# Estimator

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In statistics, an estimator is a rule for calculating an estimate of a given quantity based on observed data: thus the rule and its result (the estimate) are distinguished. This article discusses estimators and estimates that are point estimators; that is, they yield single-valued results, although this includes the possibility of single vector-valued results and results that can be expressed as a single function. This is in contrast to an interval estimator, where the result would be a range of plausible values (or vectors or functions).

Statistical 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.

## Contents

### Background

An "estimator" or "point estimate" is a statistic (that is, a measurable function of the data) that is used to infer the value of an unknown parameter in a statistical model. The parameter being estimated is sometimes called the estimand.[citation needed] It can be either finite-dimensional (in parametric and semi-parametric models), or infinite-dimensional (semi-nonparametric and non-parametric models).[citation needed] If the parameter is denoted θ then the estimator is typically written by adding a “hat” over the symbol: $\scriptstyle\hat\theta$. 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”. The attractiveness of different estimators can be judged by looking at their properties, such as unbiasedness, mean square error, consistency, asymptotic distribution, etc.. The construction and comparison of estimators are the subjects of the estimation theory. In the context of decision theory, an estimator is a type of decision rule, and its performance may be evaluated through the use of loss functions.

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