### - Art Gallery -

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 } T onto $${\displaystyle \{X_{1},\dots ,X_{n}\}}$$is given by[1]

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

Properties

Hájek projection $$\hat{T}$$ is an $$L^{2}$$ projection of T onto a linear subspace of all random variables of the form $${\displaystyle \sum _{i=1}^{n}g_{i}(X_{i})}$$, where g i : R d → R {\displaystyle g_{i}:\mathbb {R} ^{d}\to \mathbb {R} } {\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 $$i=1,\dots ,nv$$
$${\displaystyle \operatorname {E} ({\hat {T}}\mid X_{i})=\operatorname {E} (T\mid X_{i})}$$ 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.

References

Vaart, Aad W. van der (1959-....). (2012). Asymptotic statistics. Cambridge University Press. ISBN 9780511802256. OCLC 928629884.