Inverse sequential detection of parameter changes in developing time series

Abstract

Progressive values of two probabilities are obtained for parameter estimates derived from an existing set of values and from the same set enlarged by one or more new values, respectively. One probability is that of erroneously preferring the second of these estimates for the existing data ('type 1 error'), while the second probability is that of erroneously accepting their estimates for the enlarged test ('type 2 error'). A more stable combined 'no change' probability which always falls between 0.5 and 0 is derived from the (logarithmic) width of the uncertainty region of an equivalent 'inverted' sequential probability ratio test (SPRT, Wald 1945) in which the error probabilities are calculated rather than prescribed. A parameter change is indicated when the compound probability undergoes a progressive decrease. The test is explicitly formulated and exemplified for Gaussian samples.