# Detection of two-sided alternatives in a Brownian motion model - Computer Science Information Theory

Detection of two-sided alternatives in a Brownian motion model - Computer Science Information Theory - Descarga este documento en PDF. Documentación en PDF para descargar gratis. Disponible también para leer online.

Abstract: This work examines the problem of sequential detection of a change in thedrift of a Brownian motion in the case of two-sided alternatives. Applicationsto real life situations in which two-sided changes can occur are discussed.Traditionally, 2-CUSUM stopping rules have been used for this problem due totheir asymptotically optimal character as the mean time between false alarmstends to $\infty$. In particular, attention has focused on 2-CUSUM harmonicmean rules due to the simplicity in calculating their first moments. In thispaper, we derive closed-form expressions for the first moment of a general2-CUSUM stopping rule. We use these expressions to obtain explicit upper andlower bounds for it. Moreover, we derive an expression for the rate of changeof this first moment as one of the threshold parameters changes. Based on theseexpressions we obtain explicit upper and lower bounds to this rate of change.Using these expressions we are able to find the best 2-CUSUM stopping rule withrespect to the extended Lorden criterion. In fact, we demonstrate not only theexistence but also the uniqueness of the best 2-CUSUM stopping both in the caseof a symmetric change and in the case of a non-symmetric case. Furthermore, wediscuss the existence of a modification of the 2-CUSUM stopping rule that has astrictly better performance than its classical 2-CUSUM counterpart for smallvalues of the mean time between false alarms. We conclude with a discussion onthe open problem of strict optimality in the case of two-sided alternatives.