On the detection of > of Wiener process

L. Yu. Vostrikova

Teoriâ veroâtnostej i ee primeneniâ, Tome 26 (1981) no. 2, pp. 362-368

Résumé

The drift of a multidimensional Wiener process equals to $\theta_0$ on a time interval $[0,t_0]$ and equals to $\theta_1$ on $(t_0,T]$, the values $\theta_0$, $\theta_1$ and $t_0$ are unknown. We assume that the condition $\alpha T\le t_0\le(1-\alpha)T$ holds where the number $\alpha\in(0,1/2)$. The maximum likelihood estimates of the unknown parameters $t_0/T$, $\theta_0$ and $\theta_1$ are given and their consistency is proved. We study also the test for checking the hypothesis $H_0\colon\theta_0=\theta_1$ against the alternative $H_1\colon\theta_0\ne\theta_1$ which is based on the likelihood function. An asymptotic expression for the probability of the error of the first kind is obtained.