Consider a process that produces a series of independent identically distributed vectors. A change in an underlying state may become manifest in a modification of one or more of the marginal distributions. Often, the dependence structure between coordinates is unknown, impeding surveillance based on the joint distribution. A popular approach is to construct control charts for each coordinate separately and raise an alarm the first time any (or some) of the control charts signals. The difficulty is obtaining an expression for the overall average run length to false alarm (ARL2FA). We argue that despite the dependence structure, when the process is in control, for large ARLs to false alarm, run lengths of many types of control charts run in parallel are asymptotically independent. Furthermore, often, in-control run lengths are asymptotically exponentially distributed, enabling uncomplicated asymptotic expressions for the ARL2FA. We prove this assertion for certain Cusum and Shiryaev–Roberts-type control charts and illustrate it by simulations.

中文翻译：

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经验法则：在相关流上并行运行的许多类型的控制图的错误警报的运行长度是渐近独立的

考虑一个产生一系列独立的相同分布的向量的过程。基本状态的变化可能会在边际分布的一个或多个更改中变得明显。通常，坐标之间的依存结构是未知的，从而妨碍了基于联合分布的监视。一种流行的方法是分别为每个坐标构建控制图，并在任何（或一些）控制图信号首次出现时发出警报。困难之处在于要获得针对误报警的总体平均运行时间的表达式（ARL2FA）。我们认为，尽管有依赖关系结构，但在控制过程时，对于大ARL误报，并行运行的许多类型控制图的运行长度都渐近独立。此外，通常 控制行程长度以渐近指数形式渐近分布，从而为ARL2FA提供了简单的渐近表达式。我们针对某些Cusum和Shiryaev–Roberts型控制图证明了这一主张，并通过仿真进行了说明。