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Principle Seminar of the Department of Probability Theory, Moscow State University
February 17, 2016 16:45–17:45, Moscow, MSU, auditorium 12-24
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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
$$D = \inf_{\tau\geqslant0}\sup_{\theta\geqslant0} \mathop{\mathrm{ess\,sup}}_{\omega}\mathrm{E}^{\theta}\left((\tau-\theta)^{+}|\mathcal{F}_{\theta}\right)(\omega),$$
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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