Hajek projection

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

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

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

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