Noncentral Distribution Research Articles

Bivariate non-central negative binomial distribution: Another generalisation

Another bivariate generalisation (Type V) of the non-central negative binomial distribution is considered. This generalisation is constructed (i) as a latent structure model; (ii) as an extension of an accident proneness model investigated by Edwards/Gurland (1961); and (iii) as a reversible stochastic counter model. The third construction gives, as a result, an apparently new formulation of the Edwards/Gurland model. The probabilities, moments, recurrence formulas and some properties are given. An application to the data used by Holgate (1966) is considered.

Optimal bounds for the distributions of some test criteria for tests of dimensionality

Optimal upper bounds are obtained for the distributions of two functions of the m r smallest latent roots of HE1, where E and H have Wishart distributions with identical covariance matrices; E has a central distribution while H has a noncentral distribution with unknown noncentrality matrix A of rank r. These bounds are then used to investigate the chi-squared approximation for some test criteria used in tests of dimensionality.

Test of independence between tow sets of variates

In this paper asymptotic expansions of the null as well as non-null distributions of the likelihood ratio criterion for testing independence between two sets of variates are obtained. These appear to be better than the ones available in the literature . In factin the null case for p = 1 and p = 2 , t h e expansion reduces to the exact di stribution. In the non-null case, the expansion is given i n terms of non-central beta distributions and for the case when the population canonical correlation coefficients are small.

Asymptotic expansions of the non-central distribution of milks1 statistic in the complex gaussian case

In this paper, asymptotic expansions of the non-null distribution of the Wilks' statistic for Manova based on the complex multivariate Gaussian distribution is obtained for the cases (a) when the non-centrality parameter ω∼is constant and (b) when ∼Is of the same order as the sample size. No direct method is available for the case (b) and thus the partial differential equation approach considered in this paper Is particularly useful

Simple improvements on Cornfield's approximation to the mean of a noncentral hypergeometric random variable

SUMMARY This note provides some simple improvements to Cornfield's asymptotic approximation to the mean of a noncentral hypergeometric random variable. Simplicity, in the sense of avoidance of exact computation in favour of easy approximations, derives from an observation of Mantel & Hankey that in a four-fold table with all margins fixed, the ratio of expected cross-products is equal to the odds ratio parameter governing the noncentral distribution of the reference cell. The improvements often reduce the relative error of approximation by one or more powers of ten, even for tables whose total sample size is as small as two.

On the distribution of hotelling's t2and multiple correlation r2when sampling from a mixture of two normals

In this paper the non-null distribution of Hotelling's T2 and the null distribution of multiple correlation R2 are derived when the sample is taken from a mixture of two p-component multivariate normal distributions with mean vectors μ1 and μ2 respectively and common covariance matrix ∑, ∑. In a special case the non-null distribution of R2 is a l s o given, while the general noncentral distribution is given i n Awan (1981). These results have been used to study the robustness of T2 and R2 tests by Srivastava and Awan (1982), and Awan and Srivastava (1982) respectively.

Distributions of Maximal Invariants Using Quotient Measures

This paper demonstrates the use of proper actions and quotient measures in representations of non-central distributions of maximal invariants.

Inequalities for Joint Distributions of Quadratic Forms

Chebyshev inequalities are given for joint central and noncentral distributions of k quadratic forms; these are sharpened when $k = 2$ using the canonical correlations of Hotelling. Applications are noted in ballistics, in statistical quality control, and in linear models theory.

On the noncentral distribution of the ratio of the extreme roots of wishart matrix

The distribution of the ratio of the extreme latent roots of the Wishart matrix is useful in testing the sphericity hypothesis for a multivariate normal population. Let X be a p × n matrix whose columns are distributed independently as multivariate normal with zero mean vector and covariance matrix ∑. Further, let S = XX′ and let 11 > …>1p > 0 be the characteristic roots of S. Thus S has a noncentral Wishart distribution. In this paper, the exact distribution of fp = 1 − 1p/11 is derived. The density of fp is given in terms of zonal polynomials. These results have applications in nuclear physics also.

Exact Noncentral Distributions of Wilks' $\Lambda$ and Wilks-Lawley $U$ Criteria as Mixtures of Incomplete Beta Functions: for Three Tests

In this paper it is shown that in the general case the exact noncentral distributions of Wilks' $\Lambda$ and Wilks-Lawley $U$ can be obtained in a very straightforward manner. This completely eliminates the need for the more complicated inverse Mellon transform. It is first shown that any random variable whose moments satisfy Wilks' Type B integral equation (Type B random variables) has a distribution that can be represented as a mixture of incomplete beta functions. Then it is shown that the moments of Wilks' $\Lambda$ and Wilks-Lawley $U$ criteria can be written as mixtures of the moments of Type B random variables. Combining these results yields the noncentral distribution of Wilks' $\Lambda$ and Wilks-Lawley $U$ criteria as mixtures of incomplete beta functions for the following tests: equality of two dispersion matrices; MANOVA; and canonical correlation.

Corrigenda: The Exact Noncentral Distribution of Spearman's r and Other Related Correlation Coefficients

Corrigenda: The Exact Noncentral Distribution of Spearman's r and Other Related Correlation Coefficients

The Exact Noncentral Distributions of Spearman'srand other Related Correlation Coefficients

Abstract This article presents a general construction principle for the non-central distribution of Spearman's r without the usual reference to the underlying distributions of the two continuous random variables. The only distribution used is the bivariate distribution on the rank pairs in an n × n table. The noncentral distribution is generated by the calculation of the unequal probabilities of (n!)2 bipermutations. This distribution is approximated also by a skew distribution derived from the binomial distribution. Partial and multiple correlation coefficients and tricorrelation coefficients are also studied.

The Exact Noncentral Distributions of Spearman's r and Other Related Correlation Coefficients

The Exact Noncentral Distributions of Spearman's r and Other Related Correlation Coefficients

Assessment of long-term antiarrhythmic therapy: Studies on the long-term efficacy and toxicity of tocainide

Methods for testing long-term efficacy of antiarrhythmic agents are not established. Therefore, an algorithm was developed to assess the response of ventricular premature complexes and ventricular tachycardia to long-term therapy. In 21 patients tocainide, an oral lidocaine analog, was successful in acutely suppressing ventricular premature complexes and ventricular tachycardia. Suppression was assessed with use of both the arrhythmic classification of Lown and Wolf and counts of premature ventricular complexes from 24 hour ambulatory electrocardiographic recordings. After 3 to 10 months of tocainide therapy, 24 hour control recordings were obtained in 18 patients, and administration of tocainide was stopped. With use of recordings obtained 5 and 7 days later to assess variability in the count of ventricular premature complexes, the upper 95 percent confidence limit for variation in number of complexes after arcsin transformation to correct for noncentral distribution was 80 percent. When tocainide was withdrawn, arrhythmias did not reappear in 6 patients but their reappearance in 12 patients indicated a continued requirement for tocainide. Although the decrease in ventricular premature complexes was less than 80 percent in four patients, ventricular tachycardia was effectively suppressed. Four patients had adverse reactions requiring drug withdrawal: a rash (one patient), cervical muscle spasms (one patient) and central nervous system symptoms (two patients). Minor central nervous symptoms persist in six patients. Tocainide did not induce lymphocytotoxicity. It is concluded that testing of long-term antiarrhythmic drugs should include a withdrawal trial to ensure both continued drug requirement and continued drug efficacy. Because of marked variability in ventricular premature complex counts, adequate suppression is best assessed by combining an analysis that considers arrhythmia classification and the nonparametric distribution of data.

On Approximations for the Central and Noncentral Distribution of the Generalized Variance

Abstract It is shown how the noncentral probability density function of the generalized variance can be approximated by a central probability density function under certain conditions by using moments and moment generating functions. It is also shown that these density functions can be approximated by a linear function of two gamma densities, which can readily be applied to practical problems. The theoretical results are verified by a sampling experiment.

On Approximations for the Central and Noncentral Distribution of the Generalized Variance

Abstract It is shown how the noncentral probability density function of the generalized variance can be approximated by a central probability density function under certain conditions by using moments and moment generating functions. It is also shown that these density functions can be approximated by a linear function of two gamma densities, which can readily be applied to practical problems. The theoretical results are verified by a sampling experiment.

On approximating the central and noncentral multivariate gamma distributions

In this paper a finite series approximation involving Laguerre polynomials is derived for central and noncentral multivariate gamma distributions. It is shown that if one approximates the density of any k nonnegative continuous random variables by a finite series of Laguerre polynomials up to the (n1, …, nk)th degree, then all the mixed moments up to the order (n1, …, nk) of the approximated distribution equal to the mixed moments up to the same order of the random variables. Some numerical results are given for the bivariate central and noncentral multivariate gamma distributions to indicate the usefulness of the approximations.

Latent Roots and Matrix Variates: A Review of Some Asymptotic Results

The exact noncentral distributions of matrix variates and latent roots derived from normal samples involve hypergeometric functions of matrix argument. These functions can be defined as power series, by integral representations, or as solutions of differential equations, and there is no doubt that these mathematical characterizations have been a unifying influence in multivariate noncentral distribution theory, at least from an analytic point of view. From a computational and inference point of view, however, the hypergeometric functions are themselves of very limited value due primarily to the many difficulties involved in evaluating them numerically and consequently in studying the effects of population parameters on the distributions. Asymptotic results for large sample sizes or large population latent roots have so far proved to be much more useful for such problems. The purpose of this paper is to review some of the recent results obtained in these areas.

The analysis of ratios and cross-product ratios of poisson variates with application to incidence rates

The incidence of most diseases is low enough that in. large populations the number of new cases may be considered a Poisson variate. This paper explores models and methods for analyzing such data Specific cases are the estimation and testing of ratios and the cross-product ratios, both simple and stratified* We assume the Poisson means are exponential functions of the relevant parameters. The resulting sets of sufficient statistics are partitioned into a test statistic and a vector of statistics related to the nuisance parameters . The methods derived are based on the conditional distribution of the test statistic given the other sufficient statistics. The analyses of stratified cross-product ratios are seen to be analogues of the noncentral distribution associated with theanalysis of the common odds ratio in several 2×2 tables. The various methods are illustrated in numerical examples involving incidence rates of cancer in two metropolitan areas adjusting for both age and sex.

Evaluation of probabilities for the noncentral distributions and the difference of twoT-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.