Joint Test For Mean And Variance Change-points In Long-memory Time Series

Since many financial,hydrological,meteorological and other data have longmemory,many literatures have studied the change-point test in long-memory time series in recent years,but most literatures only consider one type of change-point,such as mean change-point and variance change-point.In many actual observed data,mean change-points and variance change-points tend to occur simultaneously,so it is a meaningful work to study the effect of variance(mean)change-points on the efficacy of mean(variance)change-point test.In this paper,three self-normalized methods are proposed to study the joint test of mean change-points and variance change-points in long-memory time series.According to the relative positions of mean change-points and variance change-points,the data are divided into three cases for study.The effects of long-memory parameters,sample size,jump degree and change-point position on the efficiency of statistical test were discussed through theoretical proof and numerical simulation,and the feasibility of the method was illustrated by example analysis.The main contents are as follows:(1)Based on self-normalized statistics,the joint test of mean change-points and variance change-points of long-memory time series is studied,and the limit distribution of statistics is deduced when there are mean change-points and variance change-points in long-memory time series.It is theoretically proved that this statistic is still a consistent statistic for testing mean change-points at this time.The numerical simulation results show that when there are mean change-points and variance-change points in the sequence,the statistics are still valid except when the long-memory parameter values are large.Moreover,the larger the jump degree is,the closer the relative positions of the two kinds of change points are,and the higher the test efficiency is.Finally,the feasibility of this method is illustrated by analyzing three groups of data.(2)A new self-normalized statistic is proposed based on the model fitting residue square construction,and the joint test of mean change-point and variance change-point in long-memory time series is further studied,and the asymptotic distribution of the new statistic is proved when there is no change-point in long-memory time series.The consistency of the new statistics is proved under three alternative hypotheses with both mean and variance change-points.The numerical simulation results show that the new method is still effective when there are both mean change-points and variance changepoints in the model.Since self-normalized statistics are mainly sensitive to mean change-points,the self-normalized statistics based on residual square has a higher test effect on difference change-points.By combining these two statistics,we can distinguish whether the tested change-points are mean change-points,variance changepoints,or mean change-points and variance change-points.Finally,the feasibility of this method is illustrated by analyzing three groups of actual data.(3)For the characteristic that the self-normalized method has better test effect when the long-memory parameter is small,and the self-normalized method based on the residual square has better test effect when the long-memory parameter is large,a comprehensive statistic is proposed by combining the two statistics.The limit distribution of the statistic is proved under the null hypothesis that there is no changepoint in long memory time series,and the consistency of the synthesis method is proved under the alternative hypothesis.The numerical simulation results show that the proposed comprehensive test statistics have relatively high test efficacy when the longmemory parameters are large or small.