Probability, Random Variables, and Random Processes: Theory and Signal Processing Applications

9.10 BAYES ESTIMATION

Since an estimator T of θ is a function of the samples X, it is a random variable that is characterized by a distribution which depends on the pdf of the iid samples. In addition to sufficiency for θ, we might be interested in unbiasedness and efficiency as discussed later in this chapter. We introduce some definitions that are used to describe the quality of an estimator.

Definition: Loss Function The loss function maps T = t and θ to a real number.

Various loss functions can be used such as the following examples:

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where is a weighting function that depends only on θ, and c>0 is a threshold. Some of these loss functions are illustrated in Figure 9.4. is a measure of the level of significance placed on deviations of ...