Branch of statistics to estimate models based on measured data
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:
Examples.
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.
Basics.
For a given model, several statistical "ingredients" are needed so the estimator can be implemented. The first is a statistical sample – a set of data points taken from a random vector (RV) of size "N". Put into a vector,
formula_1
Secondly, there are "M" parameters
formula_2
whose values are to be estimated. 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:
formula_3
It is also possible for the parameters themselves to have a probability distribution (e.g., Bayesian statistics). It is then necessary to define the Bayesian probability
formula_4
After the model is formed, the goal is to estimate the parameters, with the estimates commonly denoted formula_5, 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
formula_6
as the basis for optimality. This error term is then squared and the expected value of this squared value is minimized for the MMSE estimator.
Estimators.
Commonly used estimators (estimation methods) and topics related to them include:
Examples.
Unknown constant in additive white Gaussian noise.
Consider a received discrete signal, formula_7, of formula_8 independent samples that consists of an unknown constant formula_9 with additive white Gaussian noise (AWGN) formula_10 with zero mean and known variance formula_11 ("i.e.", formula_12).
Since the variance is known then the only unknown parameter is formula_9.
The model for the signal is then
formula_14
Two possible (of many) estimators for the parameter formula_9 are:
Both of these estimators have a mean of formula_9, which can be shown through taking the expected value of each estimator
formula_19
and
formula_20
At this point, these two estimators would appear to perform the same.
However, the difference between them becomes apparent when comparing the variances.
formula_21
and
formula_22
It would seem that the sample mean is a better estimator since its variance is lower for every "N"  ; 1.
Maximum likelihood.
Continuing the example using the maximum likelihood estimator, the probability density function (pdf) of the noise for one sample formula_10 is
formula_24
and the probability of formula_7 becomes (formula_7 can be thought of a formula_27)
formula_28
By independence, the probability of formula_29 becomes
formula_30
Taking the natural logarithm of the pdf
formula_31
and the maximum likelihood estimator is
formula_32
Taking the first derivative of the log-likelihood function
formula_33
and setting it to zero
formula_34
This results in the maximum likelihood estimator
formula_35
which is simply the sample mean.
From this example, it was found that the sample mean is the maximum likelihood estimator for formula_8 samples of a fixed, unknown parameter corrupted by AWGN.
Cramér–Rao lower bound.
To find the Cramér–Rao lower bound (CRLB) of the sample mean estimator, it is first necessary to find the Fisher information number
formula_37
and copying from above
formula_38
Taking the second derivative
formula_39
and finding the negative expected value is trivial since it is now a deterministic constant
formula_40
Finally, putting the Fisher information into
formula_41
results in
formula_42
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 formula_8 and formula_9.
In other words, the sample mean is the (necessarily unique) efficient estimator, and thus also the minimum variance unbiased estimator (MVUE), in addition to being the maximum likelihood estimator.
Maximum of a uniform distribution.
One of the simplest non-trivial examples of estimation is the estimation of the maximum of a uniform distribution. 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.
Given a discrete uniform distribution formula_45 with unknown maximum, the UMVU estimator for the maximum is given by
formula_46
where "m" is the sample maximum and "k" is the sample size, sampling without replacement. 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;
 ;templatestyles src="Block indent/styles.css"/ ;"The sample maximum plus the average gap between observations in the sample",
the gap being added to compensate for the negative bias of the sample maximum as an estimator for the population maximum.
This has a variance of
formula_47
so a standard deviation of approximately formula_48, the (population) average size of a gap between samples; compare formula_49 above. This can be seen as a very simple case of maximum spacing estimation.
The sample maximum is the maximum likelihood estimator for the population maximum, but, as discussed above, it is biased.
Applications.
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.
Notes.
 ;templatestyles src="Reflist/styles.css" / ;
References.
Citations.
 ;templatestyles src="Reflist/styles.css" / ;
Sources.
 ;templatestyles src="Refbegin/styles.css" / ;