Multiple Optimality Properties of the Shewhart Test

Abstract

For the problem of sequential detection of changes, we adopt the probability maximizing approach in place of the classical minimization of the average detection delay and propose modified versions of the Shiryaev, Lorden, and Pollak performance measures. For these alternative formulations, we demonstrate that the optimum sequential detection scheme is the simple Shewhart rule. Interestingly, we can also solve problems that under the classical setup have been open for many years, as optimum change detection with time-varying observations or with multiple postchange probability measures. For the latter, we also offer the exact solution for Lorden's original setup involving average detection delays, for the case where the average false alarm period is within certain limits.

Keywords:

• Change detection
• Sequential detection
• Shewhart test

Subject Classifications:

• 62L15
• 60G40
• 62F12
• 62F15

Notes

¹According to our definition, stopping before or at τ corresponds to false alarm.

²To avoid unnecessary technical complications, throughout our work we are going to assume that the cumulative distribution functions (cdfs) of all likelihood ratios ℓ _t , under both probability measures, are continuous and strictly increasing functions.

³There is a slight difference between the current definition of the prior and the one encountered in the literature. This is because in our approach, τ is the last time instant under the nominal regime, whereas in the literature τ is conventionally considered as the first instant under the alternative.

⁴Extension to more than two pdfs is straightforward.

Recommended by T. Luginbuhl