On some characteristics of bivariate chi-square distribution

Product moments of bivariate chi-square distribution have been derived in closed forms. Finite expressions have been derived for product moments of integer orders. Marginal and conditional distributions, conditional moments, coefficient of skewness and kurtosis of conditional distribution have also been discussed. Shannon entropy of the distribution is also derived. We also discuss the Bayesian estimation of a parameter of the distribution. Results match with the independent case when the variables are uncorrelated.     

Distribution of products of independently distributed pathway random variables

A lot of work has been done on products and ratios of random variables by Provost and his co-workers, see, for example, Provost [S.B. Provost, The exact distribution of the ratio of a linear combination of chi-square variables over the root of a product of chi-square variables, Canad. J. Statist. 14 (1986), pp. 61–67; S.B. Provost, The distribution function of a statistic for testing the equality of scale parameters in two gamma populations, Metrika 36 (1989), pp. 337–345]. Here, we extend this idea by introducing a pathway model. The exact density functions of the products of pathway random variables are obtained using the Mellin transform technique. Their computable series forms are derived. The particular cases of the derived results are shown to be associated with the thermonuclear functions and reaction rate probability integral in the theory of nuclear reaction rate, Krätzel integral in applied analyses and inverse Gaussian density in stochastic processes. Graphical representations of the density functions of the product of random variables for the different values of the pathway parameters are shown. The new probability model is fitted to revenue data.     

The exact distribution of the ratio of a linear combination of chi-square variables over the root of a product of chi-square variables

This article gives the exact distribution of a statistic whose numerator and denominator are independent, the former being a linear combination of independent chi-square variables and the latter being the kth root of a product of k independent chi-square variables. This structure appears in the study of multivariate linear functional relationship models. The technique of the inverse Mellin transform is used in order to obtain the density of this test statistic in a computable form.     

The Correlated Bivariate Noncentral F Distribution and Its Application

This article proposes the singly and doubly correlated bivariate noncentral F (BNCF) distributions. The probability density function (pdf) and the cumulative distribution function (cdf) of the distributions are derived for arbitrary values of the parameters. The pdf and cdf of the distributions for different arbitrary values of the parameters are computed, and their graphs are plotted by writing and implementing new R codes. An application of the correlated BNCF distribution is illustrated in the computations of the power function of the pre-test test for the multivariate simple regression model (MSRM).     

The exact distribution function of the ratio of two dependent quadratic forms

A closed-form representation of the distribution function of the ratio of two linear combinations of Chi-squared variables is derived. The ratio is of the following form R = (X + aY)/(bY + Z), where X, Y, Z are independent Chi-square variables and a, b > 0. Two methods of obtaining the distribution function of this ratio are used. The exact density function of such a ratio is then obtained by differentiation. Two numerical examples are provided.     

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     

The Standard Error of the Mean and the Difference between Means for Finite Populations

This article discusses a representation of Pearson's chi-square for independence in two-way contingency tables in terms of conditional probabilities of two categorical random variables and proposes a functional interpretation of Pearson's chi-square. This representation is suggested for use in the teaching of statistical independence between categorical variables.     

Power of analysis of covariance tests under conditional specification

The procedure-wise power functions of two strategies for balanced single-factor analysis of covariance in the presence of possibly unequal regression slopes are evaluated and illustrated. The strategies differ in the action to be taken following a re-^ jection by the preliminary test for equal slopes. The first strategy simply discards the covariate and respecifies the model as the one-way ANOVA model for testing factor effects. The second leaves the unequal slopes covariance model intact, but respecifies the factor effects hypothesis to address the factor level means adjusted to the sample average of the covariate. One additional strategy, that of testing factor effects only if the preliminary slopes test does not reject, is included for comparison purposes. Computation of the power functions requires extensive use of the results obtained in Hawkins and Han (1986) concerning the bivariate distributions of certain ratios of independent noncentral chi-square random variables.     

On the Product and Ratio of Gamma and Beta Random Variables

Summary: The distributions of the product XY and the ratio X/Y are derived when X and Y are gamma and beta random variables distributed independently of each other. Tabulations of the associated percentage points and illustrations of their practical use are also provided. * The authors would like to thank the referee and the editor for carefully reading the paper and for their help in improving the paper.     

A note on the ratio of normal and Laplace random variables

The normal and Laplace are the two earliest known continuous distributions in statistics and the two most popular models for analyzing symmetric data. In this note, the exact distribution of the ratio | X / Y | is derived when X and Y are respectively normal and Laplace random variables distributed independently of each other. A MAPLE program is provided for computing the associated percentage points. An application of the derived distribution is provided to a discriminant problem.     

Interval Estimation for the Correlation Coefficient

ABSTRACT

The correlation coefficient (CC) is a standard measure of a possible linear association between two continuous random variables. The CC plays a significant role in many scientific disciplines. For a bivariate normal distribution, there are many types of confidence intervals for the CC, such as z-transformation and maximum likelihood-based intervals. However, when the underlying bivariate distribution is unknown, the construction of confidence intervals for the CC is not well-developed. In this paper, we discuss various interval estimation methods for the CC. We propose a generalized confidence interval for the CC when the underlying bivariate distribution is a normal distribution, and two empirical likelihood-based intervals for the CC when the underlying bivariate distribution is unknown. We also conduct extensive simulation studies to compare the new intervals with existing intervals in terms of coverage probability and interval length. Finally, two real examples are used to demonstrate the application of the proposed methods.     

On Censored Bivariate Random Variables: Copula,Characterization, and Estimation

Let (X, Y) be a bivariate random vector with joint distribution function F_{X, Y}(x, y) = C(F(x), G(y)), where C is a copula and F and G are marginal distributions of X and Y, respectively. Suppose that (X_i, Y_i), i = 1, 2, …, n is a random sample from (X, Y) but we are able to observe only the data consisting of those pairs (X_i, Y_i) for which X_i ? Y_i. We denote such pairs as (X*_i, Y_i*), i = 1, 2, …, ν, where ν is a random variable. The main problem of interest is to express the distribution function F_{X, Y}(x, y) and marginal distributions F and G with the distribution function of observed random variables X* and Y*. It is shown that if X and Y are exchangeable with marginal distribution function F, then F can be uniquely determined by the distributions of X* and Y*. It is also shown that if X and Y are independent and absolutely continuous, then F and G can be expressed through the distribution functions of X* and Y* and the stress–strength reliability P{X ? Y}. This allows also to estimate P{X ? Y} with the truncated observations (X*_i, Y_i*). The copula of bivariate random vector (X*, Y*) is also derived.     

A bivariate F distribution with marginals on arbitrary numerator and denominator degrees of freedom, and related bivariate beta and t distributions

The classical bivariate F distribution arises from ratios of chi-squared random variables with common denominators. A consequent disadvantage is that its univariate F marginal distributions have one degree of freedom parameter in common. In this paper, we add a further independent chi-squared random variable to the denominator of one of the ratios and explore the extended bivariate F distribution, with marginals on arbitrary degrees of freedom, that results. Transformations linking F, beta and skew t distributions are then applied componentwise to produce bivariate beta and skew t distributions which also afford marginal (beta and skew t) distributions with arbitrary parameter values. We explore a variety of properties of these distributions and give an example of a potential application of the bivariate beta distribution in Bayesian analysis.     

ON A SHARED ALLELE TEST OF RANDOM MATING

A simple random sample is observed from a population with a large number‘K’ of alleles, to test for random mating. Of n couples, n_ijkl have female genotype ij and male genotype kl (i, j, k, l{1,…, A‘}). The large contingency table is collapsed into three counts, n₀, n₁ and n₂ where n_p is the number of couples with s alleles in common (s = 0,1, 2). The counts are estimated by np?_o where n₀, is the estimated probability of a couple having s alleles in common under the hypothesis of random mating. The usual chi-square goodness of fit statistic X² compares observed (n_s) with expected (np?) over the three categories, s = 0,1,2. An empirical observation has suggested that X² is close to having a chi-square distribution with two degrees of freedom (X) despite a large number of parameters implicitly estimated in _e. This paper gives two theorems which show that x is indeed the approximate distribution of X² for large n and K₁“, provided that no allele type over-dominates the others.     

On the Computation of Non-Central Chi-Square Distribution Function

The cumulative distribution function of the non-central chi-square is very important in calculating the power function of some statistical tests. On the other hand it involves an integral which is difficult to obtain. In literature some workers discussed the evaluation and the approximation of the c.d.f. of the non-central chi-square [see references (2)]. In the present work two computational formulae for computing the cumulative distribution function of the non-central chi-square distribution are given, the first one deals with the case of any degrees of freedom (odd and even), and the second deals with the case of odd degrees of freedom. Numerical illustrations are discussed.     

Testing Goodness of Fit for Parametric Families of Copulas—Application to Financial Data

ABSTRACT

This article suggests a chi-square test of fit for parametric families of bivariate copulas. The marginal distribution functions are assumed to be unknown and are estimated by their empirical counterparts. Therefore, the standard asymptotic theory of the test is not applicable, but we derive a rule for the determination of the appropriate degrees of freedom in the asymptotic chi-square distribution. The behavior of the test under H ₀ and for selected alternatives is investigated by Monte Carlo simulation. The test is applied to investigate the dependence structure of daily German asset returns. It turns out that the Gauss copula is inappropriate to describe the dependencies in the data. A t _ν-copula with low degrees of freedom performs better.     

Extreme value models with application to drought data

Summary The exact distributions of the productXY are derived whenX andY are independent random variables and come from the extreme value distribution of Type I, the extreme value distribution of Type II or the extreme value distribution of Type III. Of the, six possible combinations, only three yield closed-form expressions for the distribution ofXY. A detailed application of the results is provided to drought data from Nebraska. The author would like to thank the referees and the Associate Editor for carefully reading the paper and for their great help in improving the paper.     

Editor's Report

There are two common methods for statistical inference on 2 × 2 contingency tables. One is the widely taught Pearson chi-square test, which uses the well-known χ²statistic. The chi-square test is appropriate for large sample inference, and it is equivalent to the Z-test that uses the difference between the two sample proportions for the 2 × 2 case. Another method is Fisher’s exact test, which evaluates the likelihood of each table with the same marginal totals. This article mathematically justifies that these two methods for determining extreme do not completely agree with each other. Our analysis obtains one-sided and two-sided conditions under which a disagreement in determining extreme between the two tests could occur. We also address the question whether or not their discrepancy in determining extreme would make them draw different conclusions when testing homogeneity or independence. Our examination of the two tests casts light on which test should be trusted when the two tests draw different conclusions.     

Goodness of fit tests for the multiple logistic regression model

Several test statistics are proposed for the purpose of assessing the goodness of fit of the multiple logistic regression model. The test statistics are obtained by applying a chi-square test for a contingency table in which the expected frequencies are determined using two different grouping strategies and two different sets of distributional assumptions. The null distributions of these statistics are examined by applying the theory for chi-square tests of Moore Spruill (1975) and through computer simulations. All statistics are shown to have a chi-square distribution or a distribution which can be well approximated by a chi-square. The degrees of freedom are shown to depend on the particular statistic and the distributional assumptions.

The power of each of the proposed statistics is examined for the normal, linear, and exponential alternative models using computer simulations.     

Extremes of nonexchangeability

Summary For identically distributed random variables X and Y with joint distribution function H, we show that the supremum of |H(x,y)-H(y,x)| is 1/3. Using copulas, we define a measure of nonexchangeability, and study maximally nonexchangeable random variables and copulas. In particular, we show that maximally nonexchangeable random variables are negatively correlated in the sense of Spearman's rho. An erratum to this article is available at .