Geodesic
On asymptotically optimal tests for composite hypotheses under non-standard conditions
Teoriâ veroâtnostej i ee primeneniâ, Tome 21 (1976) no. 1, pp. 34-47

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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. 
@article{TVP_1976_21_1_a2,
     author = {A. V. Bern\v{s}teǐn},
     title = {On asymptotically optimal tests for composite hypotheses under non-standard conditions},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {34--47},
     publisher = {mathdoc},
     volume = {21},
     number = {1},
     year = {1976},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_1976_21_1_a2/}
}
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A. V. Bernšteǐn. On asymptotically optimal tests for composite hypotheses under non-standard conditions. Teoriâ veroâtnostej i ee primeneniâ, Tome 21 (1976) no. 1, pp. 34-47. http://geodesic.mathdoc.fr/item/TVP_1976_21_1_a2/