Hajek projection

In statistics, Hájek projection of a random variable $T$ on a set of independent random vectors $X_{1},\dots ,X_{n}$ is a particular measurable function of $X_{1},\dots ,X_{n}$ that, loosely speaking, captures the variation of $T$ in an optimal way. It is named after the Czech statistician Jaroslav Hájek .

Definition

Given a random variable $T$ and a set of independent random vectors $X_{1},\dots ,X_{n}$, the Hájek projection ${\hat {T}}$ of $T$ onto $\{X_{1},\dots ,X_{n}\}$ is given by

${\hat {T}}=\operatorname {E} (T)+\sum _{i=1}^{n}\left[\operatorname {E} (T\mid X_{i})-\operatorname {E} (T)\right]=\sum _{i=1}^{n}\operatorname {E} (T\mid X_{i})-(n-1)\operatorname {E} (T)$

Properties

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