Английская Википедия:Hajek projection

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In statistics, Hájek projection of a random variable <math>T</math> on a set of independent random vectors <math>X_1,\dots,X_n</math> is a particular measurable function of <math>X_1,\dots,X_n</math> that, loosely speaking, captures the variation of <math>T</math> in an optimal way. It is named after the Czech statistician Jaroslav Hájek .

Definition

Given a random variable <math>T</math> and a set of independent random vectors <math>X_1,\dots,X_n</math>, the Hájek projection <math>\hat{T}</math> of <math>T</math> onto <math>\{X_1,\dots,X_n\}</math> is given by

<math>\hat{T} = \operatorname{E}(T) + \sum_{i=1}^n \left[ \operatorname{E}(T\mid X_i) - \operatorname{E}(T)\right] =

\sum_{i=1}^n \operatorname{E}(T\mid X_i) - (n-1)\operatorname{E}(T)</math>

Properties

• Hájek projection <math>\hat{T}</math> is an <math>L^2</math>projection of <math>T</math> onto a linear subspace of all random variables of the form <math>\sum_{i=1}^n g_i(X_i)</math>, where <math>g_i:\mathbb{R}^d \to \mathbb{R} </math> are arbitrary measurable functions such that <math>\operatorname{E}(g_i^2(X_i))<\infty </math> for all <math>i=1,\dots,n</math>
• <math>\operatorname{E} (\hat{T}\mid X_i)=\operatorname{E}(T\mid X_i)</math> and hence <math>\operatorname{E}(\hat{T})=\operatorname{E}(T)</math>
• Under some conditions, asymptotic distributions of the sequence of statistics <math>T_n=T_n(X_1,\dots,X_n)</math> and the sequence of its Hájek projections <math>\hat{T}_n = \hat{T}_n(X_1,\dots,X_n)</math> coincide, namely, if <math>\operatorname{Var}(T_n)/\operatorname{Var}(\hat{T}_n) \to 1</math>, then <math>\frac{T_n-\operatorname{E}(T_n)}{\sqrt{\operatorname{Var}(T_n)}} - \frac{\hat{T}_n-\operatorname{E}(\hat{T}_n)}{\sqrt{\operatorname{Var}(\hat{T}_n)}}</math> converges to zero in probability.