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Principle Seminar of the Department of Probability Theory, Moscow State University
February 17, 2016 16:45, Moscow, MSU, auditorium 12-24
 


CUSUM-statistics and its optimality in Lorden's criterion

A. N. Shiryaevab

a Steklov Mathematical Institute of Russian Academy of Sciences, Moscow
b Lomonosov Moscow State University, Faculty of Mechanics and Mathematics

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Abstract: If $\mathsf{P}^{\theta}$, $\theta\in[0,\infty]$ is a family of probability measures, which are locally absolutely continuous in the measure $\mathsf{P}^{\infty}$, then the quantity $\frac{d\mathsf{P}^{\theta}}{d\mathsf{P}^{\infty}}$, the likelihood ratio, is well known. The value of
$$\gamma_t = \sup_{\theta\leqslant t}\frac{d\mathsf{P}^{\theta}}{d\mathsf{P}^{\infty}}(0,t) $$
is called the CUSUM-statistics (CUSUM = cumulative sum). For the disorder problem Lorden proposed the following criterion of optimality
$$D = \inf_{\tau\geqslant0}\sup_{\theta\geqslant0} \mathop{\mathrm{ess sup}}_{\omega}\mathrm{E}^{\theta}((\tau-\theta)^{+}|\mathcal{F}_{\theta})(\omega),$$
where $\tau$ is stopping time.
In the talk it will be discussed how, for this criterion, (in the case of the Brownian motion whose drift is changed at the moment $\theta$) the CUSUM-optimality is proved.

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