Change Point Estimation Of Normal Distribution Sequence

The change point problem is a hot issue in statistics research.It was first put forward by Page in 1954.It is mainly used to test whether the product quality is qualified.With the development of society,change point detection theory has been widely used in the fields of economy,medicine,meteorology,image processing and so on.Normal distribution is a very important continuous probability distribution in probability statistics,which is widely used in various fields,so it is very important to study the change point estimation of normal distribution series.In this paper,the change point problem of normal distribution series is studied from two aspects: mean change point and variance change point.First,based on Bayesian theory,a posteriori distribution of the position of the mean change point is derived when the variances are known and equal.Differential evolution algorithm(DE)and adaptive differential evolution algorithm(ADE)are combined with Bayesian method to estimate the position of change point.The results show that both the differential evolution algorithm and the adaptive differential evolution algorithm can estimate the position of the mean change point effectively.Among them,differential evolution algorithm converges faster than adaptive differential evolution algorithm.Secondly,the proposed differential evolution algorithm and adaptive differential evolution algorithm combined with Bayesian method are used to estimate the variance change points of normal distributed sequences with known and equal means,and the estimated results are compared with the maximum likelihood method.The results show that,In the case of small sample size,the estimates of differential evolution algorithm and adaptive differential evolution algorithm are closer to the real value,and in the case of large sample size,the results obtained by the three methods are consistent.Finally,this paper further studies the problem of change point estimation of normal distribution series when both mean and variance are unknown.Through numerical simulation,it is concluded that the estimation effect of combining differential evolution algorithm with Bayesian method is better than that of maximum likelihood method.Then,the algorithm is applied to the daily closing price sequence of Shanghai Stock Exchange Index,and the position of change point is estimated.