## Invariantly sufficient equivariant statistics and characterizations of normality in translation classes.(English)Zbl 0533.62006

Consider $${\mathbb{R}}^ n$$ for $$n\geq 2$$. For each real a, let $$T_ a$$ be the translation by (a,a,...,a) and G the group $$(T_ a$$, $$a\in R)$$. A statistic $$S:{\mathbb{R}}^ n\to {\mathbb{R}}$$ is equivariant if $$S(T_ ax)=Sx+a$$. For a probability $$P_ 0$$ on $${\mathbb{R}}^ n$$ let W be the family $$(P_ a=PT_ a\!^{-1}:$$ $$a\in R)$$. Say that S is invariantly sufficient for the family W if for any G-invariant Borel set B there is a common version of $$E(l_ B| \sigma_ n(S))$$. $$X_ 1,...,X_ n$$ denote the coordinate variables on $${\mathbb{R}}^ n$$. The following are the main theorems of the paper:
Theorem 1: If $$P_ 0$$ is a product probability then $$a\leftrightarrow b$$ holds, where (a): There is an invariantly sufficient statistic of the form $$\sum c_ jX_ j$$ where each $$c_ j$$ as well as $$\sum c_ j$$ is non zero, (b) either $$P_ 0$$ is a point mass or is nondegenerate normal. In the degenerate case put $$U=1/n\sum X_ j$$ and in the normal case put $$U=(1/c)\sum c_ jX_ j$$ where $$c_ j=1/var(X_ j)$$ and $$c=\sum c_ j$$. Then the complete sufficient statistic U is the essentially unique invariantly sufficient equivariant statistic in $$L^ 2(P_ 0).$$
Theorem 2: Suppose $$A=(a_{ij})$$ is a regular $$n\times n$$ matrix with each column sum $$a_{.j}=\sum_{i}a_{ij}\neq 0$$. Put $$Z=XA$$. Assume that under $$P_ 0 Z_ 1...Z_ n$$ are independent with mean 0 and variances nonzero finite. Then the following are equivalent: (a) There exists one and only one invariantly sufficient unbiased linear statistic. (b) The statistic $$U=c^{-1}\sum c_ jX_ j$$ where $$c_ j=\sum(1/\sigma^ 2\!_ k)a_{jk}a_{.k}$$ and $$c=\sum c_ j$$, $$\sigma^ 2\!_ k=Var(Z_ k)$$ is sufficient. (c) $$P_ 0$$ is normal. Further, in case $$n\geq 3$$ the statistic U is admissible among all unbiased estimates for the family W iff $$P_ 0$$ is normal. Proofs are well presented and relations with earlier characterizations of normality are discussed.
Reviewer: B.V.Rao

### MSC:

 62B05 Sufficient statistics and fields 62A01 Foundations and philosophical topics in statistics 62E10 Characterization and structure theory of statistical distributions 62G05 Nonparametric estimation 62C15 Admissibility in statistical decision theory
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