estimation for two-parameter generalized binomial distributions is proposed. The Polya urn model and the latent variable model were found to provide good fits to sub-binomial data given by Parkes. An extension of the latent variable model to incorporate heterogeneous response probabilities is discussed.
TL;DR: In this paper, a statistical analysis is performed on natural events which can produce important damages to insurers, based on hurricanes which have been observed in the United States between 1954 et 1986.
Abstract: A statistical analysis is performed on natural events which can produce important damages to insurers. The analysis is based on hurricanes which have been observed in the United States between 1954 et 1986. At first, independence between the number and the amount of the losses is examined. Different distributions (Poisson and negative binomial for frequency and exponential, Pareto and lognormal for severity) are tested. Along classical tests as chi-square, Kolmogorov-Smirnov and non parametric tests, a test with weights on the upper tail of the distribution is used: the Anderson – Darling test. Confidence intervals for the probability of occurrence of a claim and expected frequency for different potential levels of claims are derived. The Poisson Log-normal model gives a very good fit to the data.
TL;DR: The ratio of cumulant to factorial moments (C2F) as discussed by the authors is a new measure of multiplicity distributions and its advantages and shortcomings are discussed using the negative binomial distribution and several QCD-inspired functions as examples.
Abstract: The ratio of cumulant to factorial moments is proposed as a new measure of multiplicity distributions. Its advantages and shortcomings are discussed using the negative binomial distribution and several QCD-inspired functions as examples. Its asymptotic at large rank reveals tiny features of high multiplicity tails of distributions which can become especially important at LHC and SSC.