**MSC:**- 62F05 Asymptotic properties of tests
- 62F12 Asymptotic properties of estimators

according to $Q_{\vartheta_1}$ and $Q_{\vartheta_2}$ with

unknown parameters $\vartheta_1,\, \vartheta_2\in \cal H$

and let $\pi_3$ be a third population whose distribution

$Q_{\vartheta_3}$ coincides with that of either $\pi_1$ or

$\pi_2$. Assume there are given samples of size $n_1=n_2$

and $n_3$, it is to be decided whether

$Q_{\vartheta_3}=Q_{\vartheta_1}$ or

$Q_{\vartheta_3}=Q_{\vartheta_2}$.\par

The decision problem is described by a sequence of localized

experiments. The corresponding lower Hajek-LeCam-bound is

established and there is constructed a sequence of decision

rules based on ML-estimators which attains this bound and

is therefore considered to be asymptotic optimal.