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


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
Full Text: DOI