Randomization. pro and contra 1

We discuss a stochastic urn model in which there are two urns A and B. B is originally empty and A contains some fixed number of white and black balls. A player selects integers n>O and b>O. Balls are drawn with replacement in A and balls of the same color are put in B as long as the number of white balls in B exceeds (b-1) times the number of black balls in B. Under this condition, the player stops after drawing n+bx balls and is declared to be a winnear if urn B has x black balls. This number of black balls, x, is shown to have the generalized negative binomial distribution     

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.     

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.     

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.     

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     

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.     

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.     

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.     

Characterizations of asymptotic distributions of continuous-time Pólya processes

Abstract

We propose an elementary but effective approach to studying a general class of Poissonized tenable and balanced urns on two colors. We characterize the asymptotic behavior of the process via a partial differential equation that governs the process, coupled with the method of moments applied in a bootstrapped manner. We show that the limiting distribution of the process underlying the Bagchi-Pal urn is gamma. We also look into the tenable and balanced processes associated with randomized replacement matrix. Similar results carry over to the process, with minor modifications in the methods of proof, done mutatis mutandis.     

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.     

A weighted likelihood ratio of two related negative hypergeometric distributions

In this paper we consider some related negative hypergeometric distributions arising from the problem of sampling without replacement from an urn containing balls of different colours and in different proportions but stopping only after some specific number of balls of different colours have been obtained. With the aid of some simple recurrence relations and identities we obtain in the case of two colours the moments for the maximum negative hypergeometric distribution, the minimum negative hypergeometric distribution, the likelihood ratio negative hypergeometric distribution and consequently the likelihood proportional negative hypergeometric distribution. To the extent that the sampling scheme is applicable to modelling data as illustrated with a biological example and, in fact, many situations of estimating Bernoulli parameters for binary traits within a finite population, these are important first-step results.     

The exact distribution of the maximum,minimum and the range of Multinomial/Dirichlet and Multivariate Hypergeometric frequencies

The exact distribution of the maximum and minimum frequencies of Multinomial/Dirichlet and Multivariate Hypergeometric distributions of n balls in m urns is compactly represented as a product of stochastic matrices. This representation does not require equal urn probabilities, is invariant to urn order, and permits rapid calculation of exact probabilities. The exact distribution of the range is also obtained. These algorithms satisfy a long-standing need for routines to compute exact Multinomial/Dirichlet and Multivariate Hypergeometric maximum, minimum, and range probabilities in statistical computation libraries and software packages.     

Probability and the Game of INSTANT INSANITY

INSTANT INSANITY is a popular game originating in middle Europe several hundred years ago as a folk toy. It may be purchased at any drugstore. There are four different colors showing on each side of the row of four blocks—red, green, blue and white. To play the game one mixes the four blocks. The purpose of the game is to rearrange the blocks so that there are again four different colors showing on each side. Each of the four blocks has four colors, two or more faces of each block having the same color. No block is identical to any of the others. There are a number of problems one might be interested in solving.     

Interim analysis of clinical trials based on urn models

Clinical trials usually involve efficient and ethical objectives such as maximizing the power and minimizing the total failure number. Interim analysis is now a standard technique in practice to achieve these objectives. Randomized urn models have been extensively studied in the literature. In this paper, we propose to perform interim analysis on clinical trials based on urn models and study its properties. We show that the urn composition, allocation of patients and parameter estimators can be approximated by a joint Gaussian process. Consequently, sequential test statistics of the proposed procedure converge to a Brownian motion in distribution and the sequential test statistics asymptotically satisfy the canonical joint distribution defined in Jennison & Turnbull (Jennison & Turnbull 2000. Group Sequential Methods with Applications to Clinical Trials, Chapman and Hall/CRC). These results provide a solid foundation and open a door to perform the interim analysis on randomized clinical trials with urn models in practice. Furthermore, we demonstrate our proposal through examples and simulations by applying sequential monitoring and stochastic curtailment techniques. The Canadian Journal of Statistics 40: 550–568; 2012 © 2012 Statistical Society of Canada     

Occupancy distributions for testing marginal homogeneity

In this paper different pairs of urns are considered. We are interested in the number of ways in which N balls can be placed into the urns so that differences and ratios between the number of balls in certain pairs of urns are the same.     

Parameter estimation in the classical occupancy model

Under the classical occupancy model, balls are randomly and independently allocated into cells (by assuming that each arrangement of balls is equally probable) in such a way that the random variable of interest is the empty cell number. In some practical applications the total cell number is known and the target parameter turns out to be the number of balls which is estimated on the basis of the observed empty cell count. For instance, the classical occupancy model is commonly adopted for airborne-microorganism abundance estimation, a topic of central importance in environmental microbiology, in aerobiology and in occupational medicine. The classical occupancy model is also applied to the analysis of US National AIDS surveillance data (which are inflated by duplicate reporting) in order to estimate the true population size of AIDS cases. Since many inaccuracies and misunderstandings are present in applied literature, the aim of the present paper is to introduce a formal analysis of the inferential issues connected with the estimation of the number of balls.     

Proof of a lattice paths conjecture connected to the tennis ball problem

We give a short history of the so-called tennis ball problem, and discuss its relation to lattice path enumeration. We also prove a conjecture related to a solution of the symmetric case, namely when the number of balls removed each turn is exactly half the number inserted.     

Wavelet Thresholding Estimation in a Poissonian Interactions Model with Application to Genomic Data

This paper deals with the study of dependencies between two given events modelled by point processes. In particular, we focus on the context of DNA to detect favoured or avoided distances between two given motifs along a genome suggesting possible interactions at a molecular level. For this, we naturally introduce a so‐called reproduction function h that allows to quantify the favoured positions of the motifs and that is considered as the intensity of a Poisson process. Our first interest is the estimation of this function h assumed to be well localized. The estimator based on random thresholds achieves an oracle inequality. Then, minimax properties of on Besov balls are established. Some simulations are provided, proving the good practical behaviour of our procedure. Finally, our method is applied to the analysis of the dependence between promoter sites and genes along the genome of the Escherichia coli bacterium.