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 by[1]
![{\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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c6f12be78212a023ec6b066e22fe2aa53966bfeb)
Properties
- 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 ![{\displaystyle i=1,\dots ,n}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f3f269b2f3b2f87fec0168426652a5ea80b56112)
and hence ![{\displaystyle \operatorname {E} ({\hat {T}})=\operatorname {E} (T)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/418528ac184de89c0f3941864194dafd4841e309)
- 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.
References