Consistency of change point estimators for symmetrical stable distribution with parameters shift

Assume that the characteristic index α of stable distribution satisfies 1 < α < 2, and that the distribution is symmetrical about its mean. We consider the change point estimators for stable distribution with α or scale parameter β shift. For the one case that mean is a known constant, if α or β changes, then density function will change too. To this end, we suppose the kernel estimation for a change point. For the other case that mean is an unknown constant, we suppose to apply empirical characteristic function to estimate the change-point location. In the two cases, we consider the consistency and strong convergence rate of estimators. Furthermore, we consider the mean shift case. If mean changes, then corresponding characteristic function will change too. To this end, we also apply empirical characteristic function to estimate change point. We obtain the similar convergence rate. Finally, we consider its application on the detection of mean shift in financial market. This work was supported by the National Natural Science Foundation of China (Grant No. 10471135) and Graduate Innovation Fund of the University of Science and Technology of China (Grant No. KD2006063)