Harold Hotelling 1895–1973

It is demonstrated that important properties of the distribution of runs can be obtained immediately from the hypergeometric distribution for which results are already well-known.     

Exact Tests of Significance in Higher Dimensional Tables

We describe methods used to provide an exact test of significance of the hypothesis that all factors are mutually independent of each other in 2³ and 2⁴ contingency tables. Several numerical examples demonstrate the advantages of exact tests over approximate significance levels. We give bounds on the number of tables needed to perform this exact significance test. In four or more dimensions the number of tables in this enumeration becomes astronomical with even modest sample sizes. Inverting the characteristic function of the exact distribution has proved useful in these situations.     

An Empirical Bayes Method for Multivariate Meta-analysis with an Application in Clinical Trials

We propose an empirical Bayes method for evaluating overall and study-specific treatment effects in multivariate meta-analysis with binary outcome. Instead of modeling transformed proportions or risks via commonly used multivariate general or generalized linear models, we directly model the risks without any transformation. The exact posterior distribution of the study-specific relative risk is derived. The hyperparameters in the posterior distribution can be inferred through an empirical Bayes procedure. As our method does not rely on the choice of transformation, it provides a flexible alternative to the existing methods and in addition, the correlation parameter can be intuitively interpreted as the correlation coefficient between risks.     

Bayesian estimation of an AR(1) process with exponential white noise

The object of this paper is a Bayesian analysis of the autoregressive model X _t ?=?ρX _t?1?+?Y _t where 0?Y _t are independent random variables with an exponential distribution of parameter θ. Our study generalizes some results obtained by Turkmann (1990 Amaral Turkmann, M. A. (1990). Bayesian analysis of an autoregressive process with exponential white noise. Statistics, 4: 601–608.  [Google Scholar]). Our analysis is based on a more general non-informative prior which allows us to improve the estimators of ρ and θ.     

New polya and inverse polya distributions of order k

New Polya and inverse Polya distributions of order k are derived by means of generalized urn models and by compounding the binomial and negative binomial distributions of order k of Philippou (1986, 1983) with the beta distribution. It i s noted that the present Polpa distribution of order k includes as special cases a new hypergeometric distribution of order k, a negative one,an inverse one, and a discrete uniform of the same order. The probability generating functions, means and variances of the new distributions are obtained, and five asymptotic results are established relating them to the abovedmentioned binomial and negative binomial distributions of order k, and to the Poisson distribution of the same order of Philippou (1983).Moment estimates are also given and applications are indicated.     

Use of Dummy Variables in Testing for Equality between Sets of Coefficients in Two Linear Regressions: A Note

Elementary inductive proofs are presented for the binomial approximation to the hypergeometric distribution, the density of an order statistic, and the distribution of when X ₁, ···, X _n are a sample from N (μ, 1).     

An Improper Method of Randomization in Experimental Design

The mark-recapture method was devised by Petersen in 1896 Petersen, C. G. J. 1896. “The Yearly Immigration of Young Plaice Into the Limfjord From the German Sea,”. Report of Danish Biological Station, 6: 1–48.  [Google Scholar] to estimate the number of fish migrating into the Limfjord, and independently by Lincoln in 1930 Lincoln, F. C. 1930. “Calculating Waterfowl Abundance on the Basis of Banding Returns,”. United States Department of Agriculture Circular, 118: 1–4.  [Google Scholar] to estimate waterfowl abundance. The technique can be applied to any search for a finite number of items by two or more people or agents, allowing the number of searched-for items to be estimated. This ubiquitous problem appears in fields from ecology and epidemiology, through to mathematics, social sciences, and computing. Here, we exactly calculate the moments of the hypergeometric distribution associated with this longstanding problem, confirming that widely used estimates conjectured in 1951 are often too small. Our Bayesian approach highlights how different search strategies will modify the estimates. The estimates are applied to several examples. For some published applications, substantial errors are found to result from using the Chapman or Lincoln–Petersen estimates. Supplementary materials for this article are available online.     

Skewed Bessel function distributions with application to rainfall data

Skewed distributions have attracted significant attention in the last few years. In this article, a skewed Bessel function distribution with the probability density function (pdf) f(x)=2 g (x) G (λ x) is introduced, where g (·) and G (·) are taken, respectively, to be the (pdf) and the cumulative distribution function of the Bessel function distribution [McKay, A.T., 1932, A Bessel function distribution, Biometrica, 24, 39–44]. Several particular cases of this distribution are identified and various representations for its moments derived. Estimation procedures by the method of maximum likelihood are also derived. Finally, an application is provided to rainfall data from Orlando, Florida.     

The General Expressions for the Moments of the Stochastic Shrinkage Parameters of the Liu Type Estimator

ABSTRACT

One of the problems with the Liu estimator is the appropriate value for the unknown biasing parameter d. In this article we consider the optimum value for d and give upper bound for the expected value of the estimator of this biasing parameter. We also derive the general expressions for the moments of the stochastic shrinkage parameters of the Liu estimator and the generalized Liu estimator. Numerical calculations are carried out to illustrate the behavior of the mean and variance of the biasing parameter. Also, a numerical example is given to illustrate the effect of the biasing parameter d, on the mean square error of the Liu estimator.     

On Comparisons of Exact Powers of Bivariate GMANOVA Tests

Exact powers of four classical tests in a GMANOVA model are compared numerically when the order of the error sum of square matrix is 2. The four tests are likelihood ratio (=LR), Pillai's V, Hotelling's T ², and Roy's largest root tests. It turns out that for small sizes, there are a few cases in which Rothenberg's condition for the relative magnitude of asymptotic powers of three standard tests does not hold.