## 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

### MSC:

 62F12 Asymptotic properties of parametric estimators
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