Estimation theory

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. The first is a statistical sample – a set of data points taken from a random vector (RV) of size N. Put into a vector,

$$\backslash mathbf\{x\}\; =\; \backslash begin\{bmatrix\}\; x[0]\; \backslash \backslash \; x[1]\; \backslash \backslash \; \backslash vdots\; \backslash \backslash \; x[N-1]\; \backslash end\{bmatrix\}.$$

Secondly, there are M parameters

$$\backslash boldsymbol\{\backslash theta\}\; =\; \backslash begin\{bmatrix\}\; \backslash theta\_1\; \backslash \backslash \; \backslash theta\_2\; \backslash \backslash \; \backslash vdots\; \backslash \backslash \; \backslash theta\_M\; \backslash end\{bmatrix\},$$

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:

$$p(\backslash mathbf\{x\}\; |\; \backslash boldsymbol\{\backslash theta\}).\backslash ,$$

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

$$\backslash pi(\; \backslash boldsymbol\{\backslash theta\}).\backslash ,$$

After the model is formed, the goal is to estimate the parameters, with the estimates commonly denoted $\backslash hat\{\backslash boldsymbol\{\backslash theta\}\}$, 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

$$\backslash mathbf\{e\}\; =\; \backslash hat\{\backslash boldsymbol\{\backslash theta\}\}\; -\; \backslash boldsymbol\{\backslash theta\}$$

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.

{{main|Estimator}}

Commonly used estimators (estimation methods) and topics related to them include:

• Maximum likelihood estimators
• Bayes estimators
• 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 — see estimator bias.
• Particle filter
• Markov chain Monte Carlo (MCMC)
• Kalman filter, and its various derivatives
• Wiener filter

Consider a received discrete signal, $x[n]$, of $N$ independent samples that consists of an unknown constant $A$ with additive white Gaussian noise (AWGN) $w[n]$ with zero mean and known variance $\backslash sigma^2$ (i.e., $\backslash mathcal\{N\}(0,\; \backslash sigma^2)$).

Since the variance is known then the only unknown parameter is $A$.

The model for the signal is then

$$x[n]\; =\; A\; +\; w[n]\; \backslash quad\; n=0,\; 1,\; \backslash dots,\; N-1$$

Two possible (of many) estimators for the parameter $A$ are:

• $\backslash hat\{A\}\_1\; =\; x[0]$
• $\backslash hat\{A\}\_2\; =\; \backslash frac\{1\}\{N\}\; \backslash sum\_\{n=0\}^\{N-1\}\; x[n]$ which is the sample mean

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

$$\backslash mathrm\{E\}\backslash left[\backslash hat\{A\}\_1\backslash right]\; =\; \backslash mathrm\{E\}\backslash left[\; x[0]\; \backslash right]\; =\; A$$

and

\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

At this point, these two estimators would appear to perform the same.

However, the difference between them becomes apparent when comparing the variances.

$$\backslash mathrm\{var\}\; \backslash left(\; \backslash hat\{A\}\_1\; \backslash right)\; =\; \backslash mathrm\{var\}\; \backslash left(\; x[0]\; \backslash right)\; =\; \backslash sigma^2$$

and

\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}

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

{{main|Maximum likelihood}}

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

$$p(w[n])\; =\; \backslash frac\{1\}\{\backslash sigma\; \backslash sqrt\{2\; \backslash pi\}\}\; \backslash exp\backslash left(-\; \backslash frac\{1\}\{2\; \backslash sigma^2\}\; w[n]^2\; \backslash right)$$

and the probability of $x[n]$ becomes ($x[n]$ can be thought of a $\backslash mathcal\{N\}(A,\; \backslash sigma^2)$)

$$p(x[n];\; A)\; =\; \backslash frac\{1\}\{\backslash sigma\; \backslash sqrt\{2\; \backslash pi\}\}\; \backslash exp\backslash left(-\; \backslash frac\{1\}\{2\; \backslash sigma^2\}\; (x[n]\; -\; A)^2\; \backslash right)$$

By independence, the probability of $\backslash mathbf\{x\}$ becomes

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)

Taking the natural logarithm of the pdf

\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

and the maximum likelihood estimator is

$$\backslash hat\{A\}\; =\; \backslash arg\; \backslash max\; \backslash ln\; p(\backslash mathbf\{x\};\; A)$$

Taking the first derivative of the log-likelihood function

\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]

and setting it to zero

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

This results in the maximum likelihood estimator

$$\backslash hat\{A\}\; =\; \backslash frac\{1\}\{N\}\; \backslash sum\_\{n=0\}^\{N-1\}x[n]$$

which is simply the sample mean.

From this example, it was found that the sample mean is the maximum likelihood estimator for $N$ samples of a fixed, unknown parameter corrupted by AWGN.

{{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

\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]

and copying from above

\frac{\partial}{\partial A} \ln p(\mathbf{x}; A)

=

\frac{1}{\sigma^2} \left[ \sum_{n=0}^{N-1}x[n] - N A \right]

Taking the second derivative

\frac{\partial^2}{\partial A^2} \ln p(\mathbf{x}; A)

=

\frac{1}{\sigma^2} (- N)

=

\frac{-N}{\sigma^2}

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

-\mathrm{E}

\left[

\frac{\partial^2}{\partial A^2} \ln p(\mathbf{x}; A)

\right]

=

\frac{N}{\sigma^2}

Finally, putting the Fisher information into

\mathrm{var}\left( \hat{A} \right)

\geq

\frac{1}{\mathcal{I}}

results in

\mathrm{var}\left( \hat{A} \right)

\geq

\frac{\sigma^2}{N}

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 $N$ and $A$.

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.

{{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 $1,2,\backslash dots,N$ with unknown maximum, the UMVU estimator for the maximum is given by

$$\backslash frac\{k+1\}\{k\}\; m\; -\; 1\; =\; m\; +\; \backslash frac\{m\}\{k\}\; -\; 1$$

where m is the sample maximum and k is the sample size, sampling without replacement.{{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 }}{{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 }} 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

$$\backslash frac\{1\}\{k\}\backslash frac\{(N-k)(N+1)\}\{(k+2)\}\; \backslash approx\; \backslash frac\{N^2\}\{k^2\}\; \backslash text\{\; for\; small\; samples\; \}\; k\; \backslash ll\; N$$

so a standard deviation of approximately $N/k$, the (population) average size of a gap between samples; compare $\backslash frac\{m\}\{k\}$ 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.

Numerous fields require the use of estimation theory.

Some of these fields include:

• Interpretation of scientific experiments
• Signal processing
• Clinical trials
• Opinion polls
• Quality control
• Telecommunications
• 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 or uncertainty and it is through statistical probability that optimal solutions are sought to extract as much information from the data as possible.

{{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}}

{{NoteFoot}}

{{Reflist}}

{{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}}

• {{Commons category-inline}}

{{DSP}}

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Category:Signal processing

Category:Mathematical and quantitative methods (economics)