Estimators

Why

We have studied guessing random variables (see Estimates). What if we can use another random variable in making our estimate?

Definition

Let $(\Omega , \mathcal{A} , \mathbfsf{P} )$ be a probability space. Let $U, V$ be sets. Let $x: \Omega \to V$ and let $y: \Omega \to U$. An estimator or predictor for $x$ given $y$ is a function from $U$ to $V$. An estimate, then, corresponds to a constant estimator, and vice versa. Some authors call the selection of an estimator estimation or an estimation problem.

Error function

An error function is a function $e: U \times V \to \R $ which quantifies the cost of an error.