3 Estimation of Random Variables
3.1 ESTIMATION OF VARIABLES
In the sciences, especially engineering and physics, we are required to estimate variables that are not directly observable but are observed only through some other measurable variables—for example, estimating the amplitude, frequency, or phase of a known signal in noise. For some problems the variables can be modeled as random variables; others may not be realistically random just simply unknown deterministic variables. Naturally it is desirable to find the “best” estimates where best is in the sense of minimizing some assigned performance measure. The desired variables will be estimated through some function of the observations where the functional relationship is selected to minimize the performance measure. A logical question becomes: what are some special types of functions, and what are some realistic performance measures? Also of major importance is what statistical information is known and how this affects the procedures for finding the optimal estimators. These are the questions addressed in this chapter.
3.1.1 Basic Formulation for Estimation of Random Variables
Say we wish to estimate a random variable X by observing another random variable Z that is statistically related to X. An estimator for X, call it , is some function, g(.), of the observable Z and is written as
Thus is itself a random variable. The ...