A Unified Way to Represent and Compute the Cumulative Distribution Function for Noncentral t,F, and χ2

This short article shows an unified approach to representing and computing the cumulative distribution function for noncentral t, F, and χ². Unlike the existing algorithms, which involve different expansion and/or recurrence, the new approach consistently represents all the three noncentral cumulative distribution functions as the integral of the normal cumulative distribution function and χ² density function.     

A Note on the Noncentral Beta Distribution Function

The noncentral beta and the related noncentral F distributions have received much attention during the last decade, as is evident from the works of Norton, Lenth, Frick, Lee, Posten, Chattamvelli, and Chattamvelli and Shanmugam. This article reviews the existing algorithms for computing the cumulative distribution function (cdf) of a noncentral beta random variable, and proposes a simple algorithm, based on a sharp error bound, for computing the cdf. A variation of the noncentral beta random variable when the noncentrality is associated only with the denominator χ² and its computational details are also discussed.     

Calculation Methods for Wallenius' Noncentral Hypergeometric Distribution

Two different probability distributions are both known in the literature as “the” noncentral hypergeometric distribution. Wallenius' noncentral hypergeometric distribution can be described by an urn model without replacement with bias. Fisher's noncentral hypergeometric distribution is the conditional distribution of independent binomial variates given their sum. No reliable calculation method for Wallenius' noncentral hypergeometric distribution has hitherto been described in the literature. Several new methods for calculating probabilities from Wallenius' noncentral hypergeometric distribution are derived. Range of applicability, numerical problems, and efficiency are discussed for each method. Approximations to the mean and variance are also discussed. This distribution has important applications in models of biased sampling and in models of evolutionary systems.     

An alternative representation of noncentral beta and F distributions

In this paper the doubly noncentral beta and F distributions are represented alternatively by using the results on the product of two hypergeometric functions. Their moments and the cumulative distribution functions are also given in terms of hypergeometric functions, which can be easily calculated by the Mathematica package.     

Efficient computation of the noncentral x2 distribution

In this article we compare and contrast finite algorithms for computing the noncentral x² distribution function for odd or even degrees of freedom with the algorithms proposed recently by Ashour & Abdel-Samad (1990). We also obtain an alternative error bound for Ruben's (1974a) algorithm for even degrees of freedom and analyze the rate of convergence of two common infinite series representations for computing the cdf     

A Family Which Integrates the Generalized Charlier Series and Extended Noncentral Negative Binomial Distributions

The generalized Charlier series distribution includes the binomial distribution, and the noncentral negative binomial distribution extends the negative binomial distribution. The present article proposes a family of counting distributions, which contains both the generalized Charlier series and extended noncentral negative binomial distributions. Compound and mixture formulations of the proposed distribution are given. The probability mass function is expressible in terms of the confluent hypergeometric function as well as the Gauss hypergeometric function. Recursive formulae for probability mass function have been studied by Panjer, Sundt and Jewell, Schröter, Sundt, and Kitano et al. in the context of insurance risk. This article explores horizontal, vertical, triangular, and diagonal recursions. Recursive formulae as well as exact expressions for descending factorial moments are studied. The proposed distribution allows overdispersion or underdispersion relative to a Poisson distribution. An illustrative example of data fitting is given.     

On the computation of the noncentral <Emphasis Type="Italic">F</Emphasis> and noncentral beta distribution

Unfortunately many of the numerous algorithms for computing the comulative distribution function (cdf) and noncentrality parameter of the noncentral F and beta distributions can produce completely incorrect results as demonstrated in the paper by examples. Existing algorithms are scrutinized and those parts that involve numerical difficulties are identified. As a result, a pseudo code is presented in which all the known numerical problems are resolved. This pseudo code can be easily implemented in programming language C or FORTRAN without understanding the complicated mathematical background. Symbolic evaluation of a finite and closed formula is proposed to compute exact cdf values. This approach makes it possible to check quickly and reliably the values returned by professional statistical packages over an extraordinarily wide parameter range without any programming knowledge. This research was motivated by the fact that a very useful table for calculating the size of detectable effects for ANOVA tables contains suspect values in the region of large noncentrality parameter values compared to the values obtained by Patnaik’s 2-moment central-F approximation. The cause is identified and the corrected form of the table for ANOVA purposes is given. The accuracy of the approximations to the noncentral-F distribution is also discussed. The authors wish to thank Mr. Richárd Király for his preliminary work. The authors are grateful to the Editor and Associate Editor of STCO and the unknown reviewers for their helpful suggestions.     

Calculating the density and distribution function for the singly and doubly noncentral F

Simple, closed form saddlepoint approximations for the distribution and density of the singly and doubly noncentral F distributions are presented. Their overwhelming accuracy is demonstrated numerically using a variety of parameter values. The approximations are shown to be uniform in the right tail and the associated limitating relative error is derived. Difficulties associated with some algorithms used for exact computation of the singly noncentral F are noted.     

Bivariate distributions of some ratios of independent noncentral chi-square random variables

The bivariate distributions of three pairs of ratios of in¬dependent noncentral chi-square random variables are considered. These ratios arise in the problem of computing the joint power function of simultaneous F-tests in balanced ANOVA and ANCOVA. The distributions obtained are generalizations to the noncentral case of existing results in the literature. Of particular note is the bivariate noncentral F distribution, which generalizes a special case of Krishnaiah*s (1964,1965) bivariate central F distribution. Explicit formulae for the cdf's of these distribu¬tions are given, along with computational procedures     

Further results on the trace of a noncentral wishart matrix

In an earlier article Mathai (1980) has given compact representations for the moments and cumulants of the trace of a noncentral Wishart matrix. He has also shown that (trA-ntr;∑)/(2ntri∑²)172. is asymptotically standard normal where A is a noncentral wishart matrix with n degrees of freedom and covariance matrix [0, In the present article explicit expressions for the exact density of the trace are given in terms of confluent hypergeometric functions and in terms of zonal polynomials for the general case and as finite sums when the sample size is odd. As a consequence of some of these representations some summation formulae for zonal polynomials are also given     

Approximations to noncentral distributions

Two methods for approximating the distribution of a noncentral random variable by a central distribution in the same family are presented. The first consists of relating a stochastic expansion of a random variable to a corresponding asymptotic expansion for its distribution function. The second approximates the cumulant generating function and is used to provide central χ² and gamma approximations to the noncentral χ² and gamma distributions.     

SOME SIMPLE APPROXIMATIONS FOR THE DOUBLY NONCENTRAL z DISTRIBUTION

We give two simple approximations for evaluating the cumulative probabilities of the doubly noncentral z distribution. These can easily be used for evaluating the cumulative probabilities of the doubly noncentral F distribution as well. We compare our results with those obtained by Tiku (1965) using series expansion. An industrial situation where a quality characteristic of interest follows the doubly noncentral z distribution is also cited. However, in this case the exact probabilities could be calculated using results on the ratio of two normal variables.     

Evaluation of probabilities for the noncentral distributions and the difference of two T-variables with a desk calculator

It is demonstrated that integrals of the noncentral chi-square, noncentral F and noncentral T distributions can be evaluated on desk calculators. The same procedure can be used to compute probabilities for the distribution of the difference of two T-variables with equal degrees of freedom. The proposed method of computation can be used with any computer which yields probabilities for the chi-square and F distributions.     

New representations of the noncentral chi-square density and cumulative

Representations of noncentral chi-square cumulative distribution function and probability density function are reviewed and new repre¬sentations are given. One representation of the cdf in terms of an integral is easily computed on any machine which has an accurate algorithm for computing the normal cdf.     

On the computation of noncentral chi-squared distributions

A computational formula for computing the cumulative distribution function of noncentral chi-squared distributions with odd degrees of freedom is given.     

A general non-central hypergeometric distribution

We construct a general non-central hypergeometric distribution, which models biased sampling without replacement. Our distribution is constructed from the combined order statistics of two samples: one of independent and identically distributed random variables with absolutely continuous distribution F and the other of independent and identically distributed random variables with absolutely continuous distribution G. The distribution depends on F and G only through F○G^{( ? 1)} (F composed with the quantile function of G), and the standard hypergeometric distribution and Wallenius’ non-central hypergeometric distribution arise as special cases. We show in efficient economic markets the quantity traded has a general non-central hypergeometric distribution.     

A Nonparametric Comparison of Conditional Distributions with Nonnegligible Cure Fractions

Survival data with nonnegligible cure fractions are commonly encountered in clinical cancer clinical research. Recently, several authors (e.g. Kuk and Chen, Biometrika 79 (1992) 531; Maller and Zhou, Journal of Applied Probability, 30 (1993) 602; Peng and Dear, Biometrics, 56 (2000) 237; Sy and Taylor, Biometrics 56 (2000) 227) have proposed to use semiparametric cure models to analyze such data. Much of the existing work has been emphasized on cure detections and regression techniques. In contrast, this project focuses on the hypothesis testing in the presence of a cure fraction. Specifically, our interest lies in detecting whether there exists survival differences among noncured patients between treatment arms. For this purpose, we investigate the use of a modified Cramér-von Mises statistic for two-sample survival comparisons within the framework of cure models. Such a test has been studied by Tamura et al., (Statistics in Medicine 19, 2000, 2169) using bootstrap procedure. We will focus on developing asymptotic theory and convergent algorithms in this paper. We show that the limiting distributions of the Cramér-von Mises statistic under the null hypothesis can be represented by stochastic integrals and a weighted noncentral chi-squares. Both representations lead to concrete numerical schemes for computing the limiting distributions. The algorithms can be easily implemented for data analysis and significantly reduce computing time compared to the bootstrap approach. For illustrative purposes, we apply the proposed test to a published clinical trial.     

Efficient time/space algorithm to compute rectangular probabilities of multinomial, multivariate hypergeometric and multivariate Pólya distributions

The computation of rectangular probabilities of multivariate discrete integer distributions such as the multinomial, multivariate hypergeometric or multivariate Pólya distributions is of great interest both for statistical applications and for probabilistic modeling purpose. All these distributions are members of a broader family of multivariate discrete integer distributions for which computationaly efficient approximate methods have been proposed for the evaluation of such probabilities, but with no control over their accuracy. Recently, exact algorithms have been proposed for computing such probabilities, but they are either dedicated to a specific distribution or to very specific rectangular probabilities. We propose a new algorithm that allows to perform the computation of arbitrary rectangular probabilities in the most general case. Its accuracy matches or even outperforms the accuracy exact algorithms when the rounding errors are taken into account. In the worst case, its computational cost is the same as the most efficient exact method published so far, and is much lower in many situations of interest. It does not need any additional storage than the one for the parameters of the distribution, which allows to deal with large dimension/large counting parameter applications at no extra memory cost and with an acceptable computation time, which is a major difference with respect to the methods published so far.     

Sampling Methods for Wallenius' and Fisher's Noncentral Hypergeometric Distributions

Several methods for generating variates with univariate and multivariate Walleniu' and Fisher's noncentral hypergeometric distributions are developed. Methods for the univariate distributions include: simulation of urn experiments, inversion by binary search, inversion by chop-down search from the mode, ratio-of-uniforms rejection method, and rejection by sampling in the τ domain. Methods for the multivariate distributions include: simulation of urn experiments, conditional method, Gibbs sampling, and Metropolis-Hastings sampling. These methods are useful for Monte Carlo simulation of models of biased sampling and models of evolution and for calculating moments and quantiles of the distributions.     

A note on exact calculation of the non central hypergeometric distribution

Direct calculation of the non central hypergeometric (NH) distribution and its moments can present computational issues in both efficiency and accuracy. In response, several methods, both approximate and exact, for calculating the NH mean and variance have appeared in the literature. We add to this body of work, a straight-forward, exact method that is easily programed, efficient, and computationally stable. Specifically, by considering the logs of the values of the NH probability mass function (pmf) and then shifting the exponents so that, prior to normalization, the mode acquires a value of 1, concerns for overflow are eliminated.