# User:O18/Estimation

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## Estimation theory

Estimation theory is a branch of statistics and signal 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. An estimator 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 the Cramér-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 samples taken from a random vector (RV) of size N. Put into a vector,

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

Secondly, we have the corresponding M parameters

${\displaystyle \mathbf {\theta } ={\begin{bmatrix}\theta _{1}\\\theta _{2}\\\vdots \\\theta _{M}\end{bmatrix}},}$

which need to be established with their probability density function (pdf) or probability mass function (pmf)

${\displaystyle p(\mathbf {x} |\mathbf {\theta } ).\,}$

It is also possible for the parameters themselves to have a probability distribution (e.g., Bayesian statistics). It is then necessary to define the epistemic probability

${\displaystyle \pi (\mathbf {\theta } ).\,}$

After the model is formed, the goal is to estimate the parameters, commonly denoted ${\displaystyle {\hat {\mathbf {\theta } }}}$, 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

${\displaystyle \mathbf {e} ={\hat {\mathbf {\theta } }}-\mathbf {\theta } }$

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:

### Example: DC gain in white Gaussian noise

Consider a received discrete signal, ${\displaystyle x[n]}$, of ${\displaystyle N}$ independent samples that consists of a DC gain ${\displaystyle A}$ with Additive white Gaussian noise ${\displaystyle w[n]}$ with known variance ${\displaystyle \sigma ^{2}}$ (i.e., ${\displaystyle {\mathcal {N}}(0,\sigma ^{2})}$). Since the variance is known then the only unknown parameter is ${\displaystyle A}$.

The model for the signal is then

${\displaystyle x[n]=A+w[n]\quad n=0,1,\dots ,N-1}$

Two possible (of many) estimators are:

• ${\displaystyle {\hat {A}}_{1}=x[0]}$
• ${\displaystyle {\hat {A}}_{2}={\frac {1}{N}}\sum _{n=0}^{N-1}x[n]}$ which is the sample mean

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

${\displaystyle \mathrm {E} \left[{\hat {A}}_{1}\right]=\mathrm {E} \left[x[0]\right]=A}$

and

${\displaystyle \mathrm {E} \left[{\hat {A}}_{2}\right]=\mathrm {E} \left[{\frac {1}{N}}\sum _{n=0}^{N-1}x[n]\right]={\frac {1}{N}}\left[\sum _{n=0}^{N-1}\mathrm {E} \left[x[n]\right]\right]={\frac {1}{N}}\left[NA\right]=A}$

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

${\displaystyle \mathrm {var} \left({\hat {A}}_{1}\right)=\mathrm {var} \left(x[0]\right)=\sigma ^{2}}$

and

${\displaystyle \mathrm {var} \left({\hat {A}}_{2}\right)=\mathrm {var} \left({\frac {1}{N}}\sum _{n=0}^{N-1}x[n]\right){\overset {independence}{=}}{\frac {1}{N^{2}}}\left[\sum _{n=0}^{N-1}\mathrm {var} (x[n])\right]={\frac {1}{N^{2}}}\left[N\sigma ^{2}\right]={\frac {\sigma ^{2}}{N}}}$

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

#### Maximum likelihood

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

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

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

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

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

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

Taking the natural logarithm of the pdf

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

and the maximum likelihood estimator is

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

Taking the first derivative of the log-likelihood function

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

and setting it to zero

${\displaystyle 0={\frac {1}{\sigma ^{2}}}\left[\sum _{n=0}^{N-1}x[n]-NA\right]=\sum _{n=0}^{N-1}x[n]-NA}$

This results in the maximum likelihood estimator

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

which is simply the sample mean. From this example, it was found that the sample mean is the maximum likelihood estimator for ${\displaystyle N}$ 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 the Fisher information number

${\displaystyle {\mathcal {I}}(A)=\mathrm {E} \left(\left[{\frac {\partial }{\partial \theta }}\ln p(\mathbf {x} ;A)\right]^{2}\right)=-\mathrm {E} \left[{\frac {\partial ^{2}}{\partial \theta ^{2}}}\ln p(\mathbf {x} ;A)\right]}$

and copying from above

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

Taking the second derivative

${\displaystyle {\frac {\partial ^{2}}{\partial A^{2}}}\ln p(\mathbf {x} ;A)={\frac {1}{\sigma ^{2}}}(-N)={\frac {-N}{\sigma ^{2}}}}$

and finding the negative expected value is trivial since it is now a deterministic constant ${\displaystyle -\mathrm {E} \left[{\frac {\partial ^{2}}{\partial A^{2}}}\ln p(\mathbf {x} ;A)\right]={\frac {N}{\sigma ^{2}}}}$

Finally, putting the Fisher information into

${\displaystyle \mathrm {var} \left({\hat {A}}\right)\geq {\frac {1}{\mathcal {I}}}}$

results in

${\displaystyle \mathrm {var} \left({\hat {A}}\right)\geq {\frac {\sigma ^{2}}{N}}}$

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 ${\displaystyle N}$ and ${\displaystyle A}$. The sample mean is the minimum variance unbiased estimator (MVUE) in addition to being the maximum 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):

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.

### 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; website)
• Optimal State Estimation: Kalman, H-infinity, and Nonlinear Approaches by Dan Simon website

## Estimation

Estimation is the calculated approximation of a result which is usable even if input data may be incomplete, uncertain, or noisy.

In statistics, see estimation theory, estimator.

In mathematics, approximation or estimation typically means finding upper or lower bounds of a quantity that cannot readily be computed precisely and is also an educated guess . While initial results may be unusably uncertain, recursive input from output, can purify results to be approximately accurate, certain, complete and noise-free.

## Estimator

In statistics, an estimator is a function of the observable sample data that is used to estimate an unknown population parameter (which is called the estimand); an estimate is the result from the actual application of the function to a particular sample of data. Many different estimators are possible for any given parameter. Some criterion is used to choose between the estimators, although it is often the case that a criterion cannot be used to clearly pick one estimator over another. To estimate a parameter of interest (e.g., a population mean, a binomial proportion, a difference between two population means, or a ratio of two population standard deviation), the usual procedure is as follows:

1. Select a random sample from the population of interest.
2. Calculate the point estimate of the parameter.
3. Calculate a measure of its variability, often a confidence interval.
4. Associate with this estimate a measure of variability.

There are two types of estimators: point estimators and interval estimators.

### Point estimators

Suppose we have a fixed parameter ${\displaystyle \theta \ }$ that we wish to estimate. Then an estimator is a function that maps a sample design to a set of sample estimates. An estimator of ${\displaystyle \theta \ }$ is usually denoted by the symbol ${\displaystyle {\widehat {\theta }}}$. A sample design can be thought of as an ordered pair ${\displaystyle \ (S,p)}$ where ${\displaystyle \ S}$ is a set of samples (or outcomes), and ${\displaystyle \ p}$ is the probability density function. The probability density function maps the set ${\displaystyle \ S}$ to the closed interval [0,1], and has the property that the sum (or integral) of the values of ${\displaystyle \ p(s)}$, over all ${\displaystyle \ s}$ in ${\displaystyle \ S}$, is equal to 1. For any given subset ${\displaystyle \ A}$ of ${\displaystyle \ S}$, the sum or integral of ${\displaystyle \ p(s)}$ over all ${\displaystyle \ s}$ in ${\displaystyle \ A}$ is ${\displaystyle \Pr(A)\,}$.

For all the properties below, the value ${\displaystyle \theta \ }$, the estimation formula, the set of samples, and the set probabilities of the collection of samples, can be considered fixed. Yet since some of the definitions vary by sample (yet for the same set of samples and probabilities), we must use ${\displaystyle \ s}$ in the notation. Hence, the estimate for a given sample ${\displaystyle \ s}$ is denoted as ${\displaystyle {\widehat {\theta }}(s)}$.

We have the following definitions and attributes.

1. For a given sample ${\displaystyle s\ }$, the error of the estimator ${\displaystyle {\widehat {\theta }}}$ is defined as ${\displaystyle {\widehat {\theta }}(s)-\theta }$, where ${\displaystyle {\widehat {\theta }}(s)\ }$ is the estimate for sample ${\displaystyle \ s}$, and ${\displaystyle \theta \ }$ is the parameter being estimated. Note that the error depends not only on the estimator (the estimation formula or procedure), but on the sample.
2. The mean squared error of ${\displaystyle {\widehat {\theta }}}$ is defined as the expected value (probability-weighted average, over all samples) of the squared errors; that is, ${\displaystyle \operatorname {MSE} ({\widehat {\theta }})=\operatorname {E} [({\widehat {\theta }}-\theta )^{2}]}$. It is used to indicate how far, on average, the collection of estimates are from the single parameter being estimated. Consider the following analogy. Suppose the parameter is the bull's-eye of a target, the estimator is the process of shooting arrows at the target, and the individual arrows are estimates. Then high MSE means the average distance of the arrows from the target is high, and low MSE means the average distance from the target is low. The arrows may or may not be clustered. For example, even if all arrows hit the same point, yet grossly miss the target, the MSE is still relatively large. Note, however, that if the MSE is relatively low, then the arrows are likely more highly clustered (than highly dispersed).
3. For a given sample ${\displaystyle s\ }$, the sampling deviation of the estimator ${\displaystyle {\widehat {\theta }}}$ is defined as ${\displaystyle {\widehat {\theta }}(s)-\operatorname {E} ({\widehat {\theta }})}$, where ${\displaystyle {\widehat {\theta }}(s)\ }$ is the estimate for sample ${\displaystyle \ s}$, and ${\displaystyle \operatorname {E} ({\widehat {\theta }})}$ is the expected value of the estimator. Note that the sampling deviation depends not only on the estimator, but on the sample.
4. The variance of ${\displaystyle {\widehat {\theta }}}$ is simply the expected value of the squared sampling deviations; that is, ${\displaystyle \operatorname {var} ({\widehat {\theta }})=\operatorname {E} [({\widehat {\theta }}-\operatorname {E} ({\widehat {\theta }}))^{2}]}$. It is used to indicate how far, on average, the collection of estimates are from the expected value of the estimates. Note the difference between MSE and variance. If the parameter is the bull's-eye of a target, and the arrows are estimates, then a relatively high variance means the arrows are dispersed, and a relatively low variance means the arrows are clustered. Some things to note: even if the variance is low, the cluster of arrows may still be far off-target, and even if the variance is high, the diffuse collection of arrows may still be unbiased. Finally, note that even if all arrows grossly miss the target, if they nevertheless all hit the same point, the variance is zero.
5. The bias of ${\displaystyle {\widehat {\theta }}}$ is defined as ${\displaystyle B({\widehat {\theta }})=\operatorname {E} ({\widehat {\theta }})-\theta }$. It is the distance between the average of the collection of estimates, and the single parameter being estimated. It also is the expected value of the error, since ${\displaystyle \operatorname {E} ({\widehat {\theta }})-\theta =\operatorname {E} ({\widehat {\theta }}-\theta )}$. If the parameter is the bull's-eye of a target, and the arrows are estimates, then a relatively high absolute value for the bias means the average position of the arrows is off-target, and a relatively low absolute bias means the average position of the arrows is on target. They may be dispersed, or may be clustered. The relationship between bias and variance is analogous to the relationship between accuracy and precision.
6. ${\displaystyle {\widehat {\theta }}}$ is an unbiased estimator of ${\displaystyle \theta \ }$ if and only if ${\displaystyle B({\widehat {\theta }})=0}$. Note that bias is a property of the estimator, not of the estimate. Often, people refer to a "biased estimate" or an "unbiased estimate," but they really are talking about an "estimate from a biased estimator," or an "estimate from an unbiased estimator." Also, people often confuse the "error" of a single estimate with the "bias" of an estimator. Just because the error for one estimate is large, does not mean the estimator is biased. In fact, even if all estimates have astronomical absolute values for their errors, if the expected value of the error is zero, the estimator is unbiased. Also, just because an estimator is biased, does not preclude the error of an estimate from being zero (we may have gotten lucky). The ideal situation, of course, is to have an unbiased estimator with low variance, and also try to limit the number of samples where the error is extreme (that is, have few outliers). Yet unbiasedness is not essential. Often, if we permit just a little bias, then we can find an estimator with lower MSE and/or fewer outlier sample estimates.
7. The MSE, variance, and bias, are related: ${\displaystyle \operatorname {MSE} ({\widehat {\theta }})=\operatorname {var} ({\widehat {\theta }})+(B({\widehat {\theta }}))^{2},}$
i.e. mean squared error = variance + square of bias.

The standard deviation of an estimator of θ (the square root of the variance), or an estimate of the standard deviation of an estimator of θ, is called the standard error of θ.

### Consistency

A consistent estimator is an estimator that converges in probability to the quantity being estimated as the sample size grows without bound.

An estimator ${\displaystyle t_{n}}$ (where n is the sample size) is a consistent estimator for parameter ${\displaystyle \theta }$ if and only if, for all ${\displaystyle \epsilon >0}$, no matter how small, we have

${\displaystyle \lim _{n\to \infty }\Pr \left\{\left|t_{n}-\theta \right|<\epsilon \right\}=1.}$

It is called strongly consistent, if it converges almost surely to the true value.

### Asymptotic normality

An asymptotically normal estimator is a consistent estimator whose distribution around the true parameter ${\displaystyle \theta }$ approaches a normal distribution with standard deviation shrinking in proportion to ${\displaystyle 1/{\sqrt {n}}}$ as the sample size ${\displaystyle n}$ grows. Using ${\displaystyle {\xrightarrow {D}}}$ to denote convergence in distribution, ${\displaystyle t_{n}}$ is asymptotically normal if

${\displaystyle {\sqrt {n}}(t_{n}-\theta ){\xrightarrow {D}}N(0,V),}$

for some ${\displaystyle V}$, which is called the asymptotic variance of the estimator.

The central limit theorem implies asymptotic normality of the sample mean ${\displaystyle {\bar {x}}}$ as an estimator of the true mean. More generally, maximum likelihood estimators are asymptotically normal under fairly weak regularity conditions — see the asymptotics section of the maximum likelihood article.

### Efficiency

Two naturally desirable properties of estimators are for them to be unbiased and have minimal mean squared error (MSE). These cannot in general both be satisfied simultaneously: a biased estimator may have lower mean squared error (MSE) than any unbiased estimator: despite having bias, the estimator variance may be sufficiently smaller than that of any unbiased estimator, and it may be preferable to use, despite the bias; see estimator bias.

Among unbiased estimators, there often exists one with the lowest variance, called the MVUE. In some cases an unbiased efficient estimator exists, which, in addition to having the lowest variance among unbiased estimators, satisfies the Cramér-Rao bound, which is an absolute lower bound on variance for statistics of a variable.

Concerning such "best unbiased estimators", see also Cramér-Rao bound, Gauss-Markov theorem, Lehmann-Scheffé theorem, Rao-Blackwell theorem.

### Other properties

Sometimes, estimators should satisfy further restrictions (restricted estimators) - eg, one might require an estimated probability to be between zero and one, or an estimated variance to be nonnegative. Sometimes this conflicts with the requirement of unbiasedness, see the example in estimator bias concerning the estimation of the exponent of minus twice lambda based on a sample of size one from the Poisson distribution with mean lambda.