Geodesic
Testing of two simple hypotheses in the presence of delayed observations
Teoriâ veroâtnostej i ee primeneniâ, Tome 24 (1979) no. 3, pp. 463-474 
T. P. Mirošničenko. Testing of two simple hypotheses in the presence of delayed observations. Teoriâ veroâtnostej i ee primeneniâ, Tome 24 (1979) no. 3, pp. 463-474. http://geodesic.mathdoc.fr/item/TVP_1979_24_3_a1/
@article{TVP_1979_24_3_a1,
     author = {T. P. Miro\v{s}ni\v{c}enko},
     title = {Testing of two simple hypotheses in the presence of delayed observations},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {463--474},
     year = {1979},
     volume = {24},
     number = {3},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_1979_24_3_a1/}
}
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Voir la notice de l'article provenant de la source Math-Net.Ru

We consider the problem of testing hypotheses $H_0\colon \theta=0$ and $H_1\colon \theta=1$ for the process $\xi_t=\theta t+w_t$, $\xi_0=0$ where $w_t$ is a Wiener process, by means of a stopping rule $\delta=(\tau,d)\colon \tau$ is a Markov moment, $d$ ($d=0$ or $d=1$) is a decision function depending on the behaviour of $\xi_t$ on $[0,\tau+m]$. In a some class of rules $\Delta(\alpha,\beta)$ we find a rule $\delta\in\Delta(\alpha,\beta)$ which minimizes the functional $$ \lambda\mathbf M_0\tau+(1-\lambda)\mathbf M_1\tau $$ for a fixed $\lambda\in[0,1]$ (here $\mathbf M_i$ is the expectation corresponding $H_i$).