Hajek Projection

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 $T$ 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 $X_1,\dots,X_n$, the Hájek projection $\hat$ of $T$ onto $\$ is given by

: \hat = \operatorname(T) + \sum_^n \left \operatorname(T\mid X_i) - \operatorname(T)\right=
\sum_^n \operatorname(T\mid X_i) - (n-1)\operatorname(T)

Properties

* Hájek projection $\hat$ is an $L^2$projection of $T$ onto a linear subspace of all random variables of the form $\sum_^n g_i(X_i)$, where $g_i:\mathbb^d \to \mathbb$ are arbitrary measurable functions such that $\operatorname(g_i^2(X_i))<\infty$ for all $i=1,\dots,n$
* $\operatorname (\hat\mid X_i)=\operatorname(T\mid X_i)$ and hence $\operatorname(\hat)=\operatorname(T)$
* 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 $\hat_n = \hat_n(X_1,\dots,X_n)$ coincide, namely, if $\operatorname(T_n)/\operatorname(\hat_n) \to 1$, then $\frac - \frac$ converges to zero in probability.

References

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Asymptotic analysis
Multivariate statistics
Probability theory