Suppose that at any stage of a statistical experiment a control variable $X$ that affects the distribution of the observed data $Y$ at this stage can be used. The distribution of $Y$ depends on some unknown parameter $\theta$, and we consider the problem of testing multiple hypotheses $H_1:\theta ={\theta }_1$, $H_2:\theta ={\theta }_2,...$, $H_k:\theta ={\theta }_k$ allowing the data to be controlled by $X$, in the following sequential context. The experiment starts with assigning a value $X_1$ to the control variable and observing $Y_1$ as a response. After some analysis, another...

Suppose that the process $X=\{X_n,n\in \mathbb{N}\}$ is observed sequentially. There are two random moments of time ${\theta }_1$ and ${\theta }_2$, independent of X, and X is a Markov process given ${\theta }_1$ and ${\theta }_2$. The transition probabilities of X change for the first time at time ${\theta }_1$ and for the second time at time ${\theta }_2$. Our objective is to find a strategy which immediately detects the distribution changes with maximal probability based on observation of X. The corresponding problem of double optimal stopping is constructed. The optimal strategy...

Following Kombarov we say that $X$ is $p$-sequential, for $p\in {\alpha }^{*}$, if for every non-closed subset $A$ of $X$ there is $f\in {}^{\alpha }X$ such that $f\left(\alpha \right)\subseteq A$ and $\overline{f}\left(p\right)\in X\setminus A$. This suggests the following definition due to Comfort and Savchenko, independently: $X$ is a FU($p$)-space if for every $A\subseteq X$ and every $x\in A^{-}$ there is a function $f\in {}^{\alpha }A$ such that $\overline{f}\left(p\right)=x$. It is not hard to see that $p\le {}_{RK}q$ ($\le {}_{RK}$ denotes the Rudin–Keisler order) $\iff$ every $p$-sequential space is $q$-sequential $\iff$ every FU($p$)-space is a FU($q$)-space. We generalize the spaces $S_n$ to construct examples of...

We consider the optimal stopping problem for a discrete-time Markov process on a Borel state space $X$. It is supposed that an unknown transition probability $p(·|x)$, $x\in X$, is approximated by the transition probability $\tilde{p}(·|x)$, $x\in X$, and the stopping rule ${\tilde{\tau }}_{*}$, optimal for $\tilde{p}$, is applied to the process governed by $p$. We found an upper bound for the difference between the total expected cost, resulting when applying ${\tilde{\tau }}_{*}$, and the minimal total expected cost. The bound given is a constant times ${\displaystyle {sup}_{x\in X}\parallel p(·|x)-\tilde{p}(·|x)\parallel }$, where $\parallel ·\parallel$ is the...