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Minimax Optimality in Robust Detection of a Disorder Time in Poisson Rate

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  • Nicole El Karoui

    () (LPMA - Laboratoire de Probabilités et Modèles Aléatoires - UPMC - Université Pierre et Marie Curie - Paris 6 - UPD7 - Université Paris Diderot - Paris 7 - CNRS - Centre National de la Recherche Scientifique)

  • Stéphane Loisel

    () (SAF - Laboratoire de Sciences Actuarielle et Financière - UCBL - Université Claude Bernard Lyon 1)

  • Yahia Salhi

    () (SAF - Laboratoire de Sciences Actuarielle et Financière - UCBL - Université Claude Bernard Lyon 1)

Abstract

We consider the minimax quickest detection problem of an unobservable time of change in the rate of an inhomogeneous Poisson process. We seek a stopping rule that minimizes the robust Lorden criterion, formulated in terms of the number of events until detection, both for the worst-case delay and the false alarm constraint. In the Wiener case, such a problem has been solved using the so-called cumulative sums (cusum) strategy by Shiryaev [33, 35], or Moustakides [24] among others. In our setting, we derive the exact optimality of the cusum stopping rule by using finite variation calculus and elementary martingale properties to characterize the performance functions of the cusum stopping rule in terms of scale functions. These are solutions of some delayed differential equations that we solve elementarily. The case of detecting a decrease in the intensity is easy to study because the performance functions are continuous. In the case of an increase where the performance functions are not continuous, martingale properties require using a discontinuous local time. Nevertheless, from an identity relating the scale functions, the optimality of the cusum rule still holds. Finally, some numerical illustration are provided.

Suggested Citation

  • Nicole El Karoui & Stéphane Loisel & Yahia Salhi, 2015. "Minimax Optimality in Robust Detection of a Disorder Time in Poisson Rate," Working Papers hal-01149749, HAL.
  • Handle: RePEc:hal:wpaper:hal-01149749 Note: View the original document on HAL open archive server: https://hal.archives-ouvertes.fr/hal-01149749
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    References listed on IDEAS

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    1. Nick Hanley & Louis Dupuy & Eoin McLaughlin, 2015. "Genuine Savings And Sustainability," Journal of Economic Surveys, Wiley Blackwell, vol. 29(4), pages 779-806, September.
    2. Greasley, David & Hanley, Nick & Kunnas, Jan & McLaughlin, Eoin & Oxley, Les & Warde, Paul, 2014. "Testing genuine savings as a forward-looking indicator of future well-being over the (very) long-run," Journal of Environmental Economics and Management, Elsevier, vol. 67(2), pages 171-188.
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    Keywords

    Change-Point; Robust Sequential Detection; Poisson Process;

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