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

$\hat{T}=\mathrm{E}(T)+\sum _{i=1}^n[\mathrm{E}(T\mid X_i)-\mathrm{E}(T)]=\sum _{i=1}^n\mathrm{E}(T\mid X_i)-(n-1)\mathrm{E}(T)$

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 $\mathrm{E}(g_i^2(X_i))<\mathrm{\infty }$ for all $i=1,\dots ,n$
• $\mathrm{E}(\hat{T}\mid X_i)=\mathrm{E}(T\mid X_i)$ and hence $\mathrm{E}(\hat{T})=\mathrm{E}(T)$
• 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 $\mathrm{Var}(T_n)/\mathrm{Var}({\hat{T}}_n)\to 1$, then $\frac{T_n-\mathrm{E}(T_n)}{\sqrt{\mathrm{Var}(T_n)}}-\frac{{\hat{T}}_n-\mathrm{E}({\hat{T}}_n)}{\sqrt{\mathrm{Var}({\hat{T}}_n)}}$ converges to zero in probability.