Sufficient statistic

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. 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. 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. The Kolmogorov structure function deals with individual finite data; the related notion there is the algorithmic sufficient statistic. The concept is due to Sir Ronald Fisher in 1920. Stephen Stigler noted in 1973 that the concept of sufficiency had 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 remained very important in theoretical work. Roughly, given a set X {displaystyle mathbf {X} } of independent identically distributed data conditioned on an unknown parameter θ {displaystyle heta } , a sufficient statistic is a function T ( X ) {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 T ( X ) {displaystyle T(mathbf {X} )} , the probability density can be written as f X ( x ) = h ( x ) g ( θ , T ( x ) ) {displaystyle f_{mathbf {X} }(x)=h(x),g( heta ,T(x))} . From this factorization, it can easily be seen that the maximum likelihood estimate of θ {displaystyle heta } will interact with X {displaystyle mathbf {X} } only through T ( X ) {displaystyle 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 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 statistic t = T(X) is sufficient for underlying parameter θ precisely if the conditional probability distribution of the data X, given the statistic t = T(X), does not depend on the parameter θ. As an example, the sample mean is sufficient for the mean (μ) of a normal distribution with known variance. Once the sample mean is known, no further information about μ can be obtained from the sample itself. On the other hand, for an arbitrary distribution the median is not sufficient for the mean: even if the median of the sample is known, knowing the sample itself would provide further information about the population mean. For example, if the observations that are less than the median are only slightly less, but observations exceeding the median exceed it by a large amount, then this would have a bearing on one's inference about the population mean. Fisher's factorization theorem or factorization criterion provides a convenient characterization of a sufficient statistic. If the probability density function is ƒθ(x), then T is sufficient for θ if and only if nonnegative functions g and h can be found such that