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Examples and Problems in Mathematical Statistics by Shelemyahu Zacks

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CHAPTER 9

Advanced Topics in Estimation Theory

PART I: THEORY

In the previous chapters, we discussed various classes of estimators, which attain certain optimality criteria, like minimum variance unbiased estimators (MVUE), asymptotic optimality of maximum likelihood estimators (MLEs), minimum mean–squared–error (MSE) equivariant estimators, Bayesian estimators, etc. In this chapter, we present additional criteria of optimality derived from the general statistical decision theory. We start with the game theoretic criterion of minimaxity and present some results on minimax estimators. We then proceed to discuss minimum risk equivariant and standard estimators. We discuss the notion of admissibility and present some results of Stein on the inadmissibility of some classical estimators. These examples lead to the so–called Stein–type and Shrinkage estimators.

9.1 MINIMAX ESTIMATORS

Given a class inline of estimators, the risk function associated with each d inline inline is R(d, θ), θ inline Θ. The maximal risk associated with d is R*(d) = R(d, θ). If in there is an estimator d* that minimizes R*(d) then d* is ...

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