Study Change Point Detection Of Mean Shift In Lévy Stable Process

Change point is a hot problem in statistics, it has been first referred in quality management, it has been made great progress both in theoretics and application in recent twenty years. Control charts are considered to be a useful tool of change point detection in statistical process control. The most widely used control charts are Shewhart control chart,CUSUM(cumulative sum),EWMA(exponentially weighted moving average),Cuscore(cumulative score) control charts and their improvements, that is, GLR(Generalized Likelihood Ratio),GEWMA(Generalized EWMA) and RFCuscore(Reference-Free Cuscore) control charts.As can be seen that all research work in the above literature is based on an assumption, that the observed process is subject to a normal distribution or the variance of the process is finite. In fact, many variables neither are normal or have finite variance. Such as the bankroll trading volume in the finance network, the return rate of stock market, temperature distribution in unclear reactors and the volume of annual rainfall, etc. These variables are usually subject to a stable distribution with infinite variance, which is called a Lévy stable process. Thus, it is very meaningful for us to study the change point detection of mean shift in Lévy stable process.As we know that the average run length(ARL) is an extensively used measure in statistical process control(SPC) for evaluating and comparing the detection performance of various control charts. In this paper we present the asymptotic estimation of the ARL for the Cuscore and the ARL′S up bound for EWMA control charts in Lévy stable process when characteristic exponentialα1 <α≤2;in addition,we also present the asymptotic estimation of the ARL′S up bound for the CUSUM control charts in Lévy stable process when characteristic exponentialα0 <α<1.