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**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.

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 |