On the estimation theory of the scale parameter

The problem of estimating of the scale parameter $\sigma$ based on independent samples from the population with d.f. $F(x/\sigma)$ is considered. Under the condition (10) the following results are proved (Theorems 1 and 2). The estimator $c^\circ_n\bar x$ of $\sigma$ is admissible in the class of all estimators (or $\alpha_1^{-1}\bar x$ is admissible in the class of unbiased estimators) for any two values $n=n_1$, $n=n_2$, $n_2>n_1\ge3$, of the sample size if and only if $F(x)$ is either the degenerate d.f. or the d.f. of the Gamma-distribution.