A sufficient statistic is a function T(X) of a sample X such that the conditional distribution of X given T(X) carries no information about the unknown parameter θ — all inferential content about θ is contained in T alone. The identifying mechanism is the Neyman–Fisher factorization theorem: T is sufficient for θ iff the joint density f(x|θ) factors as g(T(x), θ) · h(x), where g depends on data only through T and h is independent of θ. This property enables data reduction without loss of information, underpins the construction of uniformly most powerful tests via the Karlin–Rudin theorem, and guarantees that any estimator not using T can be Rao–Blackwell improved by conditioning on T. The concept was introduced by R. A. Fisher (1922) and formalized by Neyman and Fisher (1933). [formal: statisticum-sufficiens | substrate: mind | horizon: a life | explicit: yes | epoch: 0.01]
Accepted ontology entry
sufficient-statistic
A sufficient statistic is a function T(X) of a sample X such that the conditional distribution of X given T(X) carries no information about the unknown parameter θ — all inferential content about θ is contained in T alone. The identifying…
Definition
Why it is in scope
A human-made mathematical concept in statistical inference: a function of sample data that captures all information about an unknown parameter, eliminating the need to retain the full sample for estimation. Built to persist in statistical theory, hypothesis testing, and data reduction practice.
Names and aliases
- sufficient-statisticen · CANONICAL
Relations from this entry
- likelihoodDEPENDS_ON →
A sufficient statistic T(X) is defined by the factorization P(X|theta) = g(T(X),theta)·h(X) — the likelihood function. Remove likelihood and the concept of sufficiency has no mechanism to operate on. Present-tense operational necessity: sufficient statistics extract information from likelihood, not from data alone.
- statistical-modelDEPENDS_ON →
A sufficient statistic captures all information in the data about a parameter of the model. Remove the statistical model (the parameterized family and its structure) and sufficient statistics have no parameter space to be sufficient for — the concept ceases to operate.
- cmrwglr5a0045soact3r2g3ouSERVES →
A sufficient statistic is constructed for the sake of data reduction without loss of information for parameter estimation. It exists to serve estimation by summarizing the sample into a minimal statistic that retains all inferential content about θ, enabling Rao-Blackwell improvement and uniformly most powerful tests. Servant points at master estimation.
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Record identity
- Created
- Sep 3, 2026, 6:41 PM UTC
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- be56fc2c01141dbf8755820d6c8dec219ea8f694fffa527fd71ca7dc8c20e326