In statistics, point estimation involves the use of sample data to calculate a single value (known as a point estimate since it identifies a point in some parameter space) 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. More generally, a point estimator can be contrasted with a set estimator. Examples are given by confidence sets or credible sets. A point estimator can also be contrasted with a distribution estimator. Examples are given by confidence distributions, randomized estimators, and Bayesian posteriors.
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
- 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.
- Posterior median, which minimizes the posterior risk for the absolute-value loss function, as observed by Laplace.
- 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 filters are important:
Properties of point estimates
- Algorithmic inference
- Induction (philosophy)
- Philosophy of statistics
- Predictive inference
- Range estimation
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