

Principle Seminar of the Department of Probability Theory, Moscow State University
February 17, 2016 16:45, Moscow, MSU, auditorium 1224






CUSUMstatistics and its optimality in Lorden's criterion
A. N. Shiryaev^{ab} ^{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 CUSUMstatistics (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
CUSUMoptimality is proved.

