On asymptotically optimal tests for composite hypotheses under non-standard conditions

Let $X_1,\dots,X_n$ be independent identically distributed random variables from a distribution dependent on the parameters $\theta=(\theta_1,\dots,\theta_m)$ and $\xi$. The hypothesis $H_0\colon\xi=0$ is to be tested against the alternative $\xi>0$.
In [1], optimal asymptotic tests were obtained under the condition that the logarithmic derivatives of the density with respect to $\theta_r$, $r=1,\dots,m$, and $\xi$ at the point $\xi=0$ are linearly independent. In this paper, optimal asymptotic tests are constructed in the case when this condition is not satisfied. Also some results are obtained for the usual $C(\alpha)$-tests.