Two stochastic models for the geeta distribution

The first stochastic model is based upon two urns A and B, where A contains a fixed number of white and black balls and B is empty. The player selects an integer β ≥, 2 and draws the balls one by one (with replacement) from urn A and balls of the same colour are put in urn B. The process is continued as long as the number of white balls in B exceeds (β-1) times the number of black balls in B. The player stops after drawing β(x-1) balls and is declared to be a winner if urn B has (x-1) black balls. It is shown that x has the Geeta distribution.

Assuming that the mean μ is a function of two parameters θ and β it has been shown that for small changes inthe value of θ there exists a difference-differential equation which leads to the Geeta distribution.     

A q-Pólya urn model and the q-Pólya and inverse q-Pólya distributions

A q-Pólya urn model is introduced by assuming that the probability of drawing a white ball at a drawing varies geometrically, with rate q, both with the number of drawings and the number of white balls drawn in the previous drawings. Then, the probability mass functions and moments of (a) the number of white balls drawn in a specific number of drawings and (b) the number of black balls drawn until a specific number of white balls are drawn are derived. These two distributions turned out to be q-analogs of the Pólya and the inverse Pólya distributions, respectively. Also, the limiting distributions of the q-Pólya and the inverse q-Pólya distributions, as the number of balls in the urn tends to infinity, are shown to be a q-binomial and a negative q-binomial distribution, respectively. In addition, the positive or negative q-hypergeometric distribution is obtained as conditional distribution of a positive or negative q-binomial distribution, given its sum with another positive or negative q-binomial distribution, independent of it.     

The negative contagion reflection of the polya-eggenberger distribution

The Polya-Eggenberger distribution Involves drawing a ball from an urn containing black and white balls and, after each drawing, returning the ball together with s balls of the same color, The model represents positive contagion since the added balls are the same color as the one drawn, See Johnson and Kotz, (1977),

This paper derives and examines the probability distribution which results from the Polya-Eggenberger model with only one change namely, the s additional balls added after each drawing are of the opposite color, producing a negative contagion model.

Formulas in closed form are presented for the probability distribution function, the mean and variance, all binomial moments and, where s is greater than or equal to the number of balls in the urn at start, the mode, A formula for the mode is conjectured where s is less than the number of balls in the urn at start.

Finally, the probability of obtaining k black balls in n drawings is shown in certain instances to be equal to A_nk/n!

where A_nk are the Eulerian numbers.     

The two-color urn model: recursions for the moments

For the two-color reinforcement-depletion urn model, with balancing reinforcement and depletion held constant over cycles, a recursive formula is given from which all factorial moments (for white balls, for example) can be determined. When the reinforcement of each color is positive, the stationary distribution of white balls (infinite number of cycles) turns out to be determined by three parameters. namely (i) the total number of balls in the urn, (ii) the richness of the reinforcement, or ratio of white ball reinforcement to total reinforcement, and (iii) the size of the white ball reinforcement. In addition, the distribution mimics the binomial (with less variance and skewness (√β₁:) ) and from formulas for the exact first four moments rapidly approaches normality. On the basis of the few cases studied, an approximating Gram-Charlier distribution with a binomial nucleus is only moderately successful     

On some distributions arising from certain generalized sampling schemes

With the notion of success in a series of trials extended tD refer to a run of like outcomes, several new distributions are obtained as the result of sampling from an urn without replacement. or with additional replacements., In this context, the hy-pergeometric, negative hypergeometric, logarithmic series, generalized Waring, Polya and inverse Polya distributions are extended and their properties are studied     

Confidence interval estimation for negative binomial group distribution

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.     

A randomly reinforced urn

We study an urn containing balls of two or more colors. The urn is sequentially sampled. Each time a ball is extracted from the urn it is reintroduced in it together with a random number of balls of the same color: the distribution of the number of added balls may depend on the color extracted. We prove asymptotic results for the process of colors generated by the urn and for the process of its compositions. Applications to sequential clinical trials are considered as well as connections with response-adaptive design of experiments in a Bayesian framework.     

The estimation of the number of balls put in a sequence of urns

Let X₁,…,X_2n be independent and identically distributed copies of the non-negative integer valued random variable X distributed according to the unknown frequency function f(x). A total of 2n disjoint sequences of urns, each consisting of k urns, are given. X_j balls are placed in urn sequence j (1 ≤ j ≤ 2n). Each ball is placed in an urn of a given sequence with a certain known probability independently of the other balls. The variables X₁,…,X_2n are not observed; rather we observe whether certain pairs of urns are both empty or not. Our object is to estimate the mean μ of the number of balls X. Two different kinds of estimators of μ are investigated. One of the estimators studied is a method of moments type estimator while the other is motivated by the maximum likelihood principle. These estimators are compared on the basis of their asymptotic mean squared error as k tends to infinity. An application of these results to a problem in genetics involved with estimating codon substitution rates is discussed.     

Quota fulfillment times

An example of the classical occupancy problem is to sample with replacement from an urn containing several colours of balls and count the number of balls sampled until a given number of “quotas” are filled. This and the corresponding random variable for sampling without replacement will be referred to as quota fulfillment times. Asymptotic and exact methods for computing moments and distributions are given in this paper. Moments of quota fulfillment times are related to moments of order statistics of beta and gamma random variables. Most of the results for sampling without replacement and some of the results for sampling with replacement are believed to be new. Some other known sampling-with-replacement results are given for comparative purposes.     

EXCHANGEABLE PAIRS OF BERNOULLI RANDOM VARIABLES,KRAWTCHOUCK POLYNOMIALS,AND EHRENFEST URNS

This paper derives characterizations of bivariate binomial distributions of the Lancaster form with Krawtchouk polynomial eigenfunctions. These have been characterized by Eagleson, and we give two further characterizations with a more probabilistic flavour: the first as sums of correlated Bernoulli variables; and the second as the joint distribution of the number of balls of one colour at consecutive time points in a generalized Ehrenfest urn. We give a self‐contained development of Krawtchouck polynomials and Eagleson’s theorem.     

Exact Short Confidence Intervals from Discrete Data

Suppose that X is a discrete random variable whose possible values are {0, 1, 2,⋯} and whose probability mass function belongs to a family indexed by the scalar parameter θ . This paper presents a new algorithm for finding a 1 − α confidence interval for θ based on X which possesses the following three properties: (i) the infimum over θ of the coverage probability is 1 − α ; (ii) the confidence interval cannot be shortened without violating the coverage requirement; (iii) the lower and upper endpoints of the confidence intervals are increasing functions of the observed value x . This algorithm is applied to the particular case that X has a negative binomial distribution.     

Coupling methods in approximations

Let X and Y be two arbitrary k-dimensional discrete random vectors, for k ≥ 1. We prove that there exists a coupling method which minimizes P( X ≠ Y ). This result is used to find the least upper bound for the metric d( X, Y ) = sup_A|P( X ∈ A ) ? P( Y ∈ A )| and to derive the inequality d(Σ X _i, Σ Y _i) ≤ Σd( X _i, Y _i). We thus obtain a unified method to measure the disparity between the distributions of sums of independent random vectors. Several examples are given.     

“Faulty inspeciton” distributions

The distribution of the number of items observed to be defective in samples from a finite population. When detection of defectiveness is not certain, is obtained. The distribution of waiting time all a specified number of defective items is observed is also considered.

     

A Bayes Theorem Urn Model Optimization Problem

An urn model is a finite collection of indistinguishable urns together with an arbitrary distribution of a finite number of balls (bills) of k colors (denominations) into the urns. A Bayes theorem expectation optimization problem associated with certain urn models is discussed.     

An EM algorithm for estimating negative binomial parameters

An EM algorithm is proposed for computing estimates of parameters of the negative bi-nomial distribution; the algorithm does not involve further iterations in the M-step, in contrast with the one given in Schader & Schmid (1985). The approach can be applied to the corresponding problem in the logarithmic series distribution. The convergence of the proposed scheme is investigated by simulation, the observed Fisher information is derivedand numerical examples based on real data are presented.     

Properties of Lagrangian distributions

Abstract

Lagrange’s expansion is the power series expansion of the inverse function of an analytic function, and it leads to general Lagrangian distributions of the first kind as well as of the second kind. We present some theorems in which different sets of two analytic functions provide a Lagrangian distribution. Potential applicability of the theorem to tandem queueing process is studied.     

Normal approximation in an urn model with indistinguishable balls

For the Bose-Einstein Statistics, where n indistinguishable balls are distributed in m urns such that all the arrangements are equally likely, define the random variables

M_k = number of urns containing exactly k balls each;

N_k = number of urns containing at least k balls each.

We consider the approximation of the distributions of M_k and N_k by suitable normal distributions, for large but finite m. Estimates are found for the error in the approximation to both the probability mass function and the distribution function in each case. These results apply also to the alternative model where no urn is allowed to be empty. The results are illustrated by some numerical examples.     

Incomplete moments of modified power series districutions with applications

The recurrence relations between the incomplete moments and the factorial incomplete moments of the modified power series distributions (MPSD) are derived. These relations are employed to obtain the experessions for the incomplete moments and the incomplete factorial moments of some particular members of the MPSD class such as the generalized negative binomial, the generalized Poisson, the generalized logrithmic series, the lost game distribution and the distribution of the number of customers served in a busy period. An application of the incomplete moments of the generalized Poisson distribution is provided in the economic selection of a manufactured product. A numerical example is provided using the Poisson distribution and the Generalized Poisson distribution. The example illustrates the difference in results using the two models     

Estimation of binomial parameters when both , are unknown

We revisit the classic problem of estimation of the binomial parameters when both parameters n,p are unknown. We start with a series of results that illustrate the fundamental difficulties in the problem. Specifically, we establish lack of unbiased estimates for essentially any functions of just n or just p. We also quantify just how badly biased the sample maximum is as an estimator of n. Then, we motivate and present two new estimators of n. One is a new moment estimate and the other is a bias correction of the sample maximum. Both are easy to motivate, compute, and jackknife. The second estimate frequently beats most common estimates of n in the simulations, including the Carroll–Lombard estimate. This estimate is very promising. We end with a family of estimates for p; a specific one from the family is compared to the presently common estimate and the improvements in mean-squared error are often very significant. In all cases, the asymptotics are derived in one domain. Some other possible estimates such as a truncated MLE and empirical Bayes methods are briefly discussed.