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Concept
Estimator
Summary
In statistics, an estimator is a rule for calculating an estimate of a given quantity based on observed data: thus the rule (the estimator), the quantity of interest (the estimand) and its result (the estimate) are distinguished. For example, the sample mean is a commonly used estimator of the population mean.
There are point and interval estimators. The point estimators yield single-valued results. This is in contrast to an interval estimator, where the result would be a range of plausible values. "Single value" does not necessarily mean "single number", but includes vector valued or function valued estimators.
Estimation theory is concerned with the properties of estimators; that is, with defining properties that can be used to compare different estimators (different rules for creating estimates) for the same quantity, based on the same data. Such properties can be used to determine the best rules to use under given circumstances. However, in robust statistics, statistical theory goes on to consider the balance between having good properties, if tightly defined assumptions hold, and having less good properties that hold under wider conditions.
An "estimator" or "point estimate" is a statistic (that is, a function of the data) that is used to infer the value of an unknown parameter in a statistical model. A common way of phrasing it is "the estimator is the method selected to obtain an estimate of an unknown parameter".
The parameter being estimated is sometimes called the estimand. It can be either finite-dimensional (in parametric and semi-parametric models), or infinite-dimensional (semi-parametric and non-parametric models). If the parameter is denoted then the estimator is traditionally written by adding a circumflex over the symbol: . Being a function of the data, the estimator is itself a random variable; a particular realization of this random variable is called the "estimate". Sometimes the words "estimator" and "estimate" are used interchangeably.
The definition places virtually no restrictions on which functions of the data can be called the "estimators".
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