变点问题最早是从研究质量控制中提出来的, 在现实生活的诸多领域也会涉及, 如经济、气象以及工程等领域. 如果不先判断序列的参数是否发生改变, 那么可能会导致预测和统计推断的无效, 所以检验结构变点是至关重要的课题. 近年来, Ratio统计量因与传统累计和统计量相比不需要方差的估计, 从而成为一种检验结构变点的有效方法. 实际上, 大多相关文献考虑的是方差有限的情形, 然而大部分时间序列数据往往具有“尖峰厚尾”的特征, 这时大部分信息滞留在尾部且方差不存在, 不能用传统高斯序列来刻画, 所以考虑方差无穷序列更具有现实意义. 此外, 时间序列的相依性是普遍存在的, 例如AR(p)模型. 所以本文结合序列的尾部性质和相依性运用Ratio统计量来检验结构变点具有重要意义.

构造Ratio统计量来检验厚尾AR(p)序列的均值变点; 在原假设下证明了统计量的渐近分布是列维过程的泛函, 备择假设下得到统计量的一致性; 利用Subsampling抽样以获得更为准确的临界值同时也能避免参数的估计, 并证明此方法的有效性; 数值模拟结果显示在大样本下基于Subsampling抽样方法的Ratio统计量检验均值变点有着较好的经验水平和经验势.

虽有研究趋势变点的位置估计, 但少有学者提出趋势变点的检验问题. 为此, 这里考虑了厚尾AR(p)序列的趋势变点的检验问题. 构造Ratio统计量来检验趋势变点; 在原假设下证明统计量的极限分布是列维过程的泛函, 备择假设下得到统计量的一致性; 说明Subsampling抽样方法的有效性; 数值模拟结果显示在大样本下基于Subsampling抽样的Ratio检验可以很好地控制趋势变点的经验水平和经验势.

选取英国石油 (BP) 和谷歌 (Google) 股票价格的收盘价进行实证分析. 结果表明基于Subsampling抽样的厚尾AR(p)序列结构变点的Ratio检验具有有效性和可行性.

The change-point problem was the first to extract from the quality control problem. The problem of testing for structural changes covers a broad variety of real-world fields such as economics, clinical science, and engineering. If the parameters change without consideration, the prediction and statistical inference will be invalid, so it has subsequently been an essential and interesting issue to test for the structural change point. Recently, the ratio type test statistic has become a popular method to detect change points in time series, because it does not need to be standardized by any variance estimate. In fact, lots of literature have concentrated on the observations with finite variance. However, much of empirical work has shown that the heavy-tailed phenomenon exists frequently, and the information is in the tail and can not be characterized by traditional Gaussian sequences, so it makes sense to consider infinite variance sequences. In addition, it is more general to consider the weakly dependent case, such as AR(p) process. Therefore, it is of great significance to combine two key factors of the heavy tailness and dependence of random variables to study the structural changes in time series.

We consider that issues related to the mean of heavy-tailed AR(p) series are possibly subject to change at most once at some unknown point in time and a ratio statistic is constructed to test whether unknown changes have occurred. It is shown that asymptotic distribution of this test statistic under the no-change null hypothesis is functional for Lévy process and its consistency is given under the alternative. To avoid the nuisance parameter, we provide a Subsampling method that returns more accurate critical values for this test. The validity of the Subsampling algorithm is proved. A simulation study shows the Subsampling ratio test achieves the correct sizes and comparable powers in large samples.

Researchers have proposed some estimators to analyze changes in trend, but there has been little discussion on how detecting these change points. We consider that issues related to the trend of heavy-tailed AR(p) series may occur at most one change at some unknown point in time. It is shown that asymptotic distribution of this test statistic under the no-change null hypothesis is functional for Lévy process and its consistency is given under the alternative.  A simulation study shows the Subsampling ratio test achieves the correct sizes and comparable powers in large samples.

Further, we consider two sets of the stock price data of British Petroleum and Google. These two empirical examples indicate that Subsampling ratio tests are effective and feasible to detect structural changes in time series with heavy-tailed AR(p) errors .