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The purpose of this article is to derive the necessary and sufficient conditions for the power series 〖P℘〗_(ℷ,γ)^μ (t) whose coefficients are probabilities of the beta negative binomial distribution to be in the family F(σ,τ,ϵ,η,μ,ℷ,γ) of holomorphic functions which are defined in the open unit disk. We establish a number of important geometric properties, such as, coefficient estimates, extreme points, neighborhood property, integral representation, radii of starlikeness and convexity and Hadamard product properties for functions belongs to this family. Also we determinate some differential subordination properties of the power series 〖P℘〗_(ℷ,γ)^μ (t).

Let k and n be integers with 0 ≤ k ≤ n and p ∈ ( 0 , 1 ) . We prove that the function G ( a ) = G k , n , p ( a ) = Γ ( a n + 1 ) Γ ( a k + 1 ) Γ ( a ( n − k ) + 1 ) p a k ( 1 − p ) a ( n − k ) is completely monotonic on ( 0 , ∞ ) . This extends a result of Leblanc and Johnson, who showed in 2007 that the sequence { G ( j ) } j = 1 ∞ is decreasing.

In this article, we use Stein’s method together with z -functions to give an improved bound for the total variation distance between the distribution of a non-negative integer-valued random variable X and the negative binomial distribution with parameters r ∈ R + and p = 1 − q ∈ ( 0 , 1 ) , where r q p is equal to the mean of X , E ( X ) . The improved bound is sharper than that mentioned in Teerapabolarn and Boondirek (2010). We give three examples of the negative binomial approximation to the distribution of X concerning the negative hypergeometric, Polya and negative Polya distributions.

Sharp bounds for the probability of occurrence of at least one out of n events are found when the probability distribution of the occurrences is supposed to follow monotonic distribution and first m(mn1) binomial moments are known. The bounds are found explicitly for two different cases of increasing and decreasing distributions using operations research techniques. Inverses are found for two different classes of matrices and the bounds are obtained using those inverses.

This article proposes a generalized binomial distribution, which is derived from the finite capacity queueing system with state-dependent service and arrival rates. This distribution is also genera...

It is a well known fact that for the hierarchical model of a Poisson random variable Y whose mean has an Erlang distribution, the unconditional distribution of Y is negative binomial. However, the proofs in the literature [1] [2] provide no intuitive understanding as to why this result should be true. It is the purpose of this manuscript to give a new proof of this result which provides such an understanding. The memoryless property of the exponential distribution allows one to conclude that the events in two independent Poisson processes may be regarded as Bernoulli trials, and this fact is used to achieve the research purpose. Another goal of this manuscript is to give another proof of this last fact which does not rely on the memoryless property.

In the present article measurement error effect on the power of control chart for ZTNBD is investigated based on standardized normal variate. Numerical calculations are presented so as to enable an appreciation of the consequences of measurement errors on the power curve. To examine the sensitivity of the monitoring procedure, average run length is also considered.

This distribution is used to predict the number of failures before a success occurs in a sequence of Bernoulli trials. In addition, it is used to model a distribution that shows excessive variance because some categories show an excess and some a deficiency of counts. For example, filariasis is a rare disease that might be expected to have a distribution with the number of affected patients per village following a Poisson distribution. In fact, there are an excess number of villages with no disease or with much disease, and relatively few villages have intermediate numbers. This type of distribution is referred to as clumped, contagious, or overdispersed, and is often well fitted by a negative binomial distribution. The inference drawn is that some villages are in areas without insect vectors and others where the vectors are very plentiful.

Logit models belong to the class of probability models that determine discrete probabilities over a limited number of possible outcomes. They are often called ‘Quantal Variables’ or ‘Stimulus and Response Models’ in Biological Literature. The conventional R2 measure of goodness-of-fit is problematic in logit models. This has therefore led to the proposal of several alternative goodness-of-fit measures. But researchers in this area have identified the base rate problem in using these several alternative goodness-of-fit measures. This research is an extension of work done by people in this area. Specifically, this research is aimed at investigating the goodness-of-fit performances of eight statistics using the Bernoulli and Binomial distributions as explanatory variables under various scenarios. The study will draw conclusions on the “best” fit. The data for the study was generated through simulation and analysed using the multiple correlation analysis. The findings clearly show that for the Bernoulli Distribution, the goodness-of-fit statistics to use are: RO2, RC2, RM2 and λp; and for the Binomial Distribution, the goodness-of-fit statistics to use are: and RN2 and λp. RO2 stood out as the “best” goodness-of-fit statistics.

For modeling material properties having a bounded range, the beta distribution may be adopted as the marginal distribution of a second-order non-Gaussian random field. Three aspects related to the simulation of such random field are discussed in this study. First, an unbiased and consistent estimator for the lower (and upper) bound of the beta distribution based on sample data is proposed. This estimator is shown to be generally more efficient than that given by the method of moments. Second, a simple explicit function relating the auto-correlation function of the non-Gaussian random field to that of the underlying Gaussian field is proposed. The relationship facilitates control on the scale of fluctuation of the non-Gaussian field. Third, an algorithm is proposed for generating random fields with an approximate marginal beta distribution and a prescribed cross-correlation, where the latter can range from −1 to 1. Numerical examples are given to illustrate the effectiveness and efficiency of each of the three aspects. The estimation of the lower bound of material property is exemplified through field data from a real project.

In this note, we derive the exact distribution of S by using the method of generating function and BELL polynomials, where S = X1 + X2 + ⋅⋅⋅ + Xn, and each Xi follows the negative binomial distribution with arbitrary parameters. As a particular case, we also obtain the exact distribution of the convolution of geometric random variables.

Starting from a recent paper by Murshed (Stoch Environ Res Risk Assess 25:897–911, 2011) in which a good performance of the Beta-k distribution in analyzing extreme hydrologic events is shown, in this paper, we propose the use of two new four-parameters distribution functions strongly related to the Beta-k distribution, namely the Beta-Dagum and the Beta-Singh-Maddala distributions. More in detail, the new distributions are a generalization of a reparametrization of Beta-k and Beta-p distributions, respectively. For these distributions some particular interpretations in terms of maximum and minimum of sequences of random variables can be derived and the maximal and minimal domain of attraction can be obtained. Moreover, the method of maximum likelihood, the method of moments and the method of L-moments are examined to estimate the parameters. Finally, two different applications on real data regarding maxima and minima of river flows are reported, in order to show the potentiality of these two models in the extreme events analysis.

In this paper, the modes of the negative binomial distribution of order k are studied. Firstly, the method of transition probability flow graphs is introduced to deal with the probability-generating function of the geometric distribution of order k, which is a special case of the negative binomial distribution of the same order. And then, the general negative binomial distribution of order k is investigated. By means of probability distribution function, the mode of the geometric distribution of order k is derived, i.e. mX(k)=k. Based on the Fibonacci sequence and Poly-nacci sequence, the modes of the negative binomial distribution of order k in some cases are obtained: (1) mX(2,2)=6,7,8 and mX(3,2)=16, for p=0.5; (2) mX(2,3)=13 for p=0.5. Finally, an application of negative binomial distribution of order k in continuous sampling plans is given.

This paper considers a distribution formed by convolution of binomial and negative binomial variables. The distribution has the flexibility to adapt to the model under, equi, and over dispersion. Some properties of the proposed distribution are discussed, including characterization. Three stochastic processes leading to the distribution are also considered: (1) a three-dimensional random walk; (2) a birth, death, and immigration process; and (3) a thinned stochastic process.

In this article, we compare the zero-inflated Poisson (ZIP) and negative binomial (NB) distributions based on three most important criteria: the probability of zero, the mean value, and the variance. Our results show that with same mean value and variance, the ZIP distribution always has a larger probability of zeros; with same mean value and probability of zeros, the NB distribution always has a larger variance; and with same variance and probability of zeros, the ZIP distribution always has a larger mean value. We also study the properties of Vuong test in model selection in three cases by simulations.

We prove that the expectation of the binomial distribution of order k with success probability p is monotonically increasing with respect to p for all n and k. The result is extended to the problems on exchangeable random sequences and expectations of distributions of mixtures of binomial distributions of order k are studied. If the mixing measure is stochastically increasing with respect to its parameter, the expectation of the mixture of binomial distributions of order k becomes nondecreasing. As examples of mixing measures two submodels of beta distributions are examined and the resulting expectation of the mixture distribution is monotonically strictly increasing. Further, we prove some properties on the expectation of the $$\ell $$ -overlapping 1-runs in a sequence of independent and identically distributed n trials.

This article applies the Bernoulli process (the binomial distribution) to probabilistic seismic‐hazard analysis (PSHA), and the results are identical with the Poisson process. To determine the parameter p required of the binomial distribution, the probabilities associated with the seismic hazard are divided into two categories. The first consists of spatial probabilities that represent the likelihood that the target value of site ground motion is exceeded, given that an event of exceedance of m 0 occurs somewhere within a seismic source or a specific region. The spatial probabilities are not within any period but dependent on the chosen spatial range and lower limit of magnitude m 0 . The second is referred to as temporal probability, because this type of probability is always within a period, for example, 0.2 g peak ground acceleration (PGA) with a probability of 0.1 in 50 yrs. From the point of view of the binomial distribution, a temporal probability or a seismic‐hazard curve is conditional on given spatial range and m 0 , and is calculated on the basis of the number rather than the physical temporal process of earthquakes. Applying binomial distribution to PSHA does not require the independence of earthquakes but of ground motions at the site. Therefore, the aftershocks may not be removed from the catalog. In addition, the return period can be directly defined by the spatial probability and the average annual rate of exceedance of m 0 .

This paper proposes a new model for autoregressive time series of counts in terms of a convolution of Poisson and negative binomial random variables, known as Poisson–negative binomial (PNB) distribution. The corresponding first-order integer valued time series models are developed and their properties are discussed. The geometric PNB and the geometric semi PNB distributions are also introduced and studied.

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.

Some estimates are presented for the accuracy of approximation of the negative binomial distribution by the gamma distribution. These results are applied to estimate the convergence rate to the Student distribution for the distributions of some statistics constructed from samples with random sizes.

This paper uses the Stein-Chen method and the binomial w-functions to derive a non-uniform bound for approximating the cumulative distribution function of a sum of independent binomial random variables, each with parameters mi and pi, by a cumulative Poisson distribution function with mean P n=1 mipi. With this bound, the cumulative Poisson distribution function can be used as an approximation of the cumulative distribution function of the summands when all pi are small.

The beta distribution is a basic distribution serving several purposes. It is used to model data, and also, as a more flexible version of the uniform distribution, it serves as a prior distribution for a binomial probability. The bivariate beta distribution plays a similar role for two probabilities that have a bivariate binomial distribution. We provide a new multivariate distribution with beta marginal distributions, positive probability over the unit square, and correlations over the full range. We discuss its extension to three or more dimensions., Comment: 10 pages, 1 table, 1 figure

This paper uses Stein’s method and w-functions to determine a uniform bound for the Kolmogorov distance between the cumulative distribution function of a non-negative integer-valued random variable X and the negative binomial cumulative distribution function with parameters 0 r > and 1( 0,1) pq =− ∈ , where () rq p EX = and () EX is the mean of X. Two examples are provided to

The zero inflated negative binomial-beta exponential (ZINB-BE) distribution is developed, it is an alternative distribution for the excessive zero counts with overdispersion. The ZINB-BE distribution is a mixture of two distributions which are Bernoulli and negative binomial-beta exponential distributions. In this work, some characteristics of the proposed distribution are presented, such as, mean and variance. The maximum likelihood estimation is applied to parameter estimation of the proposed distribution. Finally some results of Monte Carlo simulation study, it seems to have high-efficiency when the sample size is large.The zero inflated negative binomial-beta exponential (ZINB-BE) distribution is developed, it is an alternative distribution for the excessive zero counts with overdispersion. The ZINB-BE distribution is a mixture of two distributions which are Bernoulli and negative binomial-beta exponential distributions. In this work, some characteristics of the proposed distribution are presented, such as, mean and variance. The maximum likelihood estimation is applied to parameter estimation of the proposed distribution. Finally some results of Monte Carlo simulation study, it seems to have high-efficiency when the sample size is large.

In this note we present a modification in the EM algorithm for the destructive negative binomial cure rate model. This alteration enables us to obtain the estimates of the whole parameter vector from the complete log-likelihood function, avoiding the corresponding observed log-likelihood function, which is more involved. To achieve this goal, we resort to the mixture representation of the negative binomial distribution in terms of the Poisson and gamma distributions.

The zero-truncated negative binomial-Erlang distribution is introduced. It is developed from negative binomial-Erlang distribution. In this work, the probability mass function is derived and some properties are included. The parameters of the zero-truncated negative binomial-Erlang distribution are estimated by using the maximum likelihood estimation. Finally, the proposed distribution is applied to real data, the number of methamphetamine in the Bangkok, Thailand. Based on the results, it shows that the zero-truncated negative binomial-Erlang distribution provided a better fit than the zero-truncated Poisson, zero-truncated negative binomial, zero-truncated generalized negative-binomial and zero-truncated Poisson-Lindley distributions for this data.

This article describes a generalization of the binomial distribution. The closed form probability function for the probability of k successes out of n correlated, exchangeable Bernoulli trials depends on the number of trials and its two parameters: the common success probability and the common correlation. The distribution is derived under the assumption that the common correlation between all pairs of Bernoulli trials remains unchanged conditional on successes in all completed trials. The distribution was developed to model bond defaults but may be suited to biostatistical applications involving clusters of binary data encountered in repeated measurements or toxicity studies of families of organisms. Maximum likelihood estimates for the parameters of the distribution are found for a set of binary data from a developmental toxicity study on litters of mice.

In this paper, we introduce and study geometric distribution of order k with a reward. In a sequence of binary trials, suppose that each time a success occurs a random reward is received. The distribution of the number of trials until the sum of consecutive rewards is equal to or exceeds the level k is called geometric distribution of order k with a reward. We obtain expressions for the probability mass function of this distribution.

SYNOPTIC ABSTRACTIn this article we study the problem of selecting among k independent Bernoulli populations whose success probabilities are unknown, and the sample size for each population is assumed to follow a discrete uniform distribution with known range. We consider two goals and propose procedures for each goal: (1) selecting the best and (2) selecting the best in comparison with a standard. The “best” is defined as that having the highest success probability. We derive the probability of a correct selection and the least favorable configuration for each procedure by using the exact binomial distribution, without any approximation. Simulations and examples are provided to illustrate our procedures.

We study the distributions of the random variables Sn and Vr related to a sequence of dependent Bernoulli variables, where Sn denotes the number of successes in n trials and Vr the number of trials necessary to obtain r successes. The purpose of this article is twofold: (1) Generalizing some results on the “nature” of the binomial and negative binomial distributions we show that Sn and Vr can follow any prescribed discrete distribution. The corresponding joint distributions of the Bernoulli variables are characterized as the solutions of systems of linear equations. (2) We consider a specific type of dependence of the Bernoulli variables, where the probability of a success depends only on the number of previous successes. We develop some theory based on new closed-form representations for the probability mass functions of Sn and Vr which enable direct computations of the probabilities.

A new generalization of the logarithmic series distribution has been obtained as a limiting case of the zero-truncated Mishra’s [10] generalized negative binomial distribution (GNBD). This distribution has an advantage over the Mishra’s [9] quasi logarithmic series distribution (QLSD) as its moments appear in compact forms unlike the QLSD. This makes the estimation of parameters easier by the method of moments. The first four moments of this distribution have been obtained and the distribution has been fitted to some well known data-sets to test its goodness of fit.

The Markov-Bernoulli geometric distribution is obta ined when a generalization, as a Markov process, of the independent Bernoulli sequence of random variab les is introduced by considering the success probability changes with respect to the Markov chai n. The resulting model is called the MarkovBernoulli model and it has a wide variety of applic ation fields. In this study, some characterizations are given concerning the Markov-Bernoulli geometric distribution as the distribution of the summation inde x of independent randomly truncated non-negative inte ger valued random variables. The achieved results generalize the corresponding characterizations conc erning the usual geometric distribution.

In this paper, we give a new improved negative binomial distribution with parameters r and M N p = to approximate the negative hypergeometric distribution with parameters

In this paper a new probability density function with bounded domain is presented. The new distribution arises from the generalized Lindley distribution proposed by Zakerzadeh and Dolati (2010). This new distribution that depends on two parameters can be considered as an alternative to the classical beta distribution. It presents the advantage of not including any special function in its formulation. After studying its most important properties, some useful results regarding insurance and inventory management applications are obtained. In particular, in insurance, we suggest a special class of distorted premium principles based on this distribution and we compare it with the well-known power dual premium principle. Since the mean of the new distribution can be normalized to give a simple parameter, this new model is appropriate to be used as a regression model when the response is bounded, being therefore an alternative to the beta regression model recently proposed in the statistical literature.

There has been a considerable recent attention in modeling over dispersed binomial data occurring in toxicology, biology, clinical medicine, epidemiology and other similar fields using a class of Binomial mixture distribution such as Beta Binomial distribution (BB) and Kumaraswamy-Binomial distribution (KB). A new three-parameter binomial mixture distribution namely, McDonald Generalized Beta Binomial (McGBB) distribution has been developed which is superior to KB and BB since studies have shown that it gives a better fit than the KB and BB distribution on both real life data set and on the extended simulation study in handling over dispersed binomial data. The dispersion parameter will be treated as nuisance in the analysis of proportions since our interest is in the parameters of McGBB distribution. In this paper, we consider estimation of parameters of this MCGBB model using Quasi-likelihood (QL) and Quadratic estimating functions (QEEs) with dispersion. By varying the coefficients of the QEE’s we obtain four sets of estimating equations which in turn yield four sets of estimates. We compare small sample relative efficiencies of the estimates based on QEEs and quasi-likelihood with the maximum likelihood estimates. The comparison is performed using real life data sets arising from alcohol consumption practices and simulated data. These comparisons show that estimates based on optimal QEEs and QL are highly efficient and are the best among all estimates investigated.

This paper uses Stein’s method and the specic w-function to determine a non-uniform bound on the point metric of the beta binomial distribution with parameters n, and and a binomial distribution with parameters n and + . Numerical examples are provided to illustrate the result obtained. Mathematics Subject Classication: 62E17, 60F05

We study inference and diagnostics for count time series regression models that include a feedback mechanism. In particular, we are interested in negative binomial processes for count time series. We study probabilistic properties and quasi-likelihood estimation for this class of processes. We show that the resulting estimators are consistent and asymptotically normally distributed. These facts enable us to construct probability integral transformation plots for assessing any assumed distributional assumptions. The key observation in developing the theory is a mean parameterized form of the negative binomial distribution. For transactions data, it is seen that the negative binomial distribution offers a better fit than the Poisson distribution. This is an immediate consequence of the fact that transactions can be represented as a collection of individual activities that correspond to different trading strategies.

Zero-inflated negative binomial distribution is characterized in this paper through a linear differential equation satisfied by its probability generating function.