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Monitoring Mean and Variance Change-Points in Long-Memory Time Series

CHEN Zhanshou1,2, LI Fuxiao3, ZHU Li4, XING Yuhong1,2   

  1. 1. School of Mathematics and Statistics, Qinghai Normal University, Xining 810008, China; 2. Academy of Plateau Science and Sustainability, Xining 810008, China; 3. Department of Applied Mathematics, Xi'an University of Technology, Xi'an 710048, China; 4. College of Finance, Xingjiang University of Finance and Economics, Urumqi 830012, China 6. Academy of Plateau Science and Sustainability, Xining 810008, China
  • Received:2020-09-14 Revised:2020-12-28 Online:2022-06-25 Published:2022-06-20
  • Supported by:
    This research was supported by the Natural Science Foundation of Heilongjiang Province under Grant No. LH2020F035.

CHEN Zhanshou, LI Fuxiao, ZHU Li, XING Yuhong. Monitoring Mean and Variance Change-Points in Long-Memory Time Series[J]. Journal of Systems Science and Complexity, 2022, 35(3): 1009-1029.

This paper proposes two ratio-type statistics to sequentially detect mean and variance change-points in the long-memory time series. The limiting distributions of monitoring statistics under the no change-point null hypothesis, alternative hypothesis as well as change-point misspecified hypothesis are proved. In particular, a sieve bootstrap approximation method is proposed to determine the critical values. Simulations indicate that the new monitoring procedures have better finite sample performance than the available off-line tests when the change-point nears to the beginning time of monitoring, and can discriminate between mean and variance change-point. Finally, the authors illustrate their procedures via two real data sets:A set of annual volume of discharge data of the Nile river, and a set of monthly temperature data of northern hemisphere. The authors find a new variance change-point in the latter data.
[1] Pitarakis J Y, Least squares estimation and tests of breaks in mean and variance under misspecification, The Econometrics Journal, 2004, 7:32-54.
[2] Aue A and Horváth L, Structural breaks in time series, Journal of Time Series Analysis, 2013, 34:1-16.
[3] Jandhyala V, Fotopoulos S, MacNeill I, et al., Inference for single and multiple change-points in time series, Journal of Time Series Analysis, 2013, 34:423-446.
[4] Eichinger B and Kirch C, An MOSUM procedure for the estimation of multiple random change points, Bernoulli, 2018, 24:526-564.
[5] Hidalgo J and Robinson P M, Testing for structural change in a long-memory environment, Journal of Econometrics, 1996, 70:159-174.
[6] Horváth L and Kokoszka P, The effect of long-range dependence on change-point estimators, Journal of Statistical Planning&Inference, 1997, 64:57-81.
[7] Wang L and Wang J, Change-of variance problem for linear processes with long memory, Statistical Papers, 2006, 47:279-298.
[8] Li Y, Xu J, and Zhang L, Testing for changes in the mean or variance of long memory processes, Acta Mathematica Sinica, 2010, 26:2443-2460.
[9] Zhao W, Tian Z, and Xia Z, Ratio test for variance change point in linear process with long memory, Statistical Papers, 2010, 51:397-407.
[10] Shao X, A simple test of changes in mean in the possible presence of long-range dependence, Journal of Time Series Analysis, 2011, 32:598-606.
[11] Betken A, Testing for change-points in long-range dependent time series by means of a selfnormalized wilcoxon test, Journal of Time Series Analysis, 2016, 37:785-809.
[12] Betken A and Kulik R, Testing for change in long-memory stochastic volatility time series, Journal of Time Series Analysis, 2019, 40:707-738.
[13] Wenger K, Leschinski C, and Sibbertsen P, Change-in-mean tests in long-memory time series:A review of recent developments, ASTA Advances in Statistical Analysis, 2019, 103:237-256.
[14] Beran J and Terrin N, Testing for a change of the long-memory parameter, Biometrika, 1996, 163:186-199.
[15] Horváth L and Shao Q, Limit theorems for quadratic forms with applications to Whittle's estimates, Journal of Applied Probability, 1999, 9:146-187.
[16] Lavancier F, Leipus R, Philippe A, et al., Detection of nonconstant long memory parameter, Econometric Theory, 2013, 29:1009-1056.
[17] Iacone F and Lazarová S, Semiparametric detection of changes in long range dependence, Journal of Time Series Analysis, 2019, 40:693-706.
[18] Chu C S J and White H, Monitoring structural change, Econometrica, 1996, 64:1045-1065.
[19] Zou C, Tsung F, and Wang Z, Monitoring general linear profiles using multivariate exponentially weighted moving average schemes, Technometrics, 2007, 49:395-408.
[20] Chen Z and Tian Z, Modified procedures for change point monitoring in linear regression models, Mathematics and Computers in Simulation, 2010, 81:62-75.
[21] Kirch C and Weber S, Modified sequential change point procedures based on estimating functions, Electronic Journal of Statistics, 2018, 12:1579-1613.
[22] Zhao W, Xue Y, and Liu X, Monitoring parameter change in linear regression model based on the efficient score vector, Physica A:Statistical Mechanics and Its Applications, 2019, 527:121-135.
[23] Chen Z, Tian Z, and Xing Y, Sieve bootstrap monitoring persistence change in long memory process, Statistics and Its Interface, 2016, 9:37-45.
[24] Chen Z, Xing Y, and Li F, Sieve bootstrap monitoring for change from short to long memory, Economics Letters, 2016, 40:53-56.
[25] Chen Z, Xiao Y, and Li F, Monitoring memory parameter change-points in long-memory time series, Empirical Economics, 2021, 60:2365-2389.
[26] Bühlmann P, Sieve bootstrap for time series, Bernoulli, 1997, 3:123-148.
[27] Poskitt D S, Properties of the sieve Bootstrap for fractionally integrated and non-invertible processes, Journal of Time Series Analysis, 2008, 29:224-250.
[28] Poskitt D S, Martin G M, and Grose S, Bias correction of semiparametric long memory parameter estimators via the prefiltered sieve bootstrap, Econometric Theory, 2017, 33:578-609.
[29] Taqqu M, Weak convergence to fractional Brownian motion and to the Rosenblatt process, Z. Wahrsch. Verw. Gebiete, 1975, 31:287-302.
[30] Johansen S and Nielsen M, A necessary moment condition for the fractional functional central limit theorem, Econometric Theory, 2012, 28:671-679.
[31] Davidson J and Hashimzade N, Type I and type II fractional Brownian motions:A reconsideration, Computational Statistics&Data Analysis, 2009, 53:2089-2106.
[32] Kapetanios G, Papailias F, and Taylor A M R, A generalised fractional differencing bootstrap for long memory processes, Journal of Time Series Analysis, 2019, 40:467-492.
[33] Deo R S and Hurvich C M, Linear trend with fractionally integrated errors, Journal of Time Series Analysis, 1998, 19:379-397.
[34] Beran J and Feng Y, SEMIFAR models-A semiparametric approach to modelling trends, long-range dependence and nonstationarity, Computational Statistics and Data Analysis, 2002, 40:393-419.
[35] Wang L, Gradual changes in long memory processes with applications, Statistics, 2007, 41:221-240.
[36] Yau C and Zhao Z, Inference for multiple change points in time series via Likelihood ratio scan statistics, Journal of the Royal Statistical Society, B, 2016, 78:895-916.
[37] Bickel P J and Freedman D A, Some asymptotic theory for the bootstrap, Annals of Statistics, 1981, 9:1196-1217.
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