The Disorder Problem

Abstract

The disorder problem is traditionally a problem of hypotheses testing using the hypothesis: did it spasmodically change some of the characteristics of the observed process during the time of observation or not? Here we describe disorder problem as the problem of estimating the moment of switching of the characteristics of an observed diffusion-type process in a situation when it is known that the switching was realized during the observation. First we consider the problem of simultaneous estimation of the moment of disorder and some other parameter when the trend of the process is a smooth function of this parameter. Then we consider the problem of disorder estimation of multidimensional observations when each observed component has its own moment of switching and a generalization of this problem, when all the moments of switching are some smooth functions of another unknown parameter. The behavior of the MLE of the switching moment when the mathematical model is wrong is considered in the final part. It is interesting to note that even in this mis-specified problem, consistent estimation is possible. We will also return to disorder problems in partially observed systems (Section 6.3) and in the minimum distance estimation of switching equations (Section 7.5).