Estimation theory is a branch of statistics that deals with estimating the values of parameters based on measured empirical data that has a random component.
The parameters describe an underlying physical setting in such a way that their value affects the distribution of the measured data.
An estimator attempts to approximate the unknown parameters using the measurements.
In estimation theory, two approaches are generally considered:[1] 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 parameter sought; the estimate is based on a small random sample of voters.
Alternatively, it is desired to estimate the probability of a voter voting for a particular candidate, based on some demographic features, such as age.
Or, for example, in radar the aim is to find the range of objects (airplanes, boats, etc.)
by analyzing the two-way transit timing of received echoes of transmitted pulses.
Since the reflected pulses are unavoidably embedded in electrical noise, their measured values are randomly distributed, so that the transit time must be estimated.
As another example, in electrical communication theory, the measurements which contain information regarding the parameters of interest are often associated with a noisy signal.
For a given model, several statistical "ingredients" are needed so the estimator can be implemented.
Third, the continuous probability density function (pdf) or its discrete counterpart, the probability mass function (pmf), of the underlying distribution that generated the data must be stated conditional on the values of the parameters:
It is also possible for the parameters themselves to have a probability distribution (e.g., Bayesian statistics).
Commonly used estimators (estimation methods) and topics related to them include: Consider a received discrete signal,
independent samples that consists of an unknown constant
with additive white Gaussian noise (AWGN)
, which can be shown through taking the expected value of each estimator
Continuing the example using the maximum likelihood estimator, the probability density function (pdf) of the noise for one sample
From this example, it was found that the sample mean is the maximum likelihood estimator for
samples of a fixed, unknown parameter corrupted by AWGN.
and finding the negative expected value is trivial since it is now a deterministic constant
Comparing this to the variance of the sample mean (determined previously) shows that the sample mean is equal to the Cramér–Rao lower bound for all values of
It is used as a hands-on classroom exercise and to illustrate basic principles of estimation theory.
Further, in the case of estimation based on a single sample, it demonstrates philosophical issues and possible misunderstandings in the use of maximum likelihood estimators and likelihood functions.
[2][3] This problem is commonly known as the German tank problem, due to application of maximum estimation to estimates of German tank production during World War II.
The formula may be understood intuitively as; the gap being added to compensate for the negative bias of the sample maximum as an estimator for the population maximum.
, the (population) average size of a gap between samples; compare
This can be seen as a very simple case of maximum spacing estimation.
Numerous fields require the use of estimation theory.
Some of these fields include: Measured data are 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 as possible.