Asymptotic distribution of regression correlation coefficient for Poisson regression model

This study treats an asymptotic distribution for measures of predictive power for generalized linear models (GLMs). We focus on the regression correlation coefficient (RCC) that is one of the measures of predictive power. The RCC, proposed by Zheng and Agresti is a population value and a generalization of the population value for the coefficient of determination. Therefore, the RCC is easy to interpret and familiar. Recently, Takahashi and Kurosawa provided an explicit form of the RCC and proposed a new RCC estimator for a Poisson regression model. They also showed the validity of the new estimator compared with other estimators. This study discusses the new statistical properties of the RCC for the Poisson regression model. Furthermore, we show an asymptotic normality of the RCC estimator.     

Comparison of performance measures for multivariate discrete models

For multivariate probit models, Spiess and Tutz suggest three alternative performance measures, which are all based on the decomposition of the variation. The multivariate probit model can be seen as a special case of the discrete copula model. This paper proposes some new measures based on the value of the likelihood function and the prediction-realization table. In addition, it generalizes the measures from Spiess and Tutz for the discrete copula model. Results of a simulation study designed to compare the different measures in various situations are presented.     

Elliptically contoured model and factorization of wllks1 A

In a series of papers, Kshirsagar (1964, 1971) and McHenry and Kshirsagar (1977), factorize Wilks' A into a number of factors and find the independent null multivariate beta densities of these factors. These factors are the likelihood ratio test criteria for testing the goodness of fit of certain assigned discriminant functions or canonical variables either in the space of independent or dependent variables. Essentially the factors of Wilks' A are the factors of certain multivariate beta distributed matrix or its determinant. The Bartlett decomposition of the underlying multivariate beta distribution into independent factors determines the distribution of these factors. The present paper generalizes Kshirsagar's (1971) normal theory to the elliptically contoured model, and shows that his results are null robust for the elliptically contoured model.     

Improved deviance goodness of fit statistics for a gamma regression model

A gamma regression model with an exponential link function for the means Is considered. Moment properties of the deviance statistics based on maximum likelihood and weighted least squares fits are used to define modified deviance statistics which provide alternative global goodness of fit tests. The null distribution properties of the deviances and modified deviances are compared with those of the approximating chi-square distribution and It is shown that the use of the modified deviances gives much better control over the significance levels of the tests.     

A correlation type procedure for assessing multivariate normality

A correlation-type statistic for assessing multivariate normality is described. Its estimated finite sample distribution is tabulated, and its performance against certain alternatives is compared with that of a competing Cramer-von Mises type statistic in a Monte Carlo power study. A set of quadrivariate data is examined as illustration of the procedure.     

Book review

W.G.Cochran: Sampling Techniques, 3rd. Ed. John Wiley & Sons, New York-Santa Barbara-London-Sydney-Toronto 1977. 428 S., £ 12.50; $21.50.

H.Toutenburg: Vorhersage in linearen Modellen. Akademie-Verlag, Berlin 1975, VIII, 176b S., 3 Tab., 28,– M.

O.Kallenberg: Random Measure. Akademie-Verlag, Berlin 1975; Academic Press, London 1976. 104 pp., 28,– M.     

Smooth goodness of fit tests for the weibull distribution with singly censored data

The smooth goodness of fit tests are generalized to singly censored data and applied to the problem of testing Weibull (or extreme value) fit. Smooth tests, Pearson-type tests, and the spacings tests proposed by Mann, Schemer, and Fertig (1973) are compared on the basis of local asymptotic relative efficiency with respect to the asymptotic best test against generalized gamma alternatives, The smooth test of order one Is found to be most efficient for the generalized gamma alternatives.     

Global convergence of the log-concave MLE when the true distribution is geometric

Let X₁, …, X_n be i.i.d. from a discrete probability mass function (pmf) p. In Balabdaoui et al. [(2013), ‘Asymptotic Distribution of the Discrete Log-Concave mle and Some Applications’, JRSS-B, in press], the pointwise limit distribution of the log-concave maximum-likelihood estimator (MLE) was derived in both the well- and misspecified settings. In the well-specified setting, the geometric distribution was excluded, classified as being degenerate. In this article, we establish the global asymptotic theory of the log-concave MLE of a geometric pmf in all ?_q distances for q∈{1, 2, …}∪{∞}. We also show how these asymptotic results could be used in testing whether a pmf is geometric.     

Simulation of the p-generalized Gaussian distribution

In this paper, we introduce the p-generalized polar methods for the simulation of the p-generalized Gaussian distribution. On the basis of geometric measure representations, the well-known Box–Muller method and the Marsaglia–Bray rejecting polar method for the simulation of the Gaussian distribution are generalized to simulate the p-generalized Gaussian distribution, which fits much more flexibly to data than the Gaussian distribution and has already been applied in various fields of modern sciences. To prove the correctness of the p-generalized polar methods, we give stochastic representations, and to demonstrate their adequacy, we perform a comparison of six simulation techniques w.r.t. the goodness of fit and the complexity. The competing methods include adapted general methods and another special method. Furthermore, we prove stochastic representations for all the adapted methods.     

Data-driven tests of uniformity on product manifolds

Data-driven versions of Sobolev tests of uniformity on compact Riemannian manifolds are reviewed and their large-sample asymptotic properties are given. A variant which is suitable for product manifolds is introduced. Data-driven goodness-of-fit tests of multivariate distributions are derived from data-driven tests of uniformity on tori.