Suppose that independent normally distributed random vectors $W^{n\times 1}$ and $T^{k\times 1}$ are observed with $E(W) = 0, E(T) = \mu, \operatorname{Cov} (W) = \sigma^2 I$ and $\operatorname{Cov} (T) = \sigma^2 I$. In this paper it is shown that each member of a certain class of estimators of $\mu + \eta\sigma$ for a given vector $\eta$ is inadmissible if loss is dimension-free quadratic loss. This class includes the best invariant estimator. The proof is carried out by exhibiting, for each member, $\hat{\theta}$, of the class, an estimator depending on $\hat{\theta}$ whose risk is uniformly smaller than that of $\hat{\theta}$.