Estimation Theory - Estimation Process

Estimation Process

The entire purpose of estimation theory is to arrive at an estimator — preferably an easily implementable one. The estimator takes the measured data as input and produces an estimate of the parameters.

It is also preferable to derive an estimator that exhibits optimality. Estimator optimality usually refers to achieving minimum average error over some class of estimators, for example, a minimum variance unbiased estimator. In this case, the class is the set of unbiased estimators, and the average error measure is variance (average squared error between the value of the estimate and the parameter). However, optimal estimators do not always exist.

These are the general steps to arrive at an estimator:

• In order to arrive at a desired estimator, it is first necessary to determine a probability distribution for the measured data, and the distribution's dependence on the unknown parameters of interest. Often, the probability distribution may be derived from physical models that explicitly show how the measured data depends on the parameters to be estimated, and how the data is corrupted by random errors or noise. In other cases, the probability distribution for the measured data is simply "assumed", for example, based on familiarity with the measured data and/or for analytical convenience.
• After deciding upon a probabilistic model, it is helpful to find the theoretically acheivable (optimal) precision available to any estimator based on this model. The Cramér–Rao bound is useful for this.
• Next, an estimator needs to be developed, or applied (if an already known estimator is valid for the model). There are a variety of methods for developing estimators; maximum likelihood estimators are often the default although they may be hard to compute or even fail to exist. If possible, the theoretical performance of the estimator should be derived and compared with the optimal performace found in the last step.
• Finally, experiments or simulations can be run using the estimator to test its performance.

After arriving at an estimator, real data might show that the model used to derive the estimator is incorrect, which may require repeating these steps to find a new estimator. A non-implementable or infeasible estimator may need to be scrapped and the process started anew.

In summary, the estimator estimates the parameters of a physical model based on measured data.