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Том 26, № 3 (2017)
- Год: 2017
- Статей: 5
- URL: https://journal-vniispk.ru/1066-5307/issue/view/13894
Article
On the mean value parametrization of natural exponential families — a revisited review
Аннотация
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.
159-175
Efficient estimation of the error distribution in a varying coefficient regression model
Аннотация
It is shown that the weighted residual-based estimator of Schick, Zhu, and Du (2017) is efficient in some special cases and can be made to be efficient by adding a stochastic correction term. The efficiency is shown by deriving the efficient influence function and establishing a uniform stochastic expansion with this influence function. The correction term relies on estimators of the score function for the errors and other characteristics of the model.
176-195
Asymptotic theory of multiple-set linear canonical analysis
Аннотация
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.
196-211
Two-sample Kolmogorov-Smirnov test using a Bayesian nonparametric approach
Аннотация
In this paper, a Bayesian nonparametric approach to the two-sample problem is proposed. Given two samples \(\text{X} = {X_1}, \ldots ,{X_{m1}}\;\mathop {\text~}\limits^{i.i.d.} F\) and \(Y = {Y_1}, \ldots ,{Y_{{m_2}}}\mathop {\text~}\limits^{i.i.d.} G\), with F and G being unknown continuous cumulative distribution functions, we wish to test the null hypothesis H0: F = G. The method is based on computing the Kolmogorov distance between two posterior Dirichlet processes and comparing the results with a reference distance. The parameters of the Dirichlet processes are selected so that any discrepancy between the posterior distance and the reference distance is related to the difference between the two samples. Relevant theoretical properties of the procedure are also developed. Through simulated examples, the approach is compared to the frequentist Kolmogorov–Smirnov test and a Bayesian nonparametric test in which it demonstrates excellent performance.
212-225
Classes of improved estimators for parameters of a Pareto distribution
Аннотация
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.
226-235