A Condition to Obtain the Same Decision in the Homogeneity Testing Problem from the Frequentist and Bayesian Point of View

We develop a Bayesian procedure for the homogeneity testing problem of r populations using r × s contingency tables. The posterior probability of the homogeneity null hypothesis is calculated using a mixed prior distribution. The methodology consists of choosing an appropriate value of π₀ for the mass assigned to the null and spreading the remainder, 1 ? π₀, over the alternative according to a density function. With this method, a theorem which shows when the same conclusion is reached from both frequentist and Bayesian points of view is obtained. A sufficient condition under which the p-value is less than a value α and the posterior probability is also less than 0.5 is provided.     

ON THE USE OF X2 AS A TEST OF INDEPENDENCE IN CONTINGENCY TABLES WITH SMALL CELL EXPECTATIONS

When an r×c contingency table has many cells having very small expectations, the usual χ² approximation to the upper tail of the Pearson χ² goodness-of-fit statistic becomes very conservative. The alternatives considered in this paper are to use either a lognormal approximation, or to scale the usual χ² approximation. The study involves thousands of tables with various sample sizes, and with tables whose sizes range from 2×2 through 2×10×10. Subject to certain restrictions the new scaled χ² approximations are recommended for use with tables having an average cell expectation as small as 0·5.     

Order-preserving Estimators and an Inequality on the Integration of Zonal Polynomial

ABSTRACT

Suppose X , p × p p.d. random matrix, has the distribution which depends on a p × p p.d. parameter matrix Σ and this distribution is orthogonally invariant. The orthogonally invariant estimator of Σ which has the eigenvalues of the same order as the eigenvalues of X is called order-preserving. We conjecture that a non-order-preserving estimator is dominated by modified order-preserving estimators with respect to the entropy (Stein's) loss function. We show that an inequality on the integration of zonal polynomial is sufficient for this conjecture. We also prove this inequality for the case p = 2.     

On the Distribution of Matrix Quadratic Forms

A characterization of the distribution of the multivariate quadratic form given by X A X′, where X is a p × n normally distributed matrix and A is an n × n symmetric real matrix, is presented. We show that the distribution of the quadratic form is the same as the distribution of a weighted sum of non central Wishart distributed matrices. This is applied to derive the distribution of the sample covariance between the rows of X when the expectation is the same for every column and is estimated with the regular mean.     

The Likelihood Ratio Test with the Box–Cox Transformation for the Normal Mixture Problem: Power and Sample Size Study

Abstract

Through simulation and regression, we study the alternative distribution of the likelihood ratio test in which the null hypothesis postulates that the data are from a normal distribution after a restricted Box–Cox transformation and the alternative hypothesis postulates that they are from a mixture of two normals after a restricted (possibly different) Box–Cox transformation. The number of observations in the sample is called N. The standardized distance between components (after transformation) is D = (μ₂ ? μ₁)/σ, where μ₁ and μ₂ are the component means and σ² is their common variance. One component contains the fraction π of observed, and the other 1 ? π. The simulation results demonstrate a dependence of power on the mixing proportion, with power decreasing as the mixing proportion differs from 0.5. The alternative distribution appears to be a non-central chi-squared with approximately 2.48 + 10N ^?0.75 degrees of freedom and non-centrality parameter 0.174N(D ? 1.4)² × [π(1 ? π)]. At least 900 observations are needed to have power 95% for a 5% test when D = 2. For fixed values of D, power, and significance level, substantially more observations are necessary when π ≥ 0.90 or π ≤ 0.10. We give the estimated powers for the alternatives studied and a table of sample sizes needed for 50%, 80%, 90%, and 95% power.     

Likelihood Ratio Test Against Stochastic Order in Three Way Contingency Tables

Trend tests in dose-response have been central problems in medicine. The likelihood ratio test is often used to test hypotheses involving a stochastic order. Stratified contingency tables are common in practice. The distribution theory of likelihood ratio test has not been full developed for stratified tables and more than two stochastically ordered distributions. Under c strata of m × r tables, for testing the conditional independence against simple stochastic order alternative, this article introduces a model-free test method and gives the asymptotic distribution of the test statistic, which is a chi-bar-squared distribution. A real data set concerning an ordered stratified table will be used to show the validity of this test method.     

Optimal Configuration of a Square Array Group Testing Algorithm

We consider the optimal configuration of a square array group testing algorithm (denoted A2) to minimize the expected number of tests per specimen. For prevalence greater than 0.2498, individual testing is shown to be more efficient than A2. For prevalence less than 0.2498, closed form lower and upper bounds on the optimal group sizes for A2 are given. Arrays of dimension 2 × 2, 3 × 3, and 4 × 4 are shown to never be optimal. The results are illustrated by considering the design of a specimen pooling algorithm for detection of recent HIV infections in Malawi.     

Testing for and against a set of linear inequality constraints in a multinomial setting

There are numerous situations in categorical data analysis where one wishes to test hypotheses involving a set of linear inequality constraints placed upon the cell probabilities. For example, it may be of interest to test for symmetry in k × k contingency tables against one-sided alternatives. In this case, the null hypothesis imposes a set of linear equalities on the cell probabilities (namely p_ij = P_ji ×i > j), whereas the alternative specifies directional inequalities. Another important application (Robertson, Wright, and Dykstra 1988) is testing for or against stochastic ordering between the marginals of a k × k contingency table when the variables are ordinal and independence holds. Here we extend existing likelihood-ratio results to cover more general situations. To be specific, we consider testing H_t,0 against H₁ - H₀ and H₁ against H₂ - H ₁ when H₀:k × i=1 p_ix_ji = 0, j = 1,…, s, H₁:k × i=1 p_ix_ji × 0, j = 1,…, s, and does not impose any restrictions on p. The x_ji's are known constants, and s × k - 1. We show that the asymptotic distributions of the likelihood-ratio tests are of chi-bar-square type, and provide expressions for the weighting values.     

On the Parameter Estimation of the Asymmetric Multivariate Laplace Distribution

This article examines a family of three-parameter multivariate Laplace distributions ML _p (a, μ, Σ) which is closed under constant shifts. Parameter vectors a and μ are called shift and shape parameter, respectively, positive definite p × p-matrix Σ is a scale parameter. The first three moments are derived and used for estimating the parameters. The behavior of the obtained estimates is explored in a simulation experiment.     

On a Class of Distributions Defined by the Relationship Between Their Density and Distribution Functions

Knowledge concerning the family of univariate continuous distributions with density function f and distribution function F defined through the relation f(x) = F ^α(x)(1 ? F(x))^β, α, β ? , is reviewed and modestly extended. Symmetry, modality, tail behavior, order statistics, shape properties based on the mode, L-moments, and—for the first time—transformations between members of the family are the general properties considered. Fully tractable special cases include all the complementary beta distributions (including uniform, power law and cosine distributions), the logistic, exponential and Pareto distributions, the Student t distribution on 2 degrees of freedom and, newly, the distribution corresponding to α = β = 5/2. The logistic distribution is central to some of the developments of the article.     

ESTIMATION OF REALIZED SIGNAL TO NOISE RATIO FOR A PAIR OF MULTIVARIATE SIGNALS

Consider the case of classifying an incoming message as one of two known p-dimension signals or as a pure noise. Let the noise co-variance matrix (assumed to be same in all the three cases) be unknown. We consider the problem of estimation of “realized signal to noise ratio matrix”, which is an index of discriminatory power, under various loss functions. Optimum estimators are obtained under these loss functions. Finally, an attempt is made to provide a lower confidence bound for the realized signal to noise ratio matrix. In the process, the probability distribution of the smaller eigenvalue of a 2 × 2 confluent hypergeometric random matrix is obtained.     

New Generalizations of Cauchy Distribution

The generalized skew-normal distribution introduced by Balakrishnan (2002 Balakrishnan , N. ( 2002 ). Discussion on ‘Skew multivariate models related to hidden truncation and/or selective reporting’ by B. C. Arnold and R. J. Beaver . Test 11 : 37 – 39 .[Web of Science ®] , [Google Scholar]) is used to obtain new generalizations of univariate Cauchy distribution with two parameters, denoted by GC _{m, n} (a, b) with m and n non-negative integer numbers and a, b ∈ R. For cases (m, n) = (1, 2), (m, n) = (2, 1), (m, n) = (0, 3) and (m, n) = (3, 0) explicit forms of the density functions are derived and compared to previous generalizations of Cauchy and skew-Cauchy distributions.     

Improved minimax estimation of a normal precision matrix

Let S_{p × p} have a Wishart distribution with parameter matrix Σ and n degrees of freedom. We consider here the problem of estimating the precision matrix Σ^?1 under the loss functions L₁(σ) tr (σ) - log |σ| and L₂(σ) = tr (σ). James-Stein-type estimators have been derived for an arbitrary p. We also obtain an orthogonal invariant and a diagonal invariant minimax estimator under both loss functions. A Monte-Carlo simulation study indicates that the risk improvement of the orthogonal invariant estimators over the James-Stein type estimators, the Haff (1979) estimator, and the “testimator” given by Sinha and Ghosh (1987) is substantial.     

The Tale of Cochran's Rule: My Contingency Table has so Many Expected Values Smaller than 5, What Am I to Do?

In an informal way, some dilemmas in connection with hypothesis testing in contingency tables are discussed. The body of the article concerns the numerical evaluation of Cochran's Rule about the minimum expected value in r × c contingency tables with fixed margins when testing independence with Pearson's X² statistic using the χ² distribution.     

A lower bound to the measure of optimality for main effect plans in the general asymmetric factorial experiments

A design d is called D-optimal if it maximizes det(M _d ) and is called MS-optimal if it maximizes tr(M _d ) and minimizes tr[(M _d )²] among those which maximize tr(M _d ), where M _d stands for the information matrix produced from d under a given model. In this paper, we establish a lower bound for tr[(M _d )²] with respect to a main effects model, where d is an s ₁×s ₂×···×s _m levels asymmetric orthogonal array of strength at least 1. Nonisomorphic asymmetrical MS-optimal orthogonal arrays of strength 1 with N=6, 8 and 12 runs are also presented.     

Characterizations based on regression assumptions of order statistics

Let X₍₁₎≤X₍₂₎≤···≤X_(n) be the order statistics from independent and identically distributed random variables {X_i, 1≤i≤n} with a common absolutely continuous distribution function. In this work, first a new characterization of distributions based on order statistics is presented. Next, we review some conditional expectation properties of order statistics, which can be used to establish some equivalent forms for conditional expectations for sum of random variables based on order statistics. Using these equivalent forms, some known results can be extended immediately.     

Statistical inference of risk difference in K correlated 2×2 tables with structural zero

K correlated 2×2 tables with structural zero are commonly encountered in infectious disease studies. A hypothesis test for risk difference is considered in K independent 2×2 tables with structural zero in this paper. Score statistic, likelihood ratio statistic and Wald‐type statistic are proposed to test the hypothesis on the basis of stratified data and pooled data. Sample size formulae are derived for controlling a pre‐specified power or a pre‐determined confidence interval width. Our empirical results show that score statistic and likelihood ratio statistic behave better than Wald‐type statistic in terms of type I error rate and coverage probability, sample sizes based on stratified test are smaller than those based on the pooled test in the same design. A real example is used to illustrate the proposed methodologies. Copyright © 2009 John Wiley & Sons, Ltd.     

A Property of Count Distributions in the Hinde–Demétrio Family

The Hinde–Demétrio (HD) family of distributions, which are discrete exponential dispersion models with an additional real index parameter p, have been recently characterized from the unit variance function μ + μ ^p . For p equals to 2, 3,…, the corresponding distributions are concentrated on non negative integers, overdispersed and zero-inflated with respect to a Poisson distribution having the same mean. The negative binomial (p = 2) and strict arcsine (p = 3) distributions are HD families; the limit case (p → ∞) is associated to a suitable Poisson distribution. Apart from these count distributions, none of the HD distributions has explicit probability mass functions p _k . This article shows that the ratios r _k  = k p _k /p _k?1, k = 1,…, p ? 1, are equal and different from r _p . This new property allows, for a given count data set, to determine the integer p by some tests. The extreme situation of p = 2 is of general interest for count data. Some examples are used for illustrations and discussions.     

Maximum Likelihood Estimation in the Bivariate Binomial (0,1) Distribution:Application TO 2×2 Tables

We consider a 2×2 contingency table, with dichotomized qualitative characters (A,A) and (B,B), as a sample of size n drawn from a bivariate binomial (0,1) distribution. Maximum likelihood estimates p?₁p?₂ and p? are derived for the parameters of the two marginals p₁p₂ and the coefficient of correlation. It is found that p? is identical to Pearson's (1904)?=(χ²/n)½, where ?² is Pearson's usual chi-square for the 2×2 table. The asymptotic variance-covariance matrix of p?_lp?₂and p is also derived.