变点分析SIC,文献翻译.doc

INFORMATION CRITERION AND CHANGE POINTPROBLEM FOR REGULAR MODELSInformation criteria are commonly used for selecting competing statistical models. They do not favor the model which gives the best to the data and little interpretive value, but simpler models with good fit. Thus, model complexity is an important factor in information criteria for model selection. Existing results often equate the model complexity to the dimension of the parameter space. Although this notion is well founded in regular parametric models, it lacks some desirable properties when applied to irregular statistical models. We refine the notion of model complexity in the context of change point problems, and modify the existing information criteria. The modified criterion is found consistent in selecting the correct model and has simple limiting behavior. The resulting estimator of the location of the change point achieves the best convergence rate Op(1), and its limiting distribution is obtained. Simulation results indicate that the modified criterion has better power in detecting changes compared to other methodsIntroductionOut of several competing statistical models, we do not always use the one with the best to the data. Such models may simply interpolate the data and have little interpretive value. Information criteria, such as the Akaike information criterion and the Schwarz information criterion, are designed to select models with simple structure and good interpretive value, see Akaike (1973) and Schwarz (1978). The model complexity is often measured in terms of the dimensionality of the parameter space.Consider the problem of making inference on whether a process has undergone some changes. In the context of model selection, we want to choose between a model with a single set of parameters, or a model with two sets of parameters plus the location of change. The Akaike and the Schwarz information criteria can be readily adopted to this kind of change point problems. There have been many fruitful research done in this respect such as Hirotsu, Kuriki and Hayter (1992) and Chen and Gupta (1997), to name a few.Compared to usual model selection problems, the change point problem contains a special parameter: the location of the change. When it approaches the beginning or the end of the process, one of the two sets of the parameter becomes completely redundant. Hence, the model is un-necessarily complex. This observation motivates the notion that the model complexity also depends on the location of the change point. Consequently, we propose to generalize the Akaike and Schwarz information criteria by making the model complexity also a function of the location of the change point. The new method is shown to have a simple limiting behavior, and favourable power properties in many situations via simulation.The change point problem has been extensively discussed in the literature in recent years.The study of the change point problem dates back to Page (1954, 1955 and 1957) which tested the existence of a change point. Parametric approaches to this problem have been studied by a number of researchers, see Chernoff and Zacks (1964), Hinkley (1970), Hinkley et.al.(1980), Siegmund (1986) and Worsley (1979, 1986). Nonparametric tests and estimations have also been proposed (Brodsky and Darkhovsky, 1993; Lombard, 1987; Gombay andHuskova, 1998). Extensive discussions on the large sample behavior of likelihood ratio test statistics can be found in Gombay and Horvath (1996) and Csorgo and Horvath (1997).The detail can be found in some survey literatures such as Bhattacharya (1994), Basseville and Nikiforov (1993), Zacks (1983), and Lai (1985). The present study deviates from other studies by refining the traditional measure of the model complexity, and by determining the limiting distribution of the resulting test statistic under very general parametric model settings.In Section 2, we define and motivate the new information criterion in detail. In Section 3, we give the conditions under which the resulting test statistic has chi-square limiting distribution and the estimator of change point attains the best convergence rate. An application example and some simulation results are given in Section 4. The new method is compared to three existing methods and found to have good finite sample properties. The proofs are given in the Appendix.Main ResultsLet X1，X2， ，Xn be a sequence of independent random variables. It is suspected that Xi has density function when ik. We assume that and belong to the same parametric distribution family .The problem is to test whether this change has indeed occurred and if so, find the location of the change k. The null hypothesis is : and the alternative is : and Equivalently, we are asked to choose a model from or a model from for the data.For regular parametric (not change point) models with log likelihood function , Akaike and Schwarz information criteria are defined as:where is the maximum point of . The best model according to these criteria is the one which minimizes AIC or SIC. The Schwarz information criterion is asymptotically optimal according to certain Bayes formulation.The log likelihood function for the change point problem has the formThe Schwarz information criterion for the change point problem becomesand similarly for Akaike information criterion, where maximize for given k. See, for example, Chen and Gupta (1997). When the model complexity is the focus, we may also write it asWe suggest that the notion of = needs re-examination in the context of change point problem. When k takes values in the middle of 1 and n, both and are effective parameters. When k is near 1 or n, either or becomes redundant.Hence, k is an increasingly undesirable parameter as k getting close to 1 or n. We hence propose a modified information criterion withFor 1kn, letUnder the null model, we defineIf , we select the model with a change point and estimate the change point by such thatClearly, this procedure can be repeated when a second change point is suspected.The size of model complexity can be motivated as follows. If the change point is at k, the variance of would be proportional to and the variance of would be proportionalto. Thus, the total variance isThe specific form in (2) reflects this important fact. Thus, if a change at an early stage is suspected, relatively stronger evidence is needed to justify the change. Hence, we should place larger penalty when k is near 1 or n. This notion is shared by many researchers.The method in Inclan and Tiao (1994) scales down the statistics heavier when the suspected change point is near 1 or n. The U-statistic method in Gombay and Horvath (1995) is scaleddown by multiplying the factor k(n-k).To assess the error rates of the method, we can simulate the finite sample distribution, or find the asymptotic distribution of the related statistics. For Schwarz information criterion,the related statistic is found to have type I extreme value distribution asymptotically (Chen and Gupta, 1997; Csorgo and Horvath 1997). We show that the MIC statistic has chi-square limiting distribution for any regular distribution family, the estimator achieves the bestconvergence rate Op(1) and has a limiting distribution expressed via a random walk.Our asymptotic results under alternative model is obtained under the assumption that the location of the change point k , Thus, form a triangle array. The classical results on almost sure convergence for independent and identically distributed (iid) random variables cannot be directly applied. However, the conclusions on weak convergence will not be affected as the related probability statements are not affected by how one sequence is related to the other. Precautions will be taken on this issue but details will be omitted.Letwhere MIC(k) and MIC(n) are defined in (3) and (4). Note that this standardization removes the constant term in the difference of MIC(k) and MIC(n).常见模型信息准则和变点分析问题信息准则通常是用来选择统计模型的优劣。信息准则不仅支持优质数据模型，而且对简单模型具有良好的拟合。因此，在信息准则下选择模型，模型的复杂性是一个重要的因素。现有研究成果往往把模型的复杂性达到参数空间层面。虽然这一概念是建立在正则参数模型，当应用于不规则的统计模型时， 它缺乏一些可取的性能。我们在模型复杂性条件下定义变点问题和修正现有的信息准则。修正后的信息准则是探究所选模型一致性和简单限制行为。变点的位置的结果估计达到最佳收敛速度Op(1)，它的极限分布。仿真结果表明了与其他方法相比，改进后的信息准则在检验变点上具有较好的检测效果。介绍我们并非总是用一个从一些统计模型中找出的最好模型。这种模型也许只是简单的插入数据或者几乎没有解释一些问题的价值。信息准则是为了选择简单而且能解释问题的模型，例如Akaike信息准则（AIC）和西沃兹信息标准(SIC).就参数空间的维度而言，模型的复杂性常常是测量方面。考虑一个问题，推断在一个过程中是否发生改变。在模型选择中，我们想在单一参数模型或两个参数模型中选择其一，检测变点位置。AIC和SIC信息准则可以用来解决这种变点问题。在这些方面，已经有许多卓有成效的研究，譬如Hirotsu, Kuriki and Hayter (1992) and Chen and Gupta (1997)等，仅举几例而已。比较常规模式的选择问题,变点问题包含了一个特殊的参数:位置的改变。当变点过程接近的开始或结束是,两个参数变得十分勉强。因此，模型变得完全没有必要。模型复杂性也取决于转点的位置，这是个观察激励的概念。所以，我们建议生成AIC和SIC使得模型复杂性也是变点位置的一个功能点。新方法展示了一个简单限制行为和在许多仿真情况下合适的性能。近些年，文献中变点问题被广泛的讨论。变点问题的研究可以追溯到Page (1954, 1955 and 1957)，他检测变点的存在。许多研究者通过参数化探究变点问题，例如Chernoff and Zacks (1964), Hinkley (1970), Hinkley et.al.(1980), Siegmund (1986) and Worsley (1979, 1986).非参数检验和估计被提出，Brodsky and Darkhovsky, 1993; Lombard, 1987; Gombay and Huskova, 1998. 广泛的讨论在大样本行为的似然比检验统计,出现在Gombay、Horvath(1996)、Csorgo和Horvath(1997)论文中。其他变点的一些细节问题可以从一些调查文献中找到，譬如Bhattach arya(1994),Basseville和Nikiforov(1993),Zacks(1983),和Lai(1985)。本研究从其他研究精炼的背离传统的测量模型的复杂度,并且在非常一般的参数化模型的设置，确定的极限分布产生的统计量。在第二章中，我们在细节上定义一种新的信息准则。在第三章，给出在产生的试验统计数据卡限制分布和变化的点估计达到最好的收敛速度的条件。在第四章中，给出了一些仿真结果和一个应用实例。与现有的方法相比,发现新方法有良好的有限样本性质。在附件中给出了证明。主要结果X1，X2， ，Xn是独立随机变量，当ik时