# Point estimation

In statistics, **point estimation** involves the use of sample data to calculate a single value (known as a **point estimate** or statistic) which is to serve as a "best guess" or "best estimate" of an unknown population parameter (for example, the population mean). More formally, it is the application of a point estimator to the data to obtain a point estimate.

Point estimation can be contrasted with interval estimation: such interval estimates are typically either confidence intervals in the case of frequentist inference, or credible intervals in the case of Bayesian inference.

## Point estimators

There are a variety of point estimators, each with different properties.

- minimum-variance mean-unbiased estimator (MVUE), minimizes the risk (expected loss) of the squared-error loss-function.
- best linear unbiased estimator (BLUE)
- minimum mean squared error (MMSE)
- median-unbiased estimator, minimizes the risk of the absolute-error loss function
- maximum likelihood estimator (MLE)
- method of moments and generalized method of moments

## Bayesian point estimation

Bayesian inference is typically based on the posterior distribution. Many Bayesian point estimators are the posterior distribution's statistics of central tendency, e.g., its mean, median, or mode:

- Posterior mean, which minimizes the (posterior)
*risk*(expected loss) for a squared-error loss function; in Bayesian estimation, the risk is defined in terms of the posterior distribution, as observed by Gauss.^{[1]} - Posterior median, which minimizes the posterior risk for the absolute-value loss function, as observed by Laplace.
^{[2]}^{[3]} - maximum a posteriori (
*MAP*), which finds a maximum of the posterior distribution; for a uniform prior probability, the MAP estimator coincides with the maximum-likelihood estimator;

The MAP estimator has good asymptotic properties, even for many difficult problems, on which the maximum-likelihood estimator has difficulties.
For regular problems, where the maximum-likelihood estimator is consistent, the maximum-likelihood estimator ultimately agrees with the MAP estimator.^{[4]}^{[5]}^{[6]}
Bayesian estimators are admissible, by Wald's theorem.^{[5]}^{[7]}

The Minimum Message Length (MML) point estimator is based in Bayesian information theory and is not so directly related to the posterior distribution.

Special cases of Bayesian filters are important:

Several methods of computational statistics have close connections with Bayesian analysis:

## Properties of point estimates

## See also

- Predictive inference
- Induction (philosophy)
- Philosophy of statistics
- Algorithmic inference

## Notes

- ↑ Dodge, Yadolah, ed. (1987).
*Statistical data analysis based on the L1-norm and related methods: Papers from the First International Conference held at Neuchâtel, August 31–September 4, 1987*. Amsterdam: North-Holland Publishing Co. - ↑ Dodge, Yadolah, ed. (1987).
*Statistical data analysis based on the L1-norm and related methods: Papers from the First International Conference held at Neuchâtel, August 31–September 4, 1987*. Amsterdam: North-Holland Publishing Co. - ↑ Jaynes, E.T. (2007).
*Probability theory : the logic of science*(5. print. ed.). Cambridge [u.a.]: Cambridge Univ. Press. p. 172. ISBN 978-0-521-59271-0. - ↑ Ferguson, Thomas S (1996).
*A course in large sample theory*. Chapman & Hall. ISBN 0-412-04371-8. - 1 2 Le Cam, Lucien (1986).
*Asymptotic methods in statistical decision theory*. Springer-Verlag. ISBN 0-387-96307-3. - ↑ Ferguson, Thomas S. (1982). "An inconsistent maximum likelihood estimate".
*Journal of the American Statistical Association*.**77**(380): 831–834. doi:10.1080/01621459.1982.10477894. JSTOR 2287314. - ↑ Lehmann, E.L.; Casella, G. (1998).
*Theory of Point Estimation, 2nd ed*. Springer. ISBN 0-387-98502-6.

## Bibliography

- Bickel, Peter J. & Doksum, Kjell A. (2001).
*Mathematical Statistics: Basic and Selected Topics*.**I**(Second (updated printing 2007) ed.). Pearson Prentice-Hall. - Lehmann, Erich (1983).
*Theory of Point Estimation*. - Liese, Friedrich & Miescke, Klaus-J. (2008).
*Statistical Decision Theory: Estimation, Testing, and Selection*. Springer.