The problem of minimax estimation of a parameter θ when θ is restricted to a finite interval [θ₀,θ₀+m] is studied. The case of a convex loss function is considered. Sufficient conditions for existence of a minimax estimator which is a Bayes estimator with respect to a prior concentrated in two points θ₀ and θ₀+m are obtained. An example is presented.
The problem of robust Bayesian estimation in some models with an asymmetric loss function (LINEX) is considered. Some uncertainty about the prior is assumed by introducing two classes of priors. The most robust and conditional Γ-minimax estimators are constructed. The situations when those estimators coincide are presented.
The problem of posterior regret Γ-minimax estimation under LINEX loss function is considered. A general form of posterior regret Γ-minimax estimators is presented and it is applied to a normal model with two classes of priors. A situation when the posterior regret Γ-minimax estimator, the most stable estimator and the conditional Γ-minimax estimator coincide is presented.
The problem of measuring the Bayesian robustness is considered. An upper bound for the oscillation of a posterior functional in terms of the Kolmogorov distance between the prior distributions is given. The norm of the Frechet derivative as a measure of local sensitivity is presented. The problem of finding optimal statistical procedures is presented.
The concept of robustness of statistical procedures is one of the most important subject in Zielinski's papers. In this article the development of the idea of robustness as introduced in Zielinski's papers is presented. The definitions of a supermodel and a robustness function are given. The problem of the robust estimation of a scale parameter in an exponential model and the robustness of tests for comparison of means in two or more populations are described. Robustness in Bayesian statistical...
An upper bound for the Kolmogorov distance between the posterior distributions in terms of that between the prior distributions is given. For some likelihood functions the inequality is sharp. Applications to assessing Bayes robustness are presented.
The problem of robust Bayesian estimation in a normal model with asymmetric loss function (LINEX) is considered. Some uncertainty about the prior is assumed by introducing two classes of priors. The most robust and conditional Γ-minimax estimators are constructed. The situations when those estimators coincide are presented.
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