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Examples and Problems in Mathematical Statistics by Shelemyahu Zacks

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CHAPTER 8

Bayesian Analysis in Testing and Estimation

PART I: THEORY

This chapter is devoted to some topics of estimation and testing hypotheses from the point of view of statistical decision theory. The decision theoretic approach provides a general framework for both estimation of parameters and testing hypotheses. The objective is to study classes of procedures in terms of certain associated risk functions and determine the existence of optimal procedures. The results that we have presented in the previous chapters on minimum mean–squared–error (MSE) estimators and on most powerful tests can be considered as part of the general statistical decision theory. We have seen that uniformly minimum MSE estimators and uniformly most powerful tests exist only in special cases. One could overcome this difficulty by considering procedures that yield minimum average risk, where the risk is defined as the expected loss due to erroneous decision, according to the particular distribution Fθ. The MSE in estimation and the error probabilities in testing are special risk functions. The risk functions depend on the parameters θ of the parent distribution. The average risk can be defined as an expected risk according to some probability distribution on the parameter space. Statistical inference that considers the parameter(s) as random variables is called a Bayesian inference. The expected risk with respect to the distribution of θ is called in Bayesian theory the prior risk, and the probability ...

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