Hajek projection
In statistics, Hájek projection of a random variable on a set of independent random vectors is a particular measurable function of that, loosely speaking, captures the variation of in an optimal way. It is named after the Czech statistician Jaroslav Hájek.
Definition
Given a random variable and a set of independent random vectors, the Hájek projection of onto is given byProperties
- Hájek projection is an projection of onto a linear subspace of all random variables of the form, where are arbitrary measurable functions such that for all
- and hence
- Under some conditions, asymptotic distributions of the sequence of statistics and the sequence of its Hájek projections coincide, namely, if, then converges to zero in probability.