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Local sufficiency. (English) Zbl 0573.62026
Let $Z\sb 1,Z\sb 2,..$. be independent with distribution depending on an unknown real parameter $\vartheta$, and let X be a p-variate statistic depending on $Z\sb 1,...,Z\sb n$. Write L(X,$\vartheta)$ for the distribution of X when $\vartheta$ is the parameter. Fix $\vartheta\sb 0$ and assume that for real t, $L(X,\vartheta\sb 0+tn\sp{-1/2})$ can be approximated by Edgeworth expansions with an error of order $O(n\sp{- 1})$, i.e. $L(X,\vartheta\sb 0+tn\sp{-{1/2}})$ is - up to terms of order $O(n\sp{-1})$- determined by its first, 2nd, and third cumulants. Consider a transformation of $X=(X\sb 1,...,X\sb p)$ into $Y=(Y\sp{(1)},...,Y\sp{(p)})$ of the following kind: $$ Y\sp{(r)}=n\sp{- 1}c\sp{(r)}+\sum\sb{i}X\sb i(a\sb i\sp{(r)}+n\sp{-1/2}b\sb i\sp{(r)})+\sum\sb{i,j}X\sb iX\sb jd\sp{(r)}\sb{i,j} $$ The constants can be chosen such that (1) $L(Y\sp{(2)},...,Y\sp{(p)},\vartheta\sb 0)$ and $L(Y\sp{(2)},...,Y\sp{(p)},\vartheta\sb 0+tn\sp{-1/2})$ differ by terms of order $O(n\sp{-1})$ only (ancillarity), (2) $Y\sp{(1)}$ and $(Y\sp{(2)},...,Y\sp{(p)})$ are stochastically independent up to terms of order $O(n\sp{-1})$ when $\vartheta\sb 0$ is the parameter. (Conditions (1) and (2) can be translated into algebraic equations in the constants of the transformation and the cumulants of $L(X,\vartheta\sb 0).)$ Inference is based on $Y\sp{(1)}$, and $Y\sp{(1)}$ is called second-order locally sufficient, although $Y\sp{(1)}$ and $(Y\sp{(2)},...,Y\sp{(p)})$ are not independent up to order $O(n\sp{-1)})$ when the parameter is $\vartheta\sb 0+tn\sp{-1/2}$ (see formula (21)). The statistic $Y\sp{(1)}$ 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:
62F12Asymptotic properties of parametric estimators
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