The cross-covariance matrix of observation vectors in two linear statistical models need not be zero matrix. In such a case the problem is to find explicit expressions for the best linear unbiased estimators of both model parameters and estimators of variance components in the simplest structure of the covariance matrix. Univariate and multivariate forms of linear models are dealt with.
The singular mixed linear model with constraints is investigated with respect to an influence of inaccurate variance components on a decrease of the confidence level. The algorithm for a determination of the boundary of the insensitivity region is given. It is a set of all shifts of variance components values which make the tolerated decrease of the confidence level only. The problem about geometrical characterization of the confidence domain is also presented.
The problem of sequentially estimating powers of a scale parameter in a scale family and in a location-scale family is considered in the case when the observations become available at random times. Certain classes of sequential estimation procedures are derived under a scale invariant loss function and with the observation cost determined by a convex function of the stopping time and the number of observations up to that time.
This paper deals with four types of point estimators based on minimization of information-theoretic divergences between hypothetical and empirical distributions. These were introduced (i) by Liese and Vajda [9] and independently Broniatowski and Keziou [3], called here power superdivergence estimators, (ii) by Broniatowski and Keziou [4], called here power subdivergence estimators, (iii) by Basu et al. [2], called here power pseudodistance estimators, and (iv) by Vajda [18] called here Rényi pseudodistance...
After recalling previous work on probability generating functions for real valued random variables we extend to these random variables uniform laws of large numbers and functional limit theorem for the empirical probability generating function. We present an application to the study of continuous laws, namely, estimation of parameters of Gaussian, gamma and uniform laws by means of a minimum contrast estimator that uses the empirical probability generating function of the sample. We test the procedure...
This paper is concerned with the study of some properties of the distance between statistical individuals based on representations on the dual tangent space of a parametric manifold representation of a statistical model. Explicit expressions for distances are obtained for well-known families of distributions. We have also considered applications of the distance to parameter estimation, testing statistical hypotheses and discriminant analysis.
Necessary and sufficient conditions are derived for the inclusions and to be fulfilled where , and , are some classes of invariant linearly sufficient statistics (Oktaba, Kornacki, Wawrzosek (1988)) corresponding to the Gauss-Markov models and , respectively.
This paper deals with stability of stochastic optimization problems in a general setting. Objective function is defined on a metric space and depends on a probability measure which is unknown, but, estimated from empirical observations. We try to derive stability results without precise knowledge of problem structure and without measurability assumption. Moreover, -optimal solutions are considered. The setup is illustrated on consistency of a --estimator in linear regression model.