Showing papers on "Negative binomial distribution published in 1973"

Mathematical statistics with applications

Abstract: This calculus-based book presents a blend of theory and application. It focuses on inference making as the goal of studying probability and statistics, and features an emphasis on real-life applications of the mathematically divided techniques. Exercises based on documented data sets and other realistic experimental situations motivate the theoretical foundation. Updated and expanded coverage includes: the negative binomial probability distribution (optional); and explicit models for the analysis of variance of completely randomized block designs.

A Generalization of the Poisson Distribution

Abstract: A new generalization of the Poisson distribution, with two parameters λ1 and λ2, is obtained as a limiting form of the generalized negative binomial distribution. The variance of the distribution is greater than, equal to or smaller than the mean according as λ2 is positive, zero or negative. The distribution gives a very close fit to supposedly binomial, Poisson and negative-binomial data and provides with a model suitable to most unimodel or reverse J-shaped distributions. Diagrams showing the variations in the form of the distribution for different values of λ1 and λ2 are given.

A Consumer Purchasing Model with Erlang Inter-Purchase Times

Abstract: The gamma-Poisson form of the negative binomial distribution (NBD) model generally gives a good fit to many aspects of repeat-buying behavior for a wide range of frequently bought branded consumer goods. Nonetheless, empirical evidence suggests that purchasing a particular brand-size in successive equal time-periods tends to be more regular than Poisson. An alternative model is therefore examined in which inter-purchase times for a given consumer are described by an Erlang distribution rather than by the negative exponential distribution implicit in the Poisson assumption of the NBD model. But it is found that the NBD model is robust to this kind of departure. Because of its greater simplicity, the NBD model therefore seems preferable for practical use.

A Markov Model for beach profile changes

Abstract: An apparent complex time history of beach geometry can be described as a specific case of first-order Markov process Under the assumptions that the profile transition is controlled only by random excitations from waves and that the transition probability is identical for all the possible states of beach profile it is demonstrated that a beach profile time series contains cycles having negative binomial distribution A simplified case in which the transition probability is taken as 1/2 (ie, equal probability for either erosional or accretional transition for any profile state) is derived through both numerical simulation and theoretical derivation, the result of which shows reasonable agreement with field observations

Some approximate relations between terms of the hypergeometric, binomial and poisson distributions

Abstract: The approximations herein derived are for individual terms of the hypergeometric, binomial and Poisson distributions. They give,simple expressions for the approximate relative errors in the use of either of two binomial approximations to the hypergeometric, and also of the Poisson to either the hypergeometric or the binomial. They may be used to estimate errors involved or as corrections to provide closer approximations to desired probabilities.

Inferences on the Dispersion of Cotton Arthropods in Texas

Abstract: Discrete frequency distributions obtained from samples of various arthropods commonly found in Texas cotton fields were fitted to 6 discrete mathematical frequency-distribution models. The observed frequency distributions most frequently fit the negative binomial expectations, suggesting a clumped pattern of dispersion. These same arthropods were seldom observed to be randomly dispersed as indicated by poor agreement with the Poisson expectations.

Formulas for the number of binomial coefficients divisible by a fixed power of a prime

Abstract: Define 02(n) as the number of binomial coefficients (S) divisible by exactlypi A formula for 02(n) is found, for all n, and formulas for Oj(n) for n=apk+bpr and n=cipkl+* +Cmpkm (k1>j, ki+?-k >jfor i=1, * * , m-1) are derived

Collegiate Football Scores and the Negative Binomial Distribution

Abstract: The frequency distribution of grouped scores of individual teams in U.S. collegiate football games closely follows the negative binomial distribution.

Application of the Poisson and Negative Binomial Models to Thunderstorm and Hail Days Probabilities in Nevada

Abstract: Rare events such as thunderstorm or hail days often fit one of two distributions, the Poisson or the negative binomial. These two models were tested on monthly and annual thunderstorm days as well as on annual hail days at five locations in Nevada. A procedure for calculating efficient estimates of the parameters for the negative binomial distribution, utilizing the iterative process and the second-order polynomial model, is described. Results of the program applied to five sites in Nevada are discussed.

The Reliability of Systems of Independent Parallel Components When Some Components are Repeated

Abstract: Procedures are given for testing hypotheses and obtaining confidence intervals for the reliability of systems of k independent parallel components consisting of m < k component types. Assume that there are αi components of the ith component type, i = 1, 2, …, m and that pi is the failure probability of the ith component type. Then the probability of failure of the system is ρ = IIi−1 m pi αi . If m independent sequences of Bernoulli trials are conducted, one for each component type, techniques which may be employed for statistical inference concerning ρ are given. These techniques are suitable under those conditions for which the binomial distribution may be satisfactorily approximated by the Poisson distribution.

Confidence Intervals for Some Functions of Several Bernoulli Parameters with Reliability Applications

Abstract: The estimation of functions of Bernoulli parameters is of interest in reliability theory. In this article, the theory of exponential families is used to obtain exact confidence limits for products and quotients of Bernoulli parameters when negative binomial observations are available from each population. Sampling methods based on compound Poisson distribution are suggested for estimating more general functions such as sums of products of Bernoulli parameters. Finally, it is shown that exact confidence limits for products can be obtained by a discrete analog of the Lieberman-Ross procedure, which exploits the independence of the minimum and difference of geometric variates.

Model of earthquake occurrence in the European area

Abstract: The time distribution of earthquake occurrence in the European area is investigated by statistical laws The original data of shallow-focus earthquakes are taken from the European catalogue 1901–1967 Evidence is given that the process with the negative binomial entries as a model describing the occurrence of shallow-focus earthquakes is better than the Poisson process Further, the influence of magnitude classes and magnitude threshold value on the time distribution of earthquake occurrence is examined

Measuring the Permanent Component of a Series for Serially Correlated Observations

Abstract: Alternative systems of negative binomial and binomial weights are proposed for measuring the permanent component of a series as a weighted average of the observed series. If the observed series is generated by a stable first-order autoregressive process it is shown that there are alternative sets of weights using either weighting scheme for which the permanent and transitory components are uncorrelated.

Exponents of primes in generalized binomial coefficients.

Abstract: together with evidence for generalizations, in [1]. This conjecture was generalized and proved in [2]. An independent proof is given in [3]. Here we extend these results and give formulas for the exponents of primes in generalized factorials and generalized binomial coefficients. ; Let D be an integral domain in which the is a divisor of\" relation y \\ χ has the properties described in Section l below. In terms of a given sequence w1? w2 , . . . of nonzero elements of D, generalized factorials fn are defined by fQ = l and fn = u1u2 · · ·ιιη for n >0. If (fkfn_k) \\ fn for Ο <Ξ k ^ rc, we call {un} a Raney sequence and define generalized binomial coefficients by L

A generalised incomplete beta function and its application to multi-line stock control

Abstract: The distribution function for the negative binomial distribution is known to be an incomplete beta function. Here, some of the properties of the family of distribution functions for multivariate negative binomial distributions are explored. These properties are then used in deriving the expected cost per unit time for a multi-line joint-reordering system with Poisson demands. Policies are considered for which the quantity of any particular line in stock is the same at the beginning of every cycle. A method which gives good approximations to the optimal values of these quantities is described.

A Characterization of the Multivariate Geometric Distribution

Abstract: Publisher Summary This chapter presents the multivariate negative binomial distribution, which was introduced by Bates and Neyman who derived a number of properties of this distribution. The chapter also presents the derivation of a characterization theorem for the p-variate geometric distribution. It presents a characterization theorem based on a regression property. It focuses on functions that depend either on all p variables t1, t2, …, tp or only on a part of these variables.

Spatial distribution of the green peach aphid used in estimating the populations of gynoparae

Abstract: In 2 years, during the initial invasion of peach leaves by the green peach aphid,Myzus persicae (Sulzer), the number of gynoparae was low, and the distribution on leaves was random. Then as the mean number increased, the distribution became intermediate and could not be distinguished from either a Poisson or a negative binomial. Finally, as the mean continued to increase, the variance increased rapidly, and the population was found to fit a negative binomial distribution. Thus the aggregation response was verified because the dispersion pattern fitted a contagious distribution. A sampling plan was devised by which the dispersion parameterk was used to estimate the density of aphids per leaf based on the percentage of leaves infested. Sampling the third year of the study confirmed the validity of the sampling parameter that had been calculated from data for the 2 previous years.

Bounding poissoh approximations to the binomial

Abstract: We develop Poisson approximations to the binomial distribution which bound the value of the Poisson parameter divine, equality between binomial and Poisson partial sums.

Information through sampling from a binomial distribution

Abstract: Lindley's measure of experimental information is utilized to determine the optimal sample-size that is required to obtain a prescribed level of accuracy about the parameter of a binomial distribution. The measure is expressed through its approximate limiting relationship to the posterior variance of the parameter, and a simple application of the methodology is presented.