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## Minimum-variance unbiased estimator - Wikiwand

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## Unbiased Estimators And Their Applications: Volume 1: Univariate Case 1993

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An estimator attempts to approximate the unknown parameters using the measurements. In estimation theory, two approaches are generally considered. 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.

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.

### Supplementary files

Put into a vector ,. 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:. It is also possible for the parameters themselves to have a probability distribution e. It is then necessary to define the Bayesian probability.

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. This error term is then squared and the expected value of this squared value is minimized for the MMSE estimator. At this point, these two estimators would appear to perform the same.

However, the difference between them becomes apparent when comparing the variances. Taking the natural logarithm of the pdf. Taking the first derivative of the log-likelihood function.

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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. 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.

This has a variance of [2].

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.