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A Change-Point Problem and Composite Likelihood Inference in Time Series Models.

机译：时间序列模型中的变更点问题和综合似然推断。

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摘要

This thesis consists of essentially two parts, both of which are related to likelihood inference in time series models. The first part addresses a hypothesis testing problem about distinguishing between a short-memory time series with structural change and a long-memory time series without structural change. The second part is a study of composite likelihood inference in time series. While these two parts seem quite different, they are connected in a chapter that considers the application of composite likelihood on the problems in change point analysis.;The first part of the thesis develops a likelihood ratio based test for discriminating between a short-memory time series with structural change and a long-memory time series without structural change. Under the null hypothesis, the time series consists of two segments of short-memory time series with different means, and possibly different covariance functions. The location of the structural change is unknown. Under the alternative hypothesis the time series has no structural change but rather is long-memory. The likelihood ratio statistic is defined as the normalized log-ratio of the Whittle likelihood between the change-point model and the long-memory model, which is asymptotically normally distributed under the null. Berkes et al.(2006) proposed a test based on the CUSUM statistics for discriminating between change-point and long-memory models. The CUSUM test assumes that under the null hypothesis the time series has a shift in mean at an unknown location. Other properties of the time series such as correlation function and marginal distribution are assumed unchanged. The likelihood ratio test provides a parametric alternative to the CUSUM test proposed by Berkes et al. (2006). Moreover, the likelihood ratio test is more general than the CUSUM test in the sense that it is applicable to changes in other marginal or dependence features other than a change-in-mean. Furthermore, the likelihood ratio test tends to have higher power than the CUSUM test on detecting the long-memory Fractionally Integrated Autoregressive Moving Average (ARFIMA) model. We show its good performance in simulations and apply it to two data examples.;The second component in this thesis discusses composite likelihood inference in time series. In modern statistical analysis, one often encounters complicated models for complex data. Exact likelihood may not be a real option in these cases because of computational difficulties or the unavailability of theoretical asymptotic properties. Recently there has been considerable development in composite likelihood theory, which tackles the above problems by working on some approximations likelihood theory in some time series likelihood estimation procedures for linear time series models are described. The to the exact likelihood. This thesis attempts to study the application of composite models. The asymptotic properties of pairwise estimators are shown to be strongly consistent and asymptotically normal. This covers Autoregressive Moving Average (ARMA) as well as fractionally integrated ARMA (ARFIMA) models with the fractional integration parameter d 0.5. A comparison between using all pairs and consecutive pairs of observations in defining the composite likelihood is given. In particular, when all possible pairs of observations are used in defining the likelihood, the estimation for ARMA model is still consistent. However, the estimation for ARFIMA model is only consistent for d 0.25 but not consistent for d ≥ 0.25. Application of pairwise likelihood to a popular nonlinear model for time series of counts is also considered.;Finally, this thesis suggests some possible applications of pairwise likelihood to change-point analysis, including a connection between pairwise likelihood and distinguishing between short-memory change-point model and long-memory model. Since pairwise likelihood can be applied to complicated models where exact likelihood method is not feasible, e.g. time series with latent process, the use of pairwise likelihood can broaden the scope of applicability to problems in change-point analysis. For instance, estimation of multiple change-points in Poisson autoregressive models or testing for change-points in stochastic volatility models may be possible using a composite likelihood approach.
机译：本文主要由两部分组成，这两部分都与时间序列模型中的似然推断有关。第一部分解决了关于区分具有结构变化的短时间序列和没有结构变化的长时间序列的假设检验问题。第二部分是时间序列中复合似然推理的研究。虽然这两个部分看起来很不相同，但是它们在一个章中联系在一起，该章考虑了组合似然在变更点分析中的问题。；论文的第一部分开发了一种基于似然比的测试，用于区分短时内存时间。具有结构变化的序列和没有结构变化的长记忆时间序列。在原假设下，时间序列由两段短内存时间序列组成，它们具有不同的均值和可能不同的协方差函数。结构变化的位置未知。在替代假设下，时间序列没有结构变化，而是长记忆。似然比统计量定义为变化点模型与长内存模型之间的Whittle似然率的归一化对数比，它在零值下渐近正态分布。 Berkes等人（2006年）提出了一种基于CUSUM统计量的测试，用于区分变更点模型和长内存模型。 CUSUM检验假设在原假设下，时间序列在未知位置的均值发生了偏移。假设时间序列的其他属性（如相关函数和边际分布）不变。似然比检验提供了Berkes等人提出的CUSUM检验的参数替代方案。 （2006）。此外，在适用于除均值变化以外的其他边际或相依性变化方面，似然比检验比CUSUM检验更通用。此外，似然比测试在检测长内存分数积分自回归移动平均（ARFIMA）模型方面比CUSUM测试具有更高的功效。我们在仿真中展示了它的良好性能，并将其应用于两个数据示例。；本文的第二部分讨论了时间序列中的复合似然推断。在现代统计分析中，人们经常会遇到复杂的复杂数据模型。由于计算困难或理论渐近性质不可用，在这些情况下确切的可能性可能不是一个现实的选择。最近，复合似然理论有了长足的发展，它通过描述线性时间序列模型的某些时间序列似然估计程序中的一些近似似然理论来解决上述问题。确切的可能性。本文试图研究复合模型的应用。成对估计量的渐近性质被证明是强一致的并且渐近正态。它涵盖了分数积分参数d <0.5的自回归移动平均值（ARMA）以及分数积分ARMA（ARFIMA）模型。给出了在使用所有对和连续的观察对定义复合似然性之间的比较。特别是，当使用所有可能的观察对来定义可能性时，ARMA模型的估计仍然是一致的。但是，ARFIMA模型的估计仅在d <0.25时是一致的，而在d≥0.25时是不一致的。最后，本文还提出了成对似然法在流行的时间序列计数非线性模型中的应用。最后，本文提出了成对似然法在变化点分析中的一些可能应用，包括成对似然法与区分短时记忆变化之间的联系。点模型和长内存模型。由于成对似然可以应用于无法使用精确似然法的复杂模型，例如在具有潜在过程的时间序列中，使用成对似然可以扩大适用于变更点分析中问题的范围。例如，可以使用复合似然方法来估计泊松自回归模型中的多个变化点或测试随机波动率模型中的变化点。

著录项

  • 作者

    Yau, Chun Yip.;

  • 作者单位

    Columbia University.;

  • 授予单位 Columbia University.;
  • 学科 Statistics.
  • 学位 Ph.D.
  • 年度 2010
  • 页码 92 p.
  • 总页数 92
  • 原文格式 PDF
  • 正文语种 eng
  • 中图分类
  • 关键词

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