Locally most powerful sequential tests of a simple hypothesis vs. One-sided alternatives for independent observations

Let $X_1,X_2,\mathrm{.}\mathrm{.}\mathrm{.},X_n,\mathrm{.}\mathrm{.}\mathrm{.}$ be a stochastic process with independent values whose distribution $P_{\theta }$ depends on an unknown parameter $\theta$, $\theta \in \mathrm{\Theta }$, where $\mathrm{\Theta }$ is an open subset of the real line. The problem of testing $H_0:$ $\theta ={\theta }_0$ vs. a composite alternative $H_1:$ $\theta >{\theta }_0$ is considered, where ${\theta }_0\in \mathrm{\Theta }$ is a fixed value of the parameter. The main objective of this work is the characterization of the structure of the locally most powerful (in the sense of Berk) sequential tests in this problem.