# Estimation theory

(Redirected from Statistical estimation)

Estimation theory is a branch of statistics and signal processing that deals with estimating the values of parameters based on measured/empirical data. The parameters describe the physical scenario or object that answers a question posed by the estimator.

For example, it is desired to estimate the proportion of a population of voters who will vote for a particular candidate. That proportion is the unobservable parameter; the estimate is based on a small random sample of voters.

Or, for example, in radar the goal is to estimate the location of objects (airplanes, boats, etc.) by analyzing the received echo and a possible question to be posed is "where are the airplanes?" To answer where the airplanes are, it is necessary to estimate the distance the airplanes are at from the radar station, which can provide an absolute location if the absolute location of the radar station is known.

In estimation theory, it is assumed that the desired information is embedded into a noisy signal. Noise adds uncertainty and if there was no uncertainty then there would be no need for estimation.

## Fields that use estimation theory

There are numerous fields that require the use of estimation theory. Some of these fields include (but by no means limited to):

The measured data is likely to be subject to noise or uncertainty and it is through statistical probability that optimal solutions are sought to extract as much information from the data.

## Estimation process

The entire purpose of estimation theory is to arrive at an estimator, and preferably an implementable one that could actually be used. 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. An optimal estimator would indicate that all available information in the measured data has been extracted, for if there was unused information in the data then the estimator would not be optimal.

These are the general steps to arrive at an estimator:

• In order to arrive at a desired estimator for estimating a single or multiple parameters, it is first necessary to determine a model for the system. This model should incorporate the process being modeled as well as points of uncertainty and noise. The model describes the physical scenario in which the parameters apply.
• After deciding upon a model, it is helpful to find the limitations placed upon an estimator. This limitation, for example, can be found through the Cramér-Rao bound.
• Next, an estimator needs to be developed or applied if an already known estimator is valid for the model. The estimator needs to be tested against the limitations to determine if it is an optimal estimator (if so, then no other estimator will perform better).
• 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 start anew.

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

## Basics

To build a model, several statistical "ingredients" need to be known. These are needed to ensure the estimator has some mathematical tractability instead of being based on "good feel".

The first is a set of statistical samples taken from a random vector (RV) of size . Put into a vector,

.

Secondly, we have the corresponding  parameters

,

which need to be established with their probability density function (pdf) or probability mass function (pmf)

.

It is also possible for the parameters themselves to have a probability distribution (e.g., Bayesian statistics). It is then necessary to define the epistemic probability

.

After the model is formed, the goal is to estimate the parameters, commonly denoted , where the "hat" indicates the estimate.

One common estimator is the minimum mean squared error (MMSE) estimator, which utilizes the error between the estimated parameters and the actual value of the parameters



as the basis for optimality. This error term is then squared and minimized for the MMSE estimator.

## Estimators

This list is some of the more common estimators used, and some topics related to them:

## Example: DC gain in white Gaussian noise

Consider a received discrete signal, , of  independent samples that consists of a DC gain  with Additive white Gaussian noise  with known variance  (i.e., ). Since the variance is known then the only unknown parameter is .

The model for the signal is then



Two possible (of many) estimators are:

Both of these estimators have a mean of , which can be shown through taking the expected value of each estimator



and



At this point, these two estimators would appear to perform the same. However, the difference between them becomes apparent when comparing the variances.



and



It would seem that the sample mean is a better estimator since, as , the variance goes to zero.

### Maximum likelihood

Continuing the example using the maximum likelihood estimator, the probability density function (pdf) of the noise for one sample  is



and the probability of  becomes ( can be thought of a )



By independence, the probability of  becomes



Taking the natural logarithm of the pdf



and the maximum likelihood estimator is



Taking the first derivative of the log-likelihood function



and setting it to zero



This results in the maximum likelihood estimator



which is simply the sample mean. From this example, it was found that the sample mean is the maximum likelihood estimator for  samples of AWGN with a fixed, unknown DC gain.

### Cramér-Rao lower bounds

To find the Cramér-Rao lower bounds (CRLB) of the sample mean estimator, it is first necessary to find the Fisher information number



and copying from above



Taking the second derivative



and finding the negative expected value is trivial since it is now a deterministic constant 

Finally, putting the Fisher information into



results in



Comparing this to the variance of the sample mean (determined previously) shows that the sample mean is equal to the Cramér-Rao lower bounds for all values of  and . The sample mean is the minimum variance unbiased estimator (MVUE) in addition to being the maximum likelihood estimator.

This example of DC gain + WGN is a recurring example in Kay's Fundamentals of Statistical Signal Processing.

## References

• "Mathematical Statistics and Data Analysis" by John Rice. (ISBN 0-534-209343)
• Fundamentals of Statistical Signal Processing: Estimation Theory by Steven M. Kay (ISBN 0-13-345711-7)
• An Introduction to Signal Detection and Estimation by H. Vincent Poor (ISBN 0-387-94173-8)
• Detection, Estimation, and Modulation Theory, Part 1 by Harry L. Van Trees (ISBN 0-471-09517-6; website)
• Optimal State Estimation: Kalman, H-infinity, and Nonlinear Approaches by Dan Simon website