Editing Estimation theory

{{Short description|Branch of statistics to estimate models based on measured data}}
{{redirect-distinguish|Parameter estimation|Point estimation|Interval estimation}}
{{other uses|Estimation (disambiguation)}}
{{More footnotes|date=April 2025}}
'''Estimation theory''' is a branch of [[statistics]] that deals with estimating the values of [[Statistical parameter|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:<ref>
{{cite book |last1=Walter |first1=E. |last2=Pronzato |first2=L. |title=Identification of Parametric Models from Experimental Data |year=1997 |publisher=Springer-Verlag |location=London, England }}
</ref>

* The probabilistic approach (described in this article) assumes that the measured data is random with [[probability distribution]] dependent on the parameters of interest
* The [[set estimation|set-membership approach]] assumes that the measured data vector belongs to a set which depends on the parameter vector.

== 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 [[noise (physics)|noisy]] [[signal (electrical engineering)|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 (geometric)|vector]],
<math display="block">\mathbf{x} = \begin{bmatrix} x[0] \\ x[1] \\ \vdots \\ x[N-1] \end{bmatrix}.</math>
Secondly, there are ''M'' parameters
<math display="block">\boldsymbol{\theta} = \begin{bmatrix} \theta_1 \\ \theta_2 \\ \vdots \\ \theta_M \end{bmatrix},</math>
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:
<math display="block">p(\mathbf{x} | \boldsymbol{\theta}).\,</math>
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]]
<math display="block">\pi( \boldsymbol{\theta}).\,</math>
After the model is formed, the goal is to estimate the parameters, with the estimates commonly denoted <math>\hat{\boldsymbol{\theta}}</math>, 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
<math display="block">\mathbf{e} = \hat{\boldsymbol{\theta}} - \boldsymbol{\theta}</math>
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==
{{main|Estimator}}

Commonly used estimators (estimation methods) and topics related to them include:
*[[Maximum likelihood]] estimators
*[[Bayes estimator]]s
*[[method of moments (statistics)|Method of moments]] estimators
*[[Cramér–Rao bound]]
*[[Least squares]]
*[[Minimum mean squared error]] (MMSE), also known as Bayes least squared error (BLSE)
*[[Maximum a posteriori]] (MAP)
*[[Minimum variance unbiased estimator]] (MVUE)
*[[Nonlinear system identification]]
*[[Best linear unbiased estimator]] (BLUE)
*Unbiased estimators &mdash; see [[estimator bias]].
*[[Particle filter]]
*[[Markov chain Monte Carlo]] (MCMC)
*[[Kalman filter]], and its various derivatives
*[[Wiener filter]]

==Examples==

===Unknown constant in additive white Gaussian noise===
Consider a received [[discrete signal]], <math>x[n]</math>, of <math>N</math> [[statistical independence|independent]] [[statistical sample|samples]] that consists of an unknown constant <math>A</math> with [[additive white Gaussian noise]] (AWGN) <math>w[n]</math> with zero [[mean]] and known [[variance]] <math>\sigma^2</math> (''i.e.'', <math>\mathcal{N}(0, \sigma^2)</math>).
Since the variance is known then the only unknown parameter is <math>A</math>.

The model for the signal is then
<math display="block">x[n] = A + w[n] \quad n=0, 1, \dots, N-1</math>

Two possible (of many) estimators for the parameter <math>A</math> are:
* <math>\hat{A}_1 = x[0]</math>
* <math>\hat{A}_2 = \frac{1}{N} \sum_{n=0}^{N-1} x[n]</math> which is the [[sample mean]]

Both of these estimators have a [[mean]] of <math>A</math>, which can be shown through taking the [[expected value]] of each estimator
<math display="block">\mathrm{E}\left[\hat{A}_1\right] = \mathrm{E}\left[ x[0] \right] = A</math>
and
<math display="block">
\mathrm{E}\left[ \hat{A}_2 \right]
=
\mathrm{E}\left[ \frac{1}{N} \sum_{n=0}^{N-1} x[n] \right]
=
\frac{1}{N} \left[ \sum_{n=0}^{N-1} \mathrm{E}\left[ x[n] \right] \right]
=
\frac{1}{N} \left[ N A \right]
=
A
</math>

At this point, these two estimators would appear to perform the same.
However, the difference between them becomes apparent when comparing the variances.
<math display="block">\mathrm{var} \left( \hat{A}_1 \right) = \mathrm{var} \left( x[0] \right) = \sigma^2</math>
and
<math display="block">
\mathrm{var} \left( \hat{A}_2 \right)
=
\mathrm{var} \left( \frac{1}{N} \sum_{n=0}^{N-1} x[n] \right)
\overset{\text{independence}}{=}
\frac{1}{N^2} \left[ \sum_{n=0}^{N-1} \mathrm{var} (x[n]) \right]
=
\frac{1}{N^2} \left[ N \sigma^2 \right]
=
\frac{\sigma^2}{N}
</math>

It would seem that the sample mean is a better estimator since its variance is lower for every&nbsp;''N''&nbsp;>&nbsp;1.

====Maximum likelihood====
{{main|Maximum likelihood}}

Continuing the example using the [[maximum likelihood]] estimator, the [[probability density function]] (pdf) of the noise for one sample <math>w[n]</math> is
<math display="block">p(w[n]) = \frac{1}{\sigma \sqrt{2 \pi}} \exp\left(- \frac{1}{2 \sigma^2} w[n]^2 \right)</math>
and the probability of <math>x[n]</math> becomes (<math>x[n]</math> can be thought of a <math>\mathcal{N}(A, \sigma^2)</math>)
<math display="block">p(x[n]; A) = \frac{1}{\sigma \sqrt{2 \pi}} \exp\left(- \frac{1}{2 \sigma^2} (x[n] - A)^2 \right)</math>
By [[statistical independence|independence]], the probability of <math>\mathbf{x}</math> becomes
<math display="block">
p(\mathbf{x}; A)
=
\prod_{n=0}^{N-1} p(x[n]; A)
=
\frac{1}{\left(\sigma \sqrt{2\pi}\right)^N}
\exp\left(- \frac{1}{2 \sigma^2} \sum_{n=0}^{N-1}(x[n] - A)^2 \right)
</math>
Taking the [[natural logarithm]] of the pdf
<math display="block">
\ln p(\mathbf{x}; A)
=
-N \ln \left(\sigma \sqrt{2\pi}\right)
- \frac{1}{2 \sigma^2} \sum_{n=0}^{N-1}(x[n] - A)^2
</math>
and the maximum likelihood estimator is
<math display="block">\hat{A} = \arg \max \ln p(\mathbf{x}; A)</math>

Taking the first [[derivative]] of the log-likelihood function
<math display="block">
\frac{\partial}{\partial A} \ln p(\mathbf{x}; A)
=
\frac{1}{\sigma^2} \left[ \sum_{n=0}^{N-1}(x[n] - A) \right]
=
\frac{1}{\sigma^2} \left[ \sum_{n=0}^{N-1}x[n] - N A \right]
</math>
and setting it to zero
<math display="block">
0
=
\frac{1}{\sigma^2} \left[ \sum_{n=0}^{N-1}x[n] - N A \right]
=
\sum_{n=0}^{N-1}x[n] - N A
</math>

This results in the maximum likelihood estimator
<math display="block">\hat{A} = \frac{1}{N} \sum_{n=0}^{N-1}x[n]</math>
which is simply the sample mean.
From this example, it was found that the sample mean is the maximum likelihood estimator for <math>N</math> samples of a fixed, unknown parameter corrupted by AWGN.

====Cramér–Rao lower bound====
{{further|Cramér–Rao bound}}

To find the [[Cramér–Rao lower bound]] (CRLB) of the sample mean estimator, it is first necessary to find the [[Fisher information]] number
<math display="block">
\mathcal{I}(A)
=
\mathrm{E}
\left(
 \left[
  \frac{\partial}{\partial A} \ln p(\mathbf{x}; A)
 \right]^2
\right)
=
-\mathrm{E}
\left[
 \frac{\partial^2}{\partial A^2} \ln p(\mathbf{x}; A)
\right]
</math>
and copying from above
<math display="block">
\frac{\partial}{\partial A} \ln p(\mathbf{x}; A)
=
\frac{1}{\sigma^2} \left[ \sum_{n=0}^{N-1}x[n] - N A \right]
</math>

Taking the second derivative
<math display="block">
\frac{\partial^2}{\partial A^2} \ln p(\mathbf{x}; A)
=
\frac{1}{\sigma^2} (- N)
=
\frac{-N}{\sigma^2}
</math>
and finding the negative expected value is trivial since it is now a deterministic constant
<math>
-\mathrm{E}
\left[
 \frac{\partial^2}{\partial A^2} \ln p(\mathbf{x}; A)
\right]
=
\frac{N}{\sigma^2}
</math>

Finally, putting the Fisher information into
<math display="block">
\mathrm{var}\left( \hat{A} \right)
\geq
\frac{1}{\mathcal{I}}
</math>
results in
<math display="block">
\mathrm{var}\left( \hat{A} \right)
\geq
\frac{\sigma^2}{N}
</math>

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 <math>N</math> and <math>A</math>.
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===
{{main|German tank problem}}

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]] <math>1,2,\dots,N</math> with unknown maximum, the [[UMVU]] estimator for the maximum is given by
<math display="block">\frac{k+1}{k} m - 1 = m + \frac{m}{k} - 1</math>
where ''m'' is the [[sample maximum]] and ''k'' is the [[sample size]], sampling without replacement.<ref name="Johnson">{{citation | last=Johnson | first=Roger | title=Estimating the Size of a Population | year=1994 | journal=Teaching Statistics | volume=16 | issue=2 (Summer) | doi=10.1111/j.1467-9639.1994.tb00688.x | pages = 50–52 }}</ref><ref name="Johnson2">{{citation | last=Johnson | first=Roger | contribution=Estimating the Size of a Population | title=Getting the Best from Teaching Statistics | year=2006 | url=http://www.rsscse.org.uk/ts/gtb/contents.html | contribution-url=http://www.rsscse.org.uk/ts/gtb/johnson.pdf | url-status=dead | archive-url=https://web.archive.org/web/20081120085633/http://www.rsscse.org.uk/ts/gtb/contents.html | archive-date=November 20, 2008 }}</ref> 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;
{{block indent | em = 1.5 | text = "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.{{NoteTag|The sample maximum is never more than the population maximum, but can be less, hence it is a [[biased estimator]]: it will tend to ''underestimate'' the population maximum.}}

This has a variance of<ref name="Johnson" />
<math display="block">\frac{1}{k}\frac{(N-k)(N+1)}{(k+2)} \approx \frac{N^2}{k^2} \text{ for small samples } k \ll N</math>
so a standard deviation of approximately <math>N/k</math>, the (population) average size of a gap between samples; compare <math>\frac{m}{k}</math> 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:
* Interpretation of scientific [[experiment]]s
* [[Signal processing]]
* [[Clinical trial]]s
* [[Opinion poll]]s
* [[Quality control]]
* [[Telecommunication]]s
* [[Project management]]
* [[Software engineering]]
* [[Control theory]] (in particular [[Adaptive control]])
* [[Network intrusion detection system]]
* [[Orbit determination]]

Measured data are likely to be subject to [[noise (physics)|noise]] or uncertainty and it is through statistical [[probability]] that [[optimization (mathematics)|optimal]] solutions are sought to extract as much [[Fisher information|information]] from the data as possible.

== See also ==
{{Main category|Estimation theory}}
{{colbegin}}
* [[Best linear unbiased estimator]] (BLUE)
* [[Completeness (statistics)]]
* [[Detection theory]]
* [[Efficiency (statistics)]]
* [[Expectation-maximization algorithm]] (EM algorithm)
* [[Fermi problem]]
* [[Grey box model]]
* [[Information theory]]
* [[Least-squares spectral analysis]]
* [[Matched filter]]
* [[Maximum entropy spectral estimation]]
* [[Nuisance parameter]]
* [[Parametric equation]]
* [[Pareto principle]]
* [[Rule of three (statistics)]]
* [[State estimator]]
* [[Statistical signal processing]]
* [[Sufficiency (statistics)]]
{{colend}}

== Notes ==
{{NoteFoot}}

==References==
===Citations===
{{Reflist}}

===Sources===
{{refbegin}}
* {{cite book | title = Theory of Point Estimation | author1 = E.L. Lehmann | author2 = G. Casella | publisher=Springer|date=1998| isbn = 0387985026 | name-list-style = amp}}
* {{cite book | title = Systems Cost Engineering | author = Dale Shermon | date = 2009 | publisher = Gower Publishing | isbn = 978-0-566-08861-2}}
* {{cite book | title = Mathematical Statistics and Data Analysis | author = John Rice | date = 1995 | publisher = Duxbury Press | isbn = 0-534-209343}}
* {{cite book | title = Fundamentals of Statistical Signal Processing: Estimation Theory | author = Steven M. Kay | date = 1993 | publisher = PTR Prentice-Hall | isbn = 0-13-345711-7}}
* {{cite book | title = An Introduction to Signal Detection and Estimation | author = H. Vincent Poor | date = 1998 | publisher = Springer | isbn = 0-387-94173-8}}
* {{cite book | title = Detection, Estimation, and Modulation Theory, Part 1 | author = Harry L. Van Trees | date = 2001 | publisher = Wiley | isbn = 0-471-09517-6 | url = http://gunston.gmu.edu/demt/demtp1/ | archive-url = https://web.archive.org/web/20050428233957/http://gunston.gmu.edu/demt/demtp1/ | archive-date = 2005-04-28 }}
* {{cite book | title = Optimal State Estimation: Kalman, H-infinity, and Nonlinear Approaches | publisher=Wiley|date=2006| author = Dan Simon | url = http://academic.csuohio.edu/simond/estimation/ | archive-url = https://web.archive.org/web/20101230101133/http://academic.csuohio.edu/simond/estimation/ | archive-date=2010-12-30 }}
* {{cite book | author = Ali H. Sayed |author-link = Ali H. Sayed | title = Fundamentals of Adaptive Filtering | publisher = Wiley | location = NJ | year = 2003 | isbn = 0-471-46126-1}}
* {{Cite book
  | author1 = Yaakov Bar-Shalom
  | author1-link = Yaakov Bar-Shalom
  | author2 = X. Rong Li
  | author3 = Thiagalingam Kirubarajan
  | title = Estimation with Applications to Tracking and Navigation: Theory Algorithms and Software
  | publisher = Wiley
  | year = 2004
}}
{{refend}}

==External links==
*{{Commons category-inline}}

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[[Category:Estimation theory| ]]
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[[Category:Mathematical and quantitative methods (economics)]]