The following version of the change-point problem is considered: Given a,b \((a<1/2<b)\), let \[ x^ N(n)=m+\xi (n)+h\cdot I\{n>[\theta N]\},\quad n=1,...,N, \] where \(m\in R^ 1\), \(\xi\) (n) are i.i.d. \(N(0,\sigma^ 2)\), \(h\neq 0\), and \(\theta\in [a,b]\). The problem consists in estimating \(\theta\). The procedure is as follows:
A family \(Z_ N(\delta)\), \(0\leq \delta \leq 1\), of statistics is constructed. If \(| Z_ N(\delta)| <c\), then the decision is that no change point has appeared; otherwise \({\hat \theta}{}_ N(\delta)\) is an estimator of \(\theta\). Let \(\alpha_{\delta}(N)=P_ 0\{| Z_ N(\delta)| <c\}\) (“error of the first kind”), \(\beta_{\delta}(N)=P_{\theta}\{| Z_ N(\delta)| <c\}\) (“error of the second kind”), and \[ \gamma (x,N)=P_{\theta}\{| {\hat \theta}_ N(\delta)-\theta | >x\}. \] Asymptotic (as \(N\to \infty)\) estimates for \(\alpha_{\delta}(N)\), \(\beta_{\delta}(N)\), and \(\gamma\) (x,N) are given. The asymptotically best values of \(\delta\) are explicitly stated.