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Sequential Change-Point Detection in GARCH(p,q) Models

We suggest a sequential monitoring scheme to detect changes in the parameters of a GARCH(p,q) sequence. The procedure is based on quasi-likelihood scores and does not use model residuals. Unlike for linear regression models, the squared residuals of nonlinear time series models such as generalized autoregressive conditional heteroskedasticity (GARCH) do not satisfy a functional central limit theorem with a Wiener process as a limit, so its boundary crossing probabilities cannot be used. Our procedure nevertheless has an asymptotically controlled size, and, moreover, the conditions on the boundary function are very simple; it can be chosen as a constant. We establish the asymptotic properties of our monitoring scheme under both the null of no change in parameters and the alternative of a change in parameters and investigate its finite-sample behavior by means of a small simulation study.

An application of the maximum likelihood test to the change-point problem

A maximum-likelihood-type statistic is derived for testing a sequence of observations for no change in the parameter against a possible change. We prove that the limit distribution of the suitably normalized and centralized statistic is double exponential under the null hypothesis.

Monitoring parameter change in AR(p) time series models

Sequential tests that are generalizations of Pages CUSUM tests are proposed for detecting an abrupt change in any parameter, or in any collection of parameters of an autoregressive time series model. These tests accommodate nuisance parameters. They are based on large sample approximations to the efficient score vector under the null hypothesis of no change and under the alternative. The empirical power of the tests is evaluated in a simulation study. The new method performs better than the existing ones found in the literature if the criterion is the type I error probability, which can be unacceptably high for methods that minimize the expected value of the reaction time.

Sequential Change-Point Detection and Estimation

Abstract Two groups of sequential testing procedures are proposed to detect an abrupt change in the distribution of a sequence of observations: truncated and open ended. They are based on large sample strong approximations of the efficient score vector under the null hypothesis of no change and under the alternative hypothesis. An estimator of the time of change is proposed and its approximate bias is analyzed. The estimation of the new parameters that describe the changed distribution naturally follows.

Approximations for the time of change and the power function in change-point models

Assuming that the observations are from an exponential family we obtain the asymptotic distribution of the maximum likelihood estimator of the time of change. We also prove that the maximum likelihood ratio test is asymptotically normal, if there is a change in the parameters at an unknown time.

The weighted sequential likelihood ratio

Strong approximation of the maximum-likelihood-ratio statistic by a diffusion process is given. This allows one to develop statistics using different weight functions. Sequential tests obtained include the ones earlier defined by Barbour (1979). The precision of the approximations is examined.

Limit theorems for change in linear regression

We consider some tests to detect a change-point in a multiple linear regression model. The tests are based on the maxima of the weighted cumulative sums processes. The limit distributions may be double exponential or maxima of Gaussian processes depending on the set where the maximum of the weighted cumulative sums of residuals is taken. The design-points can be fixed or random. We also give a few applications of our results.

Change‐points and bootstrap

We show that the weighted bootstrap can be used to detect possible changes in the distribution of random vectors. We illustrate our method with change-point detection in the monthly precipitation and water discharges from Malo Raztoka.

Rates of convergence for U-statistic processes and their bootstrapped versions

U-statistic processes are often used to detect a possible change in the distributions of the observations. We obtain the exact rate of convergence in some limit theorems for U-statistics. We discuss the application of the weighted bootstrap to change-point analysis. We show that the bootstrap approximation for U-statistics is as good as the large sample approximations using Gaussian processes. However, the bootstrap approximation is much better when the limit distributions are extreme values.

Sequential change-point detection with likelihood ratios

We consider the problem of sequential change-point detection when the family of distributions is exponential, and distinguish between parameters of interest, and nuisance parameters. Likelihood ratios are used as test statistics, and their large sample approximations under the alternative hypothesis of change are given. Our formulae allow type II error approximations and they suggest different schemes for change detection and change-point estimation.