Statistics

=====Statistics, estimators and pivotal quantities=====
Consider [[Independent identically distributed|independent identically distributed (IID) random variables]] with a given [[probability distribution]]: standard [[statistical inference]] and [[estimation theory]] defines a [[random sample]] as the [[random vector]] given by the [[column vector]] of these IID variables.<ref name=Piazza>Piazza Elio, Probabilità e Statistica, Esculapio 2007</ref> The [[Statistical population|population]] being examined is described by a probability distribution that may have unknown parameters.

A statistic is a random variable that is a function of the random sample, but {{em|not a function of unknown parameters}}. The probability distribution of the statistic, though, may have unknown parameters. Consider now a function of the unknown parameter: an [[estimator]] is a statistic used to estimate such function. Commonly used estimators include [[sample mean]], unbiased [[sample variance]] and [[sample covariance]].

A random variable that is a function of the random sample and of the unknown parameter, but whose probability distribution ''does not depend on the unknown parameter'' is called a [[pivotal quantity]] or pivot. Widely used pivots include the [[z-score]], the [[Chi-squared distribution#Applications|chi square statistic]] and Student's [[Student's t-distribution#How the t-distribution arises|t-value]].

Between two estimators of a given parameter, the one with lower [[mean squared error]] is said to be more [[Efficient estimator|efficient]]. Furthermore, an estimator is said to be [[Unbiased estimator|unbiased]] if its [[expected value]] is equal to the [[true value]] of the unknown parameter being estimated, and asymptotically unbiased if its expected value converges at the [[Limit (mathematics)|limit]] to the true value of such parameter.

Other desirable properties for estimators include: [[UMVUE]] estimators that have the lowest variance for all possible values of the parameter to be estimated (this is usually an easier property to verify than efficiency) and [[consistent estimator]]s which [[converges in probability]] to the true value of such parameter.

This still leaves the question of how to obtain estimators in a given situation and carry the computation, several methods have been proposed: the [[method of moments (statistics)|method of moments]], the [[maximum likelihood]] method, the [[least squares]] method and the more recent method of [[estimating equations]].