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 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. Bayesian estimators are admissible, by Wald's theorem.

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

See also

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