In statistics, Hájek projection of a random variable T {\displaystyle T} on a set of independent random vectors X 1 , … , X n {\displaystyle X_{1},\dots ,X_{n}} is a particular measurable function of X 1 , … , X n {\displaystyle X_{1},\dots ,X_{n}} that, loosely speaking, captures the variation of T {\displaystyle T} in an optimal way. It is named after the Czech statistician Jaroslav Hájek .
Definition Given a random variable T {\displaystyle T} and a set of independent random vectors X 1 , … , X n {\displaystyle X_{1},\dots ,X_{n}} , the Hájek projection T ^ {\displaystyle {\hat {T}}} of T {\displaystyle T} onto { X 1 , … , X n } {\displaystyle \{X_{1},\dots ,X_{n}\}} is given by
T ^ = E ( T ) + ∑ i = 1 n [ E ( T ∣ X i ) − E ( T ) ] = ∑ i = 1 n E ( T ∣ X i ) − ( n − 1 ) E ( T ) {\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 T ^ {\displaystyle {\hat {T}}} is an L 2 {\displaystyle L^{2}} projection of T {\displaystyle T} onto a linear subspace of all random variables of the form ∑ i = 1 n g i ( X i ) {\displaystyle \sum _{i=1}^{n}g_{i}(X_{i})} , where g i : R d → R {\displaystyle g_{i}:\mathbb {R} ^{d}\to \mathbb {R} } are arbitrary measurable functions such that E ( g i 2 ( X i ) ) < ∞ {\displaystyle \operatorname {E} (g_{i}^{2}(X_{i}))<\infty } for all i = 1 , … , n {\displaystyle i=1,\dots ,n}
E ( T ^ ∣ X i ) = E ( T ∣ X i ) {\displaystyle \operatorname {E} ({\hat {T}}\mid X_{i})=\operatorname {E} (T\mid X_{i})} and hence E ( T ^ ) = E ( T ) {\displaystyle \operatorname {E} ({\hat {T}})=\operatorname {E} (T)}
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