Vol 26, No 3 (2017)

It is well known that any natural exponential family (NEF) is characterized by its variance function on its mean domain, often much simpler than the corresponding generating probability measures. The mean value parametrization appeared to be crucial in some statistical theory, like in generalized linear models, exponential dispersion models and Bayesian framework. The main aim of the paper is to expose the mean value parametrization for possible statistical applications. The paper presents an overview of the mean value parametrization and of the characterization property of the variance function for NEF’s. In particular it introduces the relationships existing between the NEF’s generating measure, Laplace transform and variance function as well as some supplemental results concerning the mean value representation. Some classes of polynomial variance functions are revisited for illustration. The corresponding NEF’s of such classes are generated by counting probabilities on the nonnegative integers and provide Poisson-overdispersed competitors to the homogeneous Poisson distribution.

This paper deals with asymptotics for multiple-set linear canonical analysis (MSLCA). A definition of this analysis, that adapts the classical one to the context of Euclidean random variables, is given and properties of the related canonical coefficients are derived. Then, estimators of the MSLCA’s elements, based on empirical covariance operators, are proposed and asymptotics for these estimators is obtained. More precisely, we prove their consistency and we obtain asymptotic normality for the estimator of the operator that gives MSLCA, and also for the estimator of the vector of canonical coefficients. These results are then used to obtain a test for mutual non-correlation between the involved Euclidean random variables.

The problem of estimating parameters of a Pareto distribution is investigated under a general scale invariant loss function when the scale parameter is restricted to the interval (0, 1]. We consider the estimation of shape parameter when the scale parameter is unknown. Techniques for improving equivariant estimators developed by Stein, Brewster–Zidek and Kubokawa are applied to derive improved estimators. In particular improved classes of estimators are obtained for the entropy loss and a symmetric loss. Risk functions of various estimators are compared numerically using simulations. It is also shown that the technique of Kubokawa produces improved estimators for estimating the scale parameter when the shape parameter is known.