Sufficient Condition For Minimaxity Research Articles

Estimation of the scale parameter of a selected gamma population with unequal shape parameters under stein loss function

The problem of estimation of the parameter of a selected population arises when we encounter with several populations and would like to estimate the parameter of the best (worst) selected population. Suppose be independent random samples drawn from populations respectively, where observations from Π i have a -distribution with unequal known shape parameters In this paper, we use a selection rule to select the best (worst) population, and estimate the best (worst) scale parameter of the selected population under the Stein loss function. The uniformly minimum risk unbiased (UMRU) estimators of θS and θJ are obtained. A sufficient condition for inadmissibility of scale-invariant estimators of is obtained and it is shown that the UMRU estimator of is inadmissible. For k = 2, a sufficient condition for minimaxity of a given estimator of is obtained, and the generalized Bayes estimator of θS is shown to be minimax. Finally, the risk functions of the various competing estimators are compared numerically, and a real data is provided to compute the proposed estimates and their expected risks.

Admissible and minimax estimation of the parameter of the selected Pareto population under squared log error loss function

The problem of estimation after selection arises when we select a population from the given k populations by a selection rule, and estimate the parameter of the selected population. In this paper we consider the problem of estimation of the scale parameter of the selected Pareto population \(\theta _{M}\) (or \(\theta _{J}\)) under squared log error loss function. The uniformly minimum risk unbiased (UMRU) estimator of \(\theta _{M}\) and \(\theta _{J}\) are obtained. In the case of \(k=2,\) we give a sufficient condition for minimaxity of an estimator of \(\theta _{M}\) and \(\theta _{J},\) and show that the UMRU and natural estimators of \(\theta _{J}\) are minimax. Also the class of linear admissible estimators of \(\theta _{M}\) and \(\theta _{J}\) are obtained which contain the natural estimator. By using the Brewester–Ziedeck technique we find sufficient condition for inadmissibility of some scale and permutation invariant estimators of \(\theta _{J},\) and show that the UMRU estimator of \(\theta _{J}\) is inadmissible. Finally, we compare the risk of the obtained estimators numerically, and discuss the results for selected uniform population.

Admissible and Minimax Estimators of θ r with Truncated Parameter Space Under Squared-Log Error Loss Function

Admissible and minimax estimation of θ r in a subclass of the exponential family with truncated parameter space is considered under squared-log error loss function. We give a necessary and sufficient condition for minimaxity and obtain the classes of new admissible and minimax estimators.

Solution to the inverse problem of minimax control and worst case disturbance for linear continuous-time systems

In this paper we characterize the controllers and the disturbances which are possible solutions of a certain linear quadratic minimax control problem. The necessary and sufficient conditions for minimaxity are given which involve a frequency domain inequality. It is then shown that this inequality implies good robustness properties of the closed-loop system.

Estimateurs à rétrécisseur matriciel différentiable, pour un coût quadratique général

The problem of the estimation of a multivariate normal mean when the variance is known up to a multiplicative factor is considered under an arbitrary quadratic loss. We introduce shrinkage estimators with differentiable shrinking functions under weak algebraic assumptions. We deduce sufficient conditions for minimaxity from an unbiased estimator of the risk. We also consider simpler conditions in particular cases like those of shrinking functions with controlled variations.

Minimax Multiple Shrinkage Estimation

For the canonical problem of estimating a multivariate normal mean under squared-error-loss, this article addresses the problem of selecting a minimax shrinkage estimator when vague or conflicting prior information suggests that more than one estimator from a broad class might be effective. For this situation a new class of alternative estimators, called multiple shrinkage estimators, is proposed. These estimators use the data to emulate the behavior and risk properties of the most effective estimator under consideration. Unbiased estimates of risk and sufficient conditions for minimaxity are provided. Bayesian motivations link this construction to posterior means of mixture priors. To illustrate the theory, minimax multiple shrinkage Stein estimators are constructed which can adaptively shrink the data towards any number of points or subspaces.