# Estimation theory

**Estimation theory**is a branch ofstatistics andsignal processing that deals with estimating the values of parameters based on measured/empirical data. The parameters describe an underlying physical setting in such a way that the value of the parameters affects the distribution of the measured data. Anestimator attempts to approximate the unknown parameters using the measurements.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.

**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 theCramér-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 sample s taken from arandom vector (RV) of size "N". Put into a vector,: $mathbf\{x\}\; =\; egin\{bmatrix\}\; x\; [0]\; \backslash \; x\; [1]\; \backslash \; vdots\; \backslash \; x\; [N-1]\; end\{bmatrix\}.$

Secondly, we have the corresponding "M" parameters

: $mathbf\{\; heta\}\; =\; egin\{bmatrix\}\; heta\_1\; \backslash \; heta\_2\; \backslash \; vdots\; \backslash \; heta\_M\; end\{bmatrix\},$

which need to be established with their

probability density function (pdf) orprobability mass function (pmf): $p(mathbf\{x\}\; |\; mathbf\{\; heta\}).,$

It is also possible for the parameters themselves to have a probability distribution (e.g.,

Bayesian statistics ). It is then necessary to define theepistemic probability : $pi(\; mathbf\{\; heta\}).,$

After the model is formed, the goal is to estimate the parameters, commonly denoted $hat\{mathbf\{\; heta$, 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: $mathbf\{e\}\; =\; hat\{mathbf\{\; heta\; -\; mathbf\{\; heta\}$

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

**Estimators**Commonly-used estimators, and topics related to them:

*Maximum likelihood estimators

*Bayes estimator s

*Method of moments estimators

*Cramér-Rao bound

*Minimum mean squared error (MMSE), also known as Bayes least squared error (BLSE)

*Maximum a posteriori (MAP)

*Minimum variance unbiased estimator (MVUE)

*Best linear unbiased estimator (BLUE)

*Unbiased estimators — seeestimator bias .

*Particle filter

*Markov chain Monte Carlo (MCMC)

*Kalman filter

*Ensemble Kalman filter (EnKF)

*Wiener filter **Example: DC gain in white Gaussian noise**Consider a received

discrete signal , $x\; [n]$, of $N$ independent samples that consists of a DC gain $A$ withadditive white Gaussian noise $w\; [n]$ with knownvariance $sigma^2$ ("i.e.", $mathcal\{N\}(0,\; sigma^2)$).Since the variance is known then the only unknown parameter is $A$.The model for the signal is then: $x\; [n]\; =\; A\; +\; w\; [n]\; quad\; n=0,\; 1,\; dots,\; N-1$

Two possible (of many) estimators are:

* $hat\{A\}\_1\; =\; x\; [0]$

* $hat\{A\}\_2\; =\; frac\{1\}\{N\}\; sum\_\{n=0\}^\{N-1\}\; x\; [n]$ which is thesample mean Both of these estimators have a

mean of $A$, which can be shown through taking theexpected value of each estimator:$mathrm\{E\}left\; [hat\{A\}\_1\; ight]\; =\; mathrm\{E\}left\; [\; x\; [0]\; ight]\; =\; A$and:$mathrm\{E\}left\; [\; hat\{A\}\_2\; ight]\; =mathrm\{E\}left\; [\; frac\{1\}\{N\}\; sum\_\{n=0\}^\{N-1\}\; x\; [n]\; ight]\; =frac\{1\}\{N\}\; left\; [\; sum\_\{n=0\}^\{N-1\}\; mathrm\{E\}left\; [\; x\; [n]\; ight]\; ight]\; =frac\{1\}\{N\}\; left\; [\; N\; A\; ight]\; =A$

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

:$mathrm\{var\}\; left(\; hat\{A\}\_1\; ight)\; =\; mathrm\{var\}\; left(\; x\; [0]\; ight)\; =\; sigma^2$and:$mathrm\{var\}\; left(\; hat\{A\}\_2\; ight)=mathrm\{var\}\; left(\; frac\{1\}\{N\}\; sum\_\{n=0\}^\{N-1\}\; x\; [n]\; ight)overset\{independence\}\{=\}frac\{1\}\{N^2\}\; left\; [\; sum\_\{n=0\}^\{N-1\}\; mathrm\{var\}\; (x\; [n]\; )\; ight]\; =frac\{1\}\{N^2\}\; left\; [\; N\; sigma^2\; ight]\; =frac\{sigma^2\}\{N\}$

It would seem that the sample mean is a better estimator since, as $N\; o\; infty$, the variance goes to zero.

**Maximum likelihood**Continuing the example using the

maximum likelihood estimator, theprobability density function (pdf) of the noise for one sample $w\; [n]$ is:$p(w\; [n]\; )\; =\; frac\{1\}\{sigma\; sqrt\{2\; pi\; expleft(-\; frac\{1\}\{2\; sigma^2\}\; w\; [n]\; ^2\; ight)$

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

:$p(x\; [n]\; ;\; A)\; =\; frac\{1\}\{sigma\; sqrt\{2\; pi\; expleft(-\; frac\{1\}\{2\; sigma^2\}\; (x\; [n]\; -\; A)^2\; ight)$

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

:$p(mathbf\{x\};\; A)=prod\_\{n=0\}^\{N-1\}\; p(x\; [n]\; ;\; A)=frac\{1\}\{left(sigma\; sqrt\{2pi\}\; ight)^N\}expleft(-\; frac\{1\}\{2\; sigma^2\}\; sum\_\{n=0\}^\{N-1\}(x\; [n]\; -\; A)^2\; ight)$

Taking the

natural logarithm of the pdf:$ln\; p(mathbf\{x\};\; A)=-N\; ln\; left(sigma\; sqrt\{2pi\}\; ight)-\; frac\{1\}\{2\; sigma^2\}\; sum\_\{n=0\}^\{N-1\}(x\; [n]\; -\; A)^2$

and the maximum likelihood estimator is

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

Taking the first

derivative of the log-likelihood function:$frac\{partial\}\{partial\; A\}\; ln\; p(mathbf\{x\};\; A)=frac\{1\}\{sigma^2\}\; left\; [\; sum\_\{n=0\}^\{N-1\}(x\; [n]\; -\; A)\; ight]\; =frac\{1\}\{sigma^2\}\; left\; [\; sum\_\{n=0\}^\{N-1\}x\; [n]\; -\; N\; A\; ight]$

and setting it to zero

:$0=frac\{1\}\{sigma^2\}\; left\; [\; sum\_\{n=0\}^\{N-1\}x\; [n]\; -\; N\; A\; ight]\; =sum\_\{n=0\}^\{N-1\}x\; [n]\; -\; N\; A$

This results in the maximum likelihood estimator

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

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 AWGN with a fixed, unknown DC gain.

**Cramér–Rao lower bound**To find the

Cramér-Rao lower bound (CRLB) of the sample mean estimator, it is first necessary to find theFisher information number:$mathcal\{I\}(A)=mathrm\{E\}left(\; left\; [\; frac\{partial\}\{partial\; heta\}\; ln\; p(mathbf\{x\};\; A)\; ight]\; ^2\; ight)=-mathrm\{E\}left\; [\; frac\{partial^2\}\{partial\; heta^2\}\; ln\; p(mathbf\{x\};\; A)\; ight]$

and copying from above

:$frac\{partial\}\{partial\; A\}\; ln\; p(mathbf\{x\};\; A)=frac\{1\}\{sigma^2\}\; left\; [\; sum\_\{n=0\}^\{N-1\}x\; [n]\; -\; N\; A\; ight]$

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$-mathrm\{E\}left\; [\; frac\{partial^2\}\{partial\; A^2\}\; ln\; p(mathbf\{x\};\; A)\; ight]\; =frac\{N\}\{sigma^2\}$

Finally, putting the Fisher information into

:$mathrm\{var\}left(\; hat\{A\}\; ight)geqfrac\{1\}\{mathcal\{I$

results in

:$mathrm\{var\}left(\; hat\{A\}\; ight)geqfrac\{sigma^2\}\{N\}$

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 $A$.The sample mean is the

minimum variance unbiased estimator (MVUE) in addition to being themaximum likelihood estimator.**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):

* Interpretation of scientific

experiment s

*Signal processing

*Clinical trial s

*Opinion poll s

*Quality control

*Telecommunication s

*Project management

*Software engineering

*Control theory

*Network intrusion detection system 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 as possible.**ee also**

*Best linear unbiased estimator (BLUE)

*Chebyshev center

*Completeness (statistics)

*Cramér-Rao bound

*Detection theory

*Efficiency (statistics)

*Estimator ,Estimator bias

*Expectation-maximization algorithm (EM algorithm)

*Information theory

*Kalman filter

*Least-squares spectral analysis

*Markov chain Monte Carlo (MCMC)

*Matched filter

*Maximum a posteriori (MAP)

*Maximum likelihood

*Maximum entropy spectral estimation

* Method of moments,generalized method of moments

*Minimum mean squared error (MMSE)

*Minimum variance unbiased estimator (MVUE)

*Nuisance variable

*Parametric equation

*Particle filter

*Rao-Blackwell theorem

*Spectral density ,Spectral density estimation

*Statistical signal processing

*Sufficiency (statistics)

*Wiener filter **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; [*http://gunston.gmu.edu/demt/demtp1/ website*] )

* "Optimal State Estimation: Kalman, H-infinity, and Nonlinear Approaches" by Dan Simon [*http://academic.csuohio.edu/simond/estimation/ website*]

*Wikimedia Foundation.
2010.*

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