Sufficiency (statistics)

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In statistics, a sufficient statistic is a statistic which has the property of sufficiency with respect to a statistical model and its associated unknown parameter, meaning that "no other statistic which can be calculated from the same sample provides any additional information as to the value of the parameter".[1] 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.

In practical terms, given a set  \mathbf{X} of independent identically distributed data conditioned on an unknown parameter θ, a sufficient statistic is a function 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 T(\mathbf{X}), the joint distribution can be written as 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 θ will interact with \mathbf{X} only through T(\mathbf{X}). Typically, the sufficient statistic is a simple function of the data, e.g. the sum of all the data points.

More generally, the "unknown parameter" may represent a vector of unknown quantities or may represent everything about the model that is unknown or not fully specified. In such a case, the sufficient statistic may be a set of functions, called a jointly sufficient statistic. Typically, there are as many functions as there are parameters. For example, for a Gaussian distribution with unknown mean and variance, the jointly sufficient statistic, from which maximum likelihood estimates of both parameters can be estimated, consists of two functions, the sum of all data points and the sum of all squared data points (or equivalently, the sample mean and sample variance).

The concept, due to Sir Ronald Fisher, is equivalent to the statement that, conditional on the value of a sufficient statistic for a parameter, the joint probability distribution of the data does not depend on that parameter. Both the statistic and the underlying parameter can be vectors.

A related concept is that of linear sufficiency, which is weaker than sufficiency but can be applied in some cases where there is no sufficient statistic, although it is restricted to linear estimators.[2]

The concept of sufficiency has fallen out of favor in descriptive statistics because of the strong dependence on an assumption of the distributional form (see Pitman–Koopman–Darmois theorem below), but remains very important in theoretical work.[3]

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