一类长记忆时间序列趋势项变点的Wilcoxon秩检验

doi:10.3785/j.issn.1008-9497.2022.04.006

浙江大学学报（理学版）

2022, Vol. 49

Issue (4): 427-434 DOI: 10.3785/j.issn.1008-9497.2022.04.006

数学与计算机科学

一类长记忆时间序列趋势项变点的Wilcoxon秩检验

成守尧^1,²,陈占寿^1,²(

),娘毛措^1,²,汪肖阳^1,²

^1.青海师范大学数学与统计学院，青海西宁 810008
^2.藏语智能信息处理及应用国家重点实验室，青海西宁 810008

Wilcoxon rank test for change point in trend in a class of long memory time series

Shouyao CHENG^1,²,Zhanshou CHEN^1,²(

),Maocuo NIANG^1,²,Xiaoyang WANG^1,²

^1.School of Mathematics and Statistics，Qinghai Normal University，Xining 810008，China
^2.The State Key Laboratory of Tibetan Intelligent Information Processing and Application，Xining 810008，China

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摘要：

对一类带Hurst指数的分数布朗运动趋势项变点的检验问题进行了研究，提出了一种先对观测序列做一阶差分，再基于差分序列构造Wilcoxon秩统计量做检验的后验检验方法。在原假设下证得检验统计量的极限分布是标准分数布朗运动的泛函，并给出了检验统计量的临界值。数值模拟结果表明，提出的检验方法除Hurst值较大外，均能很好地控制经验水平，经验势随样本量的增多几乎能趋近于1，且在样本量较大时，对截距项变点和方差变点稳健。采用该方法分析了1854—1989年北半球月均气温数据，未检测到趋势项变点。

关键词： Wilcoxon秩检验; 趋势项变点; 分数布朗运动

Abstract:

The problem of testing the change point in trend for a class of fractional Brownian motion with Hurst exponent is studied, and a posterior testing method is proposed which firstly makes first-order difference to the observation sequence and then constructs Wilcoxon rank statistics based on the difference sequence for testing. Under the null hypothesis, it is proved that the limit distribution of the test statistic is a functional of the standard fractional Brownian motion and the critical values of the test statistic are given. The numerical simulation results show that the test method proposed in this paper can control the empirical size well except for the case where the Hurst value is too large, and the empirical power can almost be close to 1 following the increase of sample size. Moreover, the method is robust to the intercept change point and variance change point when the sample size is large. Finally, a set of monthly temperature data in the northern hemisphere is analyzed, it is found that there is no change point in trend.

Key words: Wilcoxon rank test change point in trend fractional Brownian motion

收稿日期: 2021-04-20 出版日期: 2022-07-13

CLC:

O 212

基金资助: 国家自然科学基金资助项目(12161072);青海省自然科学基金项目(2019-ZJ-920)

通讯作者: 陈占寿 E-mail: chenzhanshou@126.com

作者简介: 成守尧（1996—），ORCID：https：//orcid.org/0000-0003-4797-0330，男，硕士，主要从事时间序列变点研究.

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引用本文:

成守尧,陈占寿,娘毛措,汪肖阳. 一类长记忆时间序列趋势项变点的Wilcoxon秩检验[J]. 浙江大学学报（理学版）, 2022, 49(4): 427-434.

Shouyao CHENG,Zhanshou CHEN,Maocuo NIANG,Xiaoyang WANG. Wilcoxon rank test for change point in trend in a class of long memory time series. Journal of Zhejiang University (Science Edition), 2022, 49(4): 427-434.

链接本文:

https://www.zjujournals.com/sci/CN/10.3785/j.issn.1008-9497.2022.04.006 或 https://www.zjujournals.com/sci/CN/Y2022/V49/I4/427

$H$	上 $α$ 分位数
$H$	$α =$ 0.01	$α =$ 0.05	$α =$ 0.10
0.6	0.375 479 3	0.309 046 4	0.276 734 0
0.7	0.299 967 2	0.246 656 3	0.217 075 4
0.8	0.233 224 5	0.185 933 1	0.164 788 9
0.9	0.155 909 6	0.124 510 4	0.109 241 7

表1 极限分布的上α分位数

$n$	经验水平值
$n$	$H =$ 0.6	$H =$ 0.7	$H =$ 0.8	$H =$ 0.9
50	0.046	0.038	0.058	0.166
100	0.029	0.049	0.066	0.136
300	0.053	0.055	0.061	0.090
500	0.052	0.047	0.058	0.106

表2 统计量Wn的经验水平值

$Δ$	$n$	$λ$	经验势
$Δ$	$n$	$λ$	$H =$ 0.6	$H =$ 0.7	$H =$ 0.8	$H =$ 0.9
0.50	50	0.25	0.095	0.151	0.214	0.536
		0.50	0.212	0.306	0.517	0.915
		0.75	0.125	0.160	0.291	0.750
	100	0.25	0.420	0.595	0.807	1.000
		0.50	0.719	0.862	0.980	1.000
		0.75	0.445	0.598	0.851	1.000
	300	0.25	1.000	1.000	1.000	1.000
		0.50	1.000	1.000	1.000	1.000
		0.75	1.000	1.000	1.000	1.000
	500	0.25	1.000	1.000	1.000	1.000
		0.50	1.000	1.000	1.000	1.000
		0.75	1.000	1.000	1.000	1.000
0.10	50	0.25	0.327	0.427	0.523	0.986
		0.50	0.644	0.846	0.973	1.000
		0.75	0.404	0.613	0.794	0.999
	100	0.25	0.983	0.998	1.000	1.000
		0.50	1.000	1.000	1.000	1.000
		0.75	0.983	0.999	1.000	1.000
	300	0.25	1.000	1.000	1.000	1.000
		0.50	1.000	1.000	1.000	1.000
		0.75	1.000	1.000	1.000	1.000
	500	0.25	1.000	1.000	1.000	1.000
		0.50	1.000	1.000	1.000	1.000
		0.75	1.000	1.000	1.000	1.000

表3 统计量Wn的经验势

$β 0$	$n$	经验水平值
$β 0$	$n$	$H =$ 0.6	$H =$ 0.7	$H =$ 0.8	$H =$ 0.9
0.5	50	0.039	0.055	0.067	0.181
	100	0.046	0.061	0.056	0.131
	300	0.066	0.053	0.051	0.119
	500	0.053	0.041	0.053	0.123
2.0	50	0.039	0.049	0.067	0.180
	100	0.045	0.050	0.070	0.131
	300	0.045	0.064	0.072	0.112
	500	0.063	0.054	0.053	0.100

表4 情况1统计量Wn 的经验水平值

$σ'$	$n$	经验水平值
$σ'$	$n$	$H =$ 0.6	$H =$ 0.7	$H =$ 0.8	$H =$ 0.9
0.5	50	0.057	0.066	0.082	0.254
	100	0.066	0.063	0.074	0.213
	300	0.059	0.059	0.084	0.187
	500	0.065	0.051	0.072	0.170
2.0	50	0.056	0.058	0.087	0.238
	100	0.061	0.061	0.086	0.214
	300	0.056	0.066	0.076	0.179
	500	0.049	0.077	0.086	0.180

表5 情况2统计量Wn 的经验水平值

$β 0$	$n$	经验势
$β 0$	$n$	$H =$ 0.6	$H =$ 0.7	$H =$ 0.8	$H =$ 0.9
0.5	50	0.683	0.845	0.972	1.000
	100	1.000	1.000	1.000	1.000
	300	1.000	1.000	1.000	1.000
	500	1.000	1.000	1.000	1.000
2.0	50	0.683	0.858	0.975	1.000
	100	0.999	1.000	1.000	1.000
	300	1.000	1.000	1.000	1.000
	500	1.000	1.000	1.000	1.000

表6 情况3统计量Wn 的经验势

$σ'$	$n$	经验势
$σ'$	$n$	$H =$ 0.6	$H =$ 0.7	$H =$ 0.8	$H =$ 0.9
0.5	50	1.000	1.000	1.000	1.000
	100	1.000	1.000	1.000	1.000
	300	1.000	1.000	1.000	1.000
	500	1.000	1.000	1.000	1.000
2.0	50	0.719	0.867	0.974	1.000
	100	1.000	1.000	1.000	1.000
	300	1.000	1.000	1.000	1.000
	500	1.000	1.000	1.000	1.000

表7 情况4统计量Wn 的经验势

图1 1854—1989年北半球经季节调整的月均气温

1	MACNEILL I B. Properties of sequences of partial sums of polynomial regression residuals with applications to tests for change of regression at unknown times［J］. The Annals of Statistics， 1978， 6（2）： 422-433. doi:10.1214/aos/1176344133 doi: 10.1214/aos/1176344133
2	HUŠKOVÁ M， PICEK J. Quantile regression and related methods：Bootstrap in detection of changes in linear regression［J］. Sankhyā： The Indian Journal of Statistics， 2005， 67（2）： 200-226.
3	秦瑞兵，郭娟. 平稳过程趋势项变点的CUSUM检验［J］. 纺织高校基础科学学报，2017，30（2）：226-229. QIN R B， GUO J. CUSUM test for change point in stochastic trend with stationary process［J］. Basic Sciences Journal of Textile Universities， 2017， 30（2）： 226-229.
4	JIANG F Y， ZHAO Z F， SHAO X F. Time series analysis of COVID-19 infection curve： A change-point perspective［J/OL］. Journal of Econometrics， 2020［2020-07-30］. . DOI：10.1016/j.jeconom. 2020.07.039 doi: 10.1016/j.jeconom. 2020.07.039
5	BARANOWSKI R， CHEN Y N， FRYZLEWICZ P. Narrowest-over-threshold detection of multiple change points and change-point-like features［J］. Journal of the Royal Statistical Society Series B （Statistical Methodology）， 2019， 81（3）： 649-672. DOI：10.1111/rssb.12322 doi: 10.1111/rssb.12322
6	TAN C C， SHI X P， SUN X Y， et al. On nonparametric change point estimator based on empirical characteristic functions ［J］. Science China Mathematics， 2016， 59（12）： 2463-2484. DOI：10. 1007/s11425-016-0138-x doi: 10. 1007/s11425-016-0138-x
7	MANDELBROT B B， VANNESS J W. Fractional Brownian motions， fractional noises and applications［J］. Society for Industrical and Applied Mathematics， 1968， 10（4）： 422-437. DOI：10.1137/1010093 doi: 10.1137/1010093
8	MOLZ F J， LIU H H， SZULGA J. Fractional Brownian motion and fractional Gaussian noise in subsurface hydrology： A review， presentation of fundamental properties， and extensions［J］. Water Resources Research， 1997， 33（10）： 2273-2286. doi:10.1029/97wr01982 doi: 10.1029/97wr01982
9	NEUMAN E， ROSENBAUM M. Fractional Brownian motion with zero Hurst parameter： A rough volatility viewpoint［J］. Electronic Communications in Probability， 2018， 23： 1-12. DOI：10.1214/18-ECP158 doi: 10.1214/18-ECP158
10	GUASONI P， MISHURA Y， RÁSONYI M. High-frequency trading with fractional Brownian motion［J］. Finance and Stochastics， 2021， 25（2）： 277-310. DOI：10.1007/s00780-020-00439-y doi: 10.1007/s00780-020-00439-y
11	WENGER K， LESCHINSKI C. Fixed-bandwidth CUSUM tests under long memory［J］. Econometrics and Statistics， 2021， 20： 46-61. DOI：10.1016/j.ecosta.2019.08.001 doi: 10.1016/j.ecosta.2019.08.001
12	陈占寿. 基于Bootstrap方法的时间序列变点检测［M］. 北京：科学出版社，2020. CHEN Z S. Time Series Change Point Detection Based on Bootstrap Method［M］. Beijing： Science Press， 2020.
13	WANG L H. Change-point detection with rank statistics in long-memory time-series models［J］. Australian & New Zealand Journal of Statistics， 2008， 50（3）： 241-256. doi:10.1111/j.1467-842x.2008.00515.x doi: 10.1111/j.1467-842x.2008.00515.x
14	DEHLING H， ROOCH A， TAQQU M S. Non-parametric change-point tests for long-range dependent data［J］. Scandinavian Journal of Statistics， 2013， 40（1）： 153-173. DOI：10.1111/j.1467-9469.2012.00799.x doi: 10.1111/j.1467-9469.2012.00799.x
15	BETKEN A. Testing for change-points in long-range dependent time series by means of a self-normalized Wilcoxon test［J］. Journal of Time Series Analysis， 2016， 37（6）： 785-809. DOI：10.1111/jtsa.12187 doi: 10.1111/jtsa.12187
16	WENGER K， LESS V. A modified Wilcoxon test for change points in long-range dependent time series［J］. Economics Letters， 2020， 192（3）： 109237. DOI：10.1016/j.econlet.2020.109237 doi: 10.1016/j.econlet.2020.109237
17	BERAN J. Statistics for Long-Memory Processes［M］. London： Chapman & Hall/CRC， 1994.
18	DEHLING H， TAQQU M S. The empirical process of some long-range dependent sequences with an application to U-statistics［J］. The Annals of Statistics， 1989， 17（4）： 1767-1783. DOI：10.1214/aos/1176347394 doi: 10.1214/aos/1176347394
19	TAQQU M S. Convergence of integrated processes of arbitrary Hermite rank［J］. Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete， 1979， 50： 53-83. DOI：10.1007/BF00535674 doi: 10.1007/BF00535674
20	SHORACK G R， WELLNER J A. Empirical Processes with Applications to Statistics［M］. New York： John Wiley & Sons， 1987.
21	ROBINSON P M. Gaussian semiparametric estimation of long range dependence［J］. The Annals of Statistics， 1995， 23（5）： 1630-1661. DOI： 10. 1214/aos/1176324317 doi: 10. 1214/aos/1176324317
22	LUO Y， HUANG Y R. A new combined approach on Hurst exponent estimate and its applications in realized volatility［J］. Physica A： Statistical Mechanics and Its Applications， 2018， 492： 1364-1372. DOI：10.1016/j.physa.2017.11.063 doi: 10.1016/j.physa.2017.11.063
23	DEO R S， HURVICH C M. Linear trend with fractionally integrated errors［J］. Journal of Time Series Analysis， 1998， 19（4）： 379-397. DOI： 10. 1111/1467-9892.00099 doi: 10. 1111/1467-9892.00099
24	WANG L H. Gradual changes in long memory processes with applications［J］. Statistics， 2007， 41（3）： 221-240. DOI：10.1080/02331880701223555 doi: 10.1080/02331880701223555

[1]	林正炎. 重分数布朗运动的列维连续模 [J]. 浙江大学学报（理学版）, 2000, 27(6): 677-681.

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