Sufficient statistic

Builds on Statistical model Statistic Conditional expectation

Definition

Let $\set{P_\theta : \theta \in \Theta}$ be a statistical model on a measurable space $(S, \mathcal{S})$ , and let $X$ be the identity random element of $S$ . Let $T = T(X)$ be a statistic with values in a measurable space. The statistic $T$ is sufficient for $\theta$ if the following holds for every $A \in \mathcal{S}$ . There is a version of the conditional probability $P_\theta(X \in A \mid T)$ that is the same for all $\theta \in \Theta$ . That is, there is a function $k_A(T)$ with $k_A(T) = P_\theta(X \in A \mid T)$ $P_\theta$ -almost surely, for every $\theta$ , as a conditional expectation of $\mathbf{1}\set{X \in A}$ given $T$ .

Remarks

Example: let $X = (X_1, \dots, X_n)$ be i.i.d. Bernoulli( $p$ ), $p \in (0, 1)$ , and $T = X_1 + \dots + X_n$ . For $x \in \set{0, 1}^n$ with $x_1 + \dots + x_n = t$ , one has $P_p(X = x \mid T = t) = p^t(1 - p)^{n - t} / \left(\binom{n}{t} p^t (1 - p)^{n - t}\right) = 1/\binom{n}{t}$ , which does not depend on $p$ ; so $T$ is sufficient. Non-example: for $n \ge 2$ , the statistic $X_1$ is not sufficient, since given $X_1$ the variable $X_2$ is still Bernoulli( $p$ ), and its law depends on $p$ . The whole sample $X$ is always sufficient, and any one-to-one measurable function of a sufficient statistic with measurable inverse is sufficient.

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The rationals form a field statement
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The rationals form an ordered field statement
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Random variable with a density definition
Identically distributed random variables definition
Expected value definition
Mutual independence of events definition
Conditional expectation definition
Independent sigma-algebras definition
Independence of random variables definition
Independent and identically distributed random variables definition
Statistic definition
Sufficient statistic definition

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3 pages build on this one.

Tag	`QNMR5F525C`
URI	`m2://QNMR5F525C`
Kind	definition
Writing	W0 stub
MSC	62B05
Defines	Sufficient statistic
Uses	Statistical model, Statistic, Conditional expectation

Sufficient statistic

Definition

Remarks

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