Hajek projection explained

T on a set of independent random vectors X _{1,...,X} _{n} is a particular measurable function of X _{1,...,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,...,X} _{n} , the Hájek projection \hat{T} of T onto \{X _{1,...,X} _{n\}} is given by^[1] \hat{T} = \operatorname{E}(T) + \sum ^{n} _{i =1} \left[ \operatorname{E}(T\mid X _{i)} - \operatorname{E}(T)\right] = \sum ^{n} _{i =1} \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 \sum ^{n} _{i =1} g _{i(X} _{i)} , where g ^{d} _{i: R} \to R are arbitrary measurable functions such that \operatorname{E}(g ^{2 (X} _{i))< inf ty} for all i =1 , ... ,n \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,...,X} _{n)} and the sequence of its Hájek projections \hat{T} _{n} = \hat{T} _{n(X} _{1,...,X} _{n)} coincide, namely, if \operatorname{Var}(T _{n)/\operatorname{Var}(\hat{T}} _{n)} \to 1 , then (T _{n-\operatorname{E})/((T) _{n)}{\sqrt{\operatorname{Var}(T} _{n)}}} - (\hat{T)/(_{n-\operatorname{E}(\hat{T}}) _{n)}{\sqrt{\operatorname{Var}(\hat{T}} _{n)}}} converges to zero in probability.

Notes and References

1. Book: Vaart, Aad W. van der (1959-....).. Asymptotic statistics. 2012. Cambridge University Press. 9780511802256. 928629884.