Point estimation

In statistics, point estimation involves the use of sample data to calculate a single value (known as a statistic) which is to serve as a "best guess" or "best estimate" of an unknown (fixed or random) population parameter.

More formally, it is the application of a point estimator to the data.

In general, point estimation should 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

• 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 (ML)
• method of moments, 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:

• Kalman filter
• Wiener filter

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

• particle filter
• Markov chain Monte Carlo (MCMC)

Properties of point estimates

• bias of an estimator
• Cramér–Rao bound

See also

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

Notes

1. 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.
2. 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.
3. 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.
4. Ferguson, Thomas S (1996). A course in large sample theory. Chapman & Hall. ISBN 0-412-04371-8.
5. Le Cam, Lucien (1986). Asymptotic methods in statistical decision theory. Springer-Verlag. ISBN 0-387-96307-3.
6. 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.
7. 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.