Local sufficiency. (English) Zbl 0573.62026

Let \(Z_ 1,Z_ 2,..\). be independent with distribution depending on an unknown real parameter \(\vartheta\), and let X be a p-variate statistic depending on \(Z_ 1,...,Z_ n\). Write L(X,\(\vartheta)\) for the distribution of X when \(\vartheta\) is the parameter. Fix \(\vartheta_ 0\) and assume that for real t, \(L(X,\vartheta_ 0+tn^{-1/2})\) can be approximated by Edgeworth expansions with an error of order \(O(n^{- 1})\), i.e. \(L(X,\vartheta_ 0+tn^{-{1/2}})\) is - up to terms of order \(O(n^{-1})\)- determined by its first, 2nd, and third cumulants.
Consider a transformation of \(X=(X_ 1,...,X_ p)\) into \(Y=(Y^{(1)},...,Y^{(p)})\) of the following kind: \[ Y^{(r)}=n^{- 1}c^{(r)}+\sum_{i}X_ i(a_ i^{(r)}+n^{-1/2}b_ i^{(r)})+\sum_{i,j}X_ iX_ jd^{(r)}_{i,j} \] The constants can be chosen such that
(1) \(L(Y^{(2)},...,Y^{(p)},\vartheta_ 0)\) and \(L(Y^{(2)},...,Y^{(p)},\vartheta_ 0+tn^{-1/2})\) differ by terms of order \(O(n^{-1})\) only (ancillarity),
(2) \(Y^{(1)}\) and \((Y^{(2)},...,Y^{(p)})\) are stochastically independent up to terms of order \(O(n^{-1})\) when \(\vartheta_ 0\) is the parameter.
(Conditions (1) and (2) can be translated into algebraic equations in the constants of the transformation and the cumulants of \(L(X,\vartheta_ 0).)\)
Inference is based on \(Y^{(1)}\), and \(Y^{(1)}\) is called second-order locally sufficient, although \(Y^{(1)}\) and \((Y^{(2)},...,Y^{(p)})\) are not independent up to order \(O(n^{-1)})\) when the parameter is \(\vartheta_ 0+tn^{-1/2}\) (see formula (21)).
The statistic \(Y^{(1)}\) is - up to linear transformations of X - uniquely determined, hence the precise specification of the conditioning statistic is avoided. Explicit computations are provided in the case where the components of X are the log likelihood derivatives.
The same method can be applied to the case of multivariate \(\vartheta\). There, the uniqueness of the locally sufficient statistic is lost. The results in the univariate and multivariate case are illustrated by examples.
Reviewer: Ch.Hipp


62F12 Asymptotic properties of parametric estimators
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