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".[1] 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 the statistic, as to which of those probability distributions is the sampling distribution.
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 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.[3]
^Fisher, R.A. (1922). "On the mathematical foundations of theoretical statistics". Philosophical Transactions of the Royal Society A. 222 (594–604): 309–368. Bibcode:1922RSPTA.222..309F. doi:10.1098/rsta.1922.0009. hdl:2440/15172. JFM 48.1280.02. JSTOR 91208.
^Dodge, Y. (2003) — entry for linear sufficiency
^Stigler, Stephen (December 1973). "Studies in the History of Probability and Statistics. XXXII: Laplace, Fisher and the Discovery of the Concept of Sufficiency". Biometrika. 60 (3): 439–445. doi:10.1093/biomet/60.3.439. JSTOR 2334992. MR 0326872.
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