Abstract
In this paper, we investigate the CUSUM statistic of change point under the negatively associated (NA) sequences. By establishing the consistency estimators for mean and covariance functions respectively, the limit distribution of the CUSUM statistic is proved to be a standard Brownian bridge, which extends the results obtained under the case of an independent normal sample and the moving average processes. Finally, the finite sample properties of the CUSUM statistic are given to show the efficiency of the method by simulation studies and an application on a real data analysis.
Highlights
Detecting a change-point and estimating its location are very important problems because of its extensive applications in many fields such as quality control, economics and finance, and so on
Hsu [11] detected the shifts of parameter in gamma sequences; Bai [1] and Shi et al [23] studied the mean shift models of change point; Kokoszka and Leipus [14] considered the CUSUMtype estimator for mean shift with dependent sequence; Lee et al [15] and Na et al [17] investigated the CUSUM statistic for parameter change in time series models; Horvath and Rice [10] summarized some classical methods in change point analysis; Christian et al [6] and Oh and Lee [18] studied the change point test for the GARCH models
We investigate the asymptotic property of CUSUM statistic of change point under Negatively Associated (NA) sequences
Summary
Detecting a change-point and estimating its location are very important problems because of its extensive applications in many fields such as quality control, economics and finance, and so on. We investigate the asymptotic property of CUSUM statistic of change point under NA sequences. Inclan and Tiao [12] proposed a CUSUM statistic to test a change-point of variance as follows: Theorem 1.1 Let {Xn, n ≥ 1} be a sequence of independent, identically distributed Normal random variables with X1 ∼ N (0, σ2) and σ2 > 0. By establishing the consistency estimators for mean and covariance functions, the limit distribution of CUSUM statistic of change point is proved to be a standard Brownian bridge. By Theorem 2.1, the limit distribution for the CUSUM statistic is presented as follows. The limit distribution for the CUSUM statistic max |Tnk| is presented in Theorem 2.2. We apply our method and the results by Inclan and Tiao [12] to detect a change-point of variances for the returns of log daily prices of Dow Jones Industrial (DJI) index which caused by COVID-19 pandemic in 2020
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