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Hajek projection

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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

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Given a random variable and a set of independent random vectors , the Hájek projection of onto is given by[1]

Properties

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  • 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.

References

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  1. ^ Vaart, Aad W. van der (1959-....). (2012). Asymptotic statistics. Cambridge University Press. ISBN 9780511802256. OCLC 928629884.{{cite book}}: CS1 maint: numeric names: authors list (link)