Equivariant estimation of a mean vector [mu] of N([mu], [Sigma]) with [mu]'[Sigma]-1[mu] = 1 or [Sigma]-1/2[mu] = c or [Sigma] = [sigma]2[mu]'[mu]l

This paper considers the problems of estimating a mean vector [mu] under constraint [mu]'[Sigma]-1[mu] = 1 or [Sigma]-1/2[mu] = c and derives the best equivariant estimators under the loss (a - [mu])' [Sigma]-1(a - [mu]), which dominate the MLE's uniformly. The results are regarded as multivariate extensions of those with known coefficient of variation in a univariate case. As a particular case for [mu]'[Sigma]-1[mu] = c, the case [Sigma] = [sigma]2[mu]'[mu]I is also treated.