在变点问题研究中，相关系数变点检验是识别两个序列间相关关系强度变化的重要技术。针对现有方法在区分变点前后相关系数差异方面的不足，本文提出了一种基于移动窗框方法检验相关系数变点的创新思路。该方法巧妙地将相关系数变点问题转化为均值变点问题，从而利用均值变点检验的成熟理论来提升相关系数变点检验的统计功效。这一转换不仅简化了检验过程，而且增强了变点检测的稳健性。此外，随着大数据时代的到来，时间序列数据的复杂性显著增加，特别是具有尖峰厚尾和相依特征的厚尾混合序列为变点检测带来了新的挑战。基于此，本文以移动窗框方法为研究手段，深入探讨了正态序列和厚尾序列的相关系数变点检验问题，具体内容如下：

首先，聚焦于基于移动窗框方法的正态序列下相关系数变点检验问题。利用移动窗框方法估计两个正态序列之间的相关系数，并将估计结果引入均值变点模型作为新序列进行分析。针对累积和统计量需要估计长期方差的问题，提出了基于残差的修正Ratio统计量进行均值变点检验。同时使用最小二乘估计方法推导出统计量在原假设下的极限分布为布朗运动的泛函，并证明了在备择假设下的一致性。此外，提供了变点位置的估计方法及其一致性证明。不失一般性，为解决统计量的极限分布形式复杂难以确定临界值的问题，采用了Bootstrap抽样方法。通过蒙特卡洛数值模拟，确定了移动窗框方法中窗框和间隔的合理选择，并将基于移动窗框的方法与已有相关系数变点检验方法进行对比，证实了本文所提方法对于相关系数变点检验更为有效。

其次，本文将相关系数变点研究范围从正态序列扩展到具有厚尾特性的混合序列。鉴于厚尾序列不满足相关系数存在的条件，本文通过将序列乘以一个收敛速度来构造伪序列，并证明了伪序列的均值、方差和协方差均存在，且伪序列与原序列具有相同的相关系数。然而，随着厚尾序列尾指数的减小，使用移动窗框方法估计出的相关系数值波动加剧。因此，本文引入了基于M估计的统计量，以提高检验的稳健性，同样推导了统计量在原假设下的极限分布以及在备择假设下的一致性，给出了变点位置的一致估计。此外，考虑到厚尾混合序列的特性，采用了Block bootstrap抽样方法以确保得到精确的临界值，该方法可以保留序列之间的相依性，并在原假设下证明了其具有一致性。后续探讨了厚尾序列对窗框和间隔选择的影响，并比较了M估计方法与最小二乘估计方法在统计量的临界值、经验水平和经验势方面的差别。结果表明，M估计方法的整体表现优于最小二乘估计方法。

最后，选取两对实际数据应用到本文所提方法中进行实证分析，进一步阐明了移动窗框方法对相关系数变点检验具备有效性和可行性。这些研究成果不仅丰富了变点检验的理论体系，也为金融风险管理等领域提供了实用的技术支持，为相关系数变点问题的统计推断和实际应用提供了新的视角和方法。

In the study of change-point problems, the examination of shifts in correlation coefficients is critical for identifying changes in the relationship between two sequences. A novel moving window method is introduced in this thesis to address the limitations of current techniques in differentiating correlation coefficient changes. By cleverly converting the correlation change-point issue into a mean change-point problem, the proposed method leverages established mean change-point testing theories to improve the statistical power of correlation change-point detection. This approach not only streamlines the testing procedure but also enhances the robustness of change-point identification. With the rise of big data, the complexity of time series data, especially heavy-tailed sequences with sharp peaks and thick tails, has increased, presenting new challenges for change-point analysis. This research employs the moving window method to investigate correlation change-point detection in both normal and heavy-tailed sequences, aiming to contribute to the field with a more refined and robust detection technique. The specific content includes the following:

This paper begins by focusing on the change-point test for correlation coefficients under normal sequences using the moving window method. The moving window method is employed to estimate the correlation coefficient between two normal sequences, and the resulting estimates are incorporated into a mean change-point model for analysis as a new sequence. To address the issue of estimating long-term variance for the cumulative sum statistic, a modified Ratio statistic based on residuals is proposed for mean change-point testing. The limit distribution of the statistic under the null hypothesis is derived as a functional of Brownian motion using the LS-estimation method, and its consistency under the alternative hypothesis is demonstrated. Additionally, an estimation method for the change-point location and its consistency proof are provided. To address the complexity of determining critical values due to the intricate form of the limit distribution of the statistic, a Bootstrap sampling method is adopted. The reasonable choices for the window size and interval in the moving window method are determined through Monte Carlo numerical simulations by the paper, and the proposed method is compared with existing correlation change-point testing methods, with the enhanced efficacy of the method introduced in this paper for detecting change-points in correlation coefficients being confirmed.

Next, this paper extends the study of change-points in correlation coefficients from normal sequences to mixed sequences with heavy-tailed characteristics. Given that heavy-tailed sequences do not satisfy the conditions for the existence of correlation coefficients, this paper constructs pseudo-sequences by multiplying the original sequences by a converging speed, thereby demonstrating that the mean, variance, and covariance of the pseudo-sequences exist, and that the pseudo-sequences have the same correlation coefficient as the original sequences. However, as the tail index of the heavy-tailed sequences decreases, the estimated correlation coefficient values fluctuate more intensely when using the moving window method. To enhance the robustness of the test, this paper introduces a statistic based on M-estimation, derives the limit distribution of the statistic under the null hypothesis and its consistency under the alternative hypothesis, and provides a consistent estimate of the change-point location. Additionally, considering the characteristics of the heavy-tailed mixed sequences, a Block bootstrap sampling method is adopted to ensure accurate critical values, which retains the dependence between sequences and is proven to be consistent under the null hypothesis. The paper also explores the impact of heavy tails on the selection of window size and interval and compares the performance of the M-estimation method with the LS-estimation method in terms of critical values, empirical levels, and empirical power of the statistic. The results indicate that the M-estimation method outperforms the LS-estimation method overall.

Finally, this paper selects two pairs of real-world data sets to apply the proposed methods for empirical analysis, further elucidating the effectiveness and feasibility of the moving window method for testing change-points in correlation coefficients. These research findings not only enrich the theoretical framework of change-point testing but also offer practical technical support for fields such as financial risk management. The work provides a new perspective and methods for statistical inference and practical application of change-point problems in correlation coefficients.

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