Point estimation

It has been suggested that this article or section be merged into Estimator. (Discuss) Proposed since May 2010.

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 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.
• Posterior median, which minimizes the posterior risk for the absolute-value loss function.
• 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.^[1][2][3] Bayesian estimators are admissible, by Wald's theorem.^[4][2]

Special cases of Bayesian estimators 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

Notes

Bibliography

• Bickel, Peter J. and 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 and Miescke, Klaus-J. (2008). Statistical Decision Theory: Estimation, Testing, and Selection. Springer.