A comparison of some approximations to the Wilcoxon-Mann-Whitney distribution

The following approximations to the exact distribution of the Wilcoxon rank sum test (Mann-Whitney U-test) are compared at or near significance levels .05, .025, .01 and .005: the normal approximation with continuity correction, the simple and complete version of the Edgeworth series approximation proposed by Fix and Hodges (1955), Buckle, Kraft and van Eeden’s (1969a) uniform approximation and Iman’s (1976) recently proposed approximation. The comparison takes into account simplicity of application as well as closeness, and some preference rules are suggested for different user objectives.     

Improved approximations for the inverse moments of the positive binomial distribution

Several alternatives to the most common approximation to the inverse moments of the positive binomial distribution are obtained. The method is based on equating moments and gives considerably better approximations for some values of the parameters.     

On the beta-correlated binomial (bcb) distribution - a three parameter generalization of the binomial distribution

A modification of the beta-correlated binomial (BCB) distribution of Paul (1985) is proposed. This modification over- comes some theoretical difficulties encountered in the original BCB distribution.     

A note on LeCam’s bound for the distance between the Poisson binomial and the Poisson distribution

n possibly different success probabilities p ₁, p ₂, ..., p _n is frequently approximated by a Poisson distribution with parameter λ = p ₁ + p ₂ + ... + p _n . LeCam's bound p ² ₁ + p ² ₂ + ... + p _n ² for the total variation distance between both distributions is particularly useful provided the success probabilities are small. The paper presents an improved version of LeCam's bound if a generalized d-dimensional Poisson binomial distribution is to be approximated by a compound Poisson distribution. Received: May 10, 2000; revised version: January 15, 2001     

Patched approximations and their convergence

Abstract

Patched approximations of copulas unify ordinal sums, shuffles of Min, checkerboard, and checkmin approximations. We give a characterization of patched approximations and an error bound of the approximations in Sobolev norm. Patched approximations with uniform marginal conditional distributions are shown to arise naturally. We prove that these uniform patched approximations converge uniformly and in the Sobolev norm. The latter convergence is settled by showing the convergence almost everywhere of the first partial derivatives. We also show that the independence copula can be approximated by conditional mutual complete copulas in the Sobolev norm.     

A generalization of the binomial distribution

This paper presents a new departure in the generalization of the binomial distribution by adopting the assumption that the underlying Bernoulli trials take on the values α or β where α < β, rather than the conventional values 0 or 1. The adoption of this more general assumption renders the binomial distribution a four-parameter distribution of the form B(n,p,α,β), and requires the generalization of Romanovsky's (1923) reduction formula for central moments. This paper assesses the usefulness of B(n,p,α,β), and its reduction formula, in the numerical analysis of two problems of interest to decision theorists.     

A three-parameter generalization of the binomial distribution

A new three-parameter distribution, a generalization of the binomial, th ebeta-binomial (BB) and the correlated binomial (CB) distributions, is derived. Improvement in fit of the new distribution over the BB and the CB distributions has been found for a set of real data     

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.     

Enhancement for two commonly-used approximations for the inverse cumulative function of the normal distribution

Two commonly used approximations for the inverse distribution function of the normal distribution are Schmeiser's and Shore's. Both approximations are based on a power transformation of either the cumulative density function (CDF) or a simple function of it. In this note we demonstrate, that if these approximations are presented in the form of the classical one-parameter Box-Cox transformation, and the exponent of the transformation is expressed as a simple function of the CDF, then the accuracy of both approximations may be considerably enhanced, without losing much in algebraic simplicity. Since both approximations are special cases of more general four-parameter systems of distributions, the results presented here indicate that the accuracy of the latter, when used to represent non-normal density functions, may also be considerably enhanced.     

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.     

One mixed negative binomial distribution with application

In this paper we do some research on a three-parameter distribution which is called beta-negative binomial (BNB) distribution, a beta mixture of negative binomial (NB) distribution. The closed form and the factorial moment of the BNB distribution are derived. In addition, we present the recursion on the pdf of BNB stopped-sum distribution, and make stochastic comparison between BNB and NB distributions. Furthermore, we have shown that BNB distribution has heavier tail than NB distribution. The application of BNB distribution is carried out on one sample of insurance data. Based on the results, we have shown that the BNB provides a better fit compared to the Poisson and the NB for count data.     

Sample size determinations when two binomial proportions are very small

Sample size determination for testing the hypothesis of equality of proportions with a specified type I and type I1 error probabilitiesis of ten based on normal approximation to the binomial distribution. When the proportionsinvolved are very small, the exact distribution of the test statistic may not follow the assumed distribution. Consequently, the sample size determined by the test statistic may not result in the sespecifiederror probabilities. In this paper the author proposes a square root formula and compares it with several existing sample size approximation methods. It is found that with small proportion (p≦.01) the squar eroot formula provides the closest approximation to the exact sample sizes which attain a specified type I and type II error probabilities. Thes quare root formula is simple inform and has the advantage that equal differencesare equally detectable.     

A generalization of the beta–binomial distribution

Summary.  The paper describes a distribution generated by the Gaussian hypergeometric function that may be seen as a generalization of the beta–binomial distribution. It is expressed as a generalized beta mixture of a binomial distribution. This new mixing distribution allows the existence of a mode and an antimode, which is very useful for fitting some data sets. Two examples illustrate the greater versatility of the new distribution compared with the beta–binomial distribution.     

Exact distribution of the convolution of negative binomial random variables

In this note, we derive the exact distribution of S by using the method of generating function and BELL polynomials, where S = X₁ + X₂ + ??? + X_n, and each X_i 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.     

New approximation of the normal distribution function

Two new and simple expressions, one for 0 ≤ x ≤ 2 and another for x > 2, for the normal distribution function, are developed which can be easily computed on desk calculators. They are also comparable in accuracy to the one developed by Patry and Keller (1964).     

The inverse binomial distribution as a statistical model

A particular case of Jain and Consul's (1971) generalized neg-ative binomial distribution is studied. The name inverse binomial is suggested because of its close relation with the inverse Gaussian distribution. We develop statistical properties including conditional inference of a parameter. An application using real data is given.     

Expectation identity for the binomial distribution and its application in the calculations of high-order binomial moments

ABSTRACT

For the exponential families normal, gamma, beta, Poisson, and negative binomial, there exists an expectation identity for each of the family. For the binomial family, we discover an expectation identity, which is useful in analytical calculations of its high-order moments.     

Mixture formulations of a bivariate negative binomial distribution with applications

This paper considers further mixture formulations of the bivariate negative binomial (BNB) distribution of Edwards and Gurland (1961) and Subrahmaniam (1966). These formulations and some known ones are applied (1) to obtain a bivariate generalized negative binomial (BGNB) distribution of Bhattacharya (1966), (2) to establish a connection between the accident-proneness models given by the BNB, BGNB and Bhattacharya's bivariate distributions, and (3) to compute the grade correlation and distribution function of the Wicksell-Kibble bivariate gamma distribution.