The Annals of Mathematical Statistics

On a Quickest Detection Problem

J. A. Bather

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In a recent paper A. N. Shiryaev [1] discusses the problem of detecting the arrival of a "disorder" in an observed stochastic process, as quickly as possible subject to a limitation on the number of false alarms. He considers two versions of a simple model. In the first, the disorder arrives at a discrete instant $\theta$ according to a geometric distribution. The process disturbed by this event consists of a sequence of independent observations $\{\xi_t\}$, such that $\xi_1, \xi_2, \cdots, \xi_{\theta - 1}$ arise from a certain distribution $F_0$, whereas $\xi_\theta, \xi_{\theta + 1}, \cdots$ come from a different distribution $F_1$. In the continuous time version of the model, the a-priori distribution of $\theta$ is exponential: $P(\theta > t) = e^{-\lambda t} \quad (t \geqq 0)$ and the disorder is represented by a change in the mean drift of an observed Wiener process $\{\eta(t)\}$. More precisely, for any given value of $\theta$, this process has independent normal increments $\delta\eta = \eta(t + \delta t) - \eta(t)$, with \begin{align*}E (\delta\eta) = 0 \quad (0 \leqq t < \theta), \notag \\ E(\delta\eta) = \delta t \qquad (t \geqq \theta), \notag \\ \operatorname{Var} (\delta\eta) = \delta t \quad (t \geqq 0)\end{align*}. In both versions it may be decided at any instant $t$ to carry out a detailed inspection in order to ascertain whether or not the disturbance has occurred. Then, if it is found that $\theta < t$ the process terminates but observation must be resumed immediately after a false alarm. Within these rules it is required to find a decision procedure which determines the instants at which a thorough inspection is worthwhile. Assuming that $N$, the expected number of false alarms, is specified in advance Shiryaev establishes the general form of policy which minimizes $\tau$, the expected delay in verifying the arrival of the disorder. The a posteriori distribution of $\theta$ at any time, does not depend on anything which took place before the last false alarm. For example, in continuous time $p(t) = P(\theta \leqq t \mid \eta(t'), 0 \leqq t' \leqq t) = P(\theta \leqq t \mid \eta(t'), s \leqq t' \leqq t),$ where $s$ is the instant of the most recent false alarm. The geometric and exponential distributions have the useful property that $P(\theta > t + s \mid \theta > s) = P(\theta > t \mid \theta > 0).$ He deduces that the optimal policy for the period following any false alarm must correspond exactly with the procedure applied initially, before the first inspection. In addition, he proves the existence of a critical level $p^\ast = p^\ast(N)$ such that, in general, observations should continue so long as $0 \leqq p(t) < p^\ast$ with an immediate inspection as soon as $p(t) \geqq p^\ast$. These and other similar results are established first for the discrete time model and then extended to the continuous time version. For the latter, the paper also gives more explicit calculations: the evaluation of $\tau$ in terms of $p^\ast$, for example. But no attempt is made to determine the critical level $p^\ast(N)$ for the optimal policy. In fact, as we shall see, a very simple relation holds: $p^\ast (N) = (N + 1)^{-1}.$ However, the aim here is to show how the optimal policy can be found for a more realistic specification of the minimization problem, involving given delay and inspection costs. We shall concentrate entirely on the continuous time model and suppose that each inspection incurs an instantaneous cost $K > 0$, not depending on its outcome, whereas any delay in detecting the arrival of the disorder leads to a cost $c > 0$ per unit time. Hence the total expected cost is $K(N + 1) + c\tau$, which depends both on the decision procedure and on the initial condition $p(0) = 0$. The minimization will be based on the calculation of the minimum expected future cost $f_\ast(\varphi)$, as a function of the current state $\varphi(t) = p(t)/(1 - p(t))$, by solving a certain differential equation with special boundary conditions. A heuristic argument, in which one simply assumes that $f_\ast(\varphi)$ is suitably differentiable, can be given without much difficulty. But, strictly speaking, it is not clear that risk functions such as $f_\ast(\operatorname{var})$, each of whose values is defined as the infimum of a class of expectations, are sufficiently well behaved. This difficulty is often encountered in statistical applications of dynamic programming to processes in continuous time. Typically, it is extremely difficult to establish the required differentiability properties directly and it is necessary to seek an indirect justification by means of existence and uniqueness theorems. In our case the formal solution can be produced explicitly and, because of this, its justification is much easier. Nevertheless, the approach is complicated by the need to establish several preliminary results, and the discussion of these special properties is limited to a brief indication of the main steps, in the hope that the essential structure of the argument will be more generally useful. Section 2 is concerned with the information process $\{\varphi(t)\}$. Its relation to the observed process $\{\eta(t)\}$ is described and certain properties of its increments are collected for later use. The main argument begins in Section 3 with a discussion of sub-optimal decision procedures defined by specifying an open continuation set $\mathscr{C}$ within the space $\lbrack0, \infty)$ of possible "initial" states $\varphi$. It is shown that any risk function $f(\varphi)$ can be determined for each sub-interval of the corresponding set $\mathscr{C}$ by solving the basic differential equation appropriately. Sections 4 and 5 consider the special solution $f_\ast(\varphi)$ which represents the optimal decision procedure, and give the required verification that $f_\ast(\varphi)$ is uniformly minimal. The final section contains a brief analysis of the operating characteristics of the optimal policy and indicates the importance of evaluating the particular minimum risk $f_\ast(0)$.

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Ann. Math. Statist., Volume 38, Number 3 (1967), 711-724.

First available in Project Euclid: 27 April 2007

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Bather, J. A. On a Quickest Detection Problem. Ann. Math. Statist. 38 (1967), no. 3, 711--724. doi:10.1214/aoms/1177698864.

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