Sufficient statistics

In statistics, a statistic is sufficient with respect to a statistical model and its associated unknown parameter if "no other statistic that can be calculated from the same sample provides any additional information as to the value of the parameter". In particular, a statistic is sufficient for a family of probability distributions if the sample from which it is calculated gives no additional information than does the statistic, as to which of those probability distributions is that of the population from which the sample was taken.

Roughly, given a set ${\displaystyle \mathbf {X} }$ of independent identically distributed data conditioned on an unknown parameter ${\displaystyle \theta }$, a sufficient statistic is a function ${\displaystyle T(\mathbf {X} )}$ whose value contains all the information needed to compute any estimate of the parameter (e.g. a maximum likelihood estimate). Due to the factorization theorem (see below), for a sufficient statistic ${\displaystyle T(\mathbf {X} )}$, the joint distribution can be written as ${\displaystyle p(\mathbf {X} )=h(\mathbf {X} )\,g(\theta ,T(\mathbf {X} ))\,}$. From this factorization, it can easily be seen that the maximum likelihood estimate of ${\displaystyle \theta }$ will interact with ${\displaystyle \mathbf {X} }$ only through ${\displaystyle T(\mathbf {X} )}$. Typically, the sufficient statistic is a simple function of the data, e.g. the sum of all the data points.

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