Comparison of Risk Ratios of Shrinkage Estimators in High Dimensions

In this paper, we analyze the risk ratios of several shrinkage estimators using a balanced loss function. The James–Stein estimator is one of a group of shrinkage estimators that has been proposed in the existing literature. For these estimators, sufficient criteria for minimaxity have been established, and the James–Stein estimator’s minimaxity has been derived. We demonstrate that the James–Stein estimator’s minimaxity is still valid even when the parameter space has infinite dimension. It is shown that the positive-part version of the James–Stein estimator is substantially superior to the James–Stein estimator, and we address the asymptotic behavior of their risk ratios to the maximum likelihood estimator (MLE) when the dimensions of the parameter space are infinite. Finally, a simulation study is carried out to verify the performance evaluation of the considered estimators.

On shrinkage estimators improving the James-Stein estimator under balanced loss function

In this paper, we are interested in estimating a multivariate normal mean under the balanced loss function using the shrinkage estimators deduced from the Maximum Likelihood Estimator (MLE). First, we consider a class of estimators containing the James-Stein estimator, we then show that any estimator of this class dominates the MLE, consequently it is minimax. Secondly, we deal with shrinkage estimators which are not only minimax but also dominate the James- Stein estimator.

Limits of Risks Ratios of Shrinkage Estimators under the Balanced Loss Function

In this paper we study the estimation of a multivariate normal mean under the balanced loss function. We present here a class of shrinkage estimators which generalizes the James-Stein estimator and we are interested to establish the asymptotic behaviour of risks ratios of these estimators to the maximum likelihood estimators (MLE). Thus, in the case where the dimension of the parameter space and the sample size are large, we determine the sufficient conditions for that the estimators cited previously are minimax

On shrinkage estimators improving the positive part of James-Stein estimator

In this work, we study the estimation of the multivariate normal mean by different classes of shrinkage estimators. The risk associated with the quadratic loss function is used to compare two estimators. We start by considering a class of estimators that dominate the positive part of James-Stein estimator. Then, we treat estimators of polynomial form and prove if we increase the degree of the polynomial we can build a better estimator from the one previously constructed. Furthermore, we discuss the minimaxity property of the considered estimators.

Baranchick-type Estimators of a Multivariate Normal Mean Under the General Quadratic Loss Function

The problem of estimating the mean of a multivariate normal distribution by different types of shrinkage estimators is investigated. We established the minimaxity of Baranchick-type estimators for identity covariance matrix and the matrix associated to the loss function is diagonal. In particular the class of James-Stein estimator is presented. The general situation for both matrices cited above is discussed

A normative definition of a Bayesian prior

<p>Applications of the Bayesian statistics require specifying a prior distribution for each unknown parameter to be estimated. The commonly used definition of a Bayesian prior distribution, information about an uncertain parameter, does not provide guidance on how to derive and formulate a prior distribution. In practice, we often use "non-informative" priors or priors based on mathematical convenience. I present a normative definition of the prior based on the shared features of the James-Stein estimator, the empirical Bayes method, and the Bayesian hierarchical model. I use the word "normative" to mean "prescriptive". It also reflects the meaning that the definition can be inconsistent with one another insofar as different types of parameters. I present two case studies where this definition guided me to formulate the modeling processes: one on modeling and predicting cyanobacterial toxin concentration in Lake Erie using chlorophyll-a concentrations (Lake Erie example) and the other on improving the accuracy of calibration-curve-based chemical measurement method (calibration-curve example). The Lake Erie example illustrates temporal exchangeable units, while the calibration-curve example showcases the ubiquity of such exchangeable units.</p>