하제크 투영법

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 ${\d$ 최적의 방법으로 $T$ $isplaystyle T}.$ 그것은 체코의 통계학자 야로슬라프 하제크의 이름을 따서 지어졌다.

정의

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^[1]

{\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)

특성.

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 ${\displaystyle g_$ ${i:\mathb {R} ^{d}\to \mathb {R}}$ 은(는) $\operatorname {E} (g_{i}^{2}(X_{i}))<\infty$ $\operatorname {E} (g_{i}^{2}(X_{i}))<\infty$ ( $\operatorname {E} (g_{i}^{2}(X_{i}))<\infty$ $\operatorname {E} (g_{i}^{2}(X_{i}))<\infty$ 2 $\operatorname {E} (g_{i}^{2}(X_{i}))<\infty$ ( $\operatorname {E} (g_{i}^{2}(X_{i}))<\infty$ $\operatorname {E} (g_{i}^{2}(X_{i}))<\infty$ ) < $\operatorname {E} (g_{i}^{2}(X_{i}))<\infty$ ${\displaysty \operatorname {E}(g_{i}^{2$ })( $X_{i})$ 와 같은 임의의 측정 가능한 함수들이다 $g_{i}:\mathbb {R} ^{d}\to \mathbb {R}$ . $i=1,\dots ,n$ $i=1,\dots ,n$ = $i=1,\dots ,n$ , $i=1,\dots ,n$ n $i=1,\displaystyle,n$ 에 대해 $\operatorname {E} (g_{i}^{2}(X_{i}))<\infty$ $<\infully }$
$\operatorname {E} ({\hat {T}}\mid X_{i})=\operatorname {E} (T\mid X_{i})$ 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 ${\displaystyle {\hat {T}}_{n}={\hat {T}}_{n}(X_{1},\dots ,X_{n})$ $}$ coincide, namely, if $\operatorname {Var} (T_{n})/\operatorname {Var} ({\hat {T}}_{n})\to 1$ , then ${\displaystyle {\frac {T_{n}-\operato$ $rname {E} (T_{n})}{\sqrt {\operatorname {Var} (T_{n})}}}-{\frac {{\hat {T}}_{n}-\operatorname {E} ({\hat {T}}_{n})}{\sqrt {\operatorname {Var} ({\hat {T}}_{n})}}}}$ converges to zero in probability.