Alternative boundaries for CUSUM tests

Alternative boundaries for the common Recursive (or Standard) CUSUM test and the OLS-based CUSUM test for structural change are suggested and their properties are examined by simulation of expectedp values. The poor power of the tests for early and late structural changes can be improved for the OLS-based version, whereas this weakness of the Recursive CUSUM test cannot be overcome by the new boundaries. Research supported by the Austrian Science Foundation (FWF) under grant SFB#010 (‘Adaptive Information Systems and Modeling in Economics and Management Science’).     

Multivariate process dispersion monitoring without subgrouping

The memory-type adaptive and non-adaptive control charts are among the best control charts for detecting small-to-moderate changes in the process parameter(s). In this paper, we propose the Crosier CUSUM (CCUSUM), EWMA, adaptive CCUSUM (ACCUSUM) and adaptive EWMA (AEWMA) charts for efficiently monitoring the changes in the covariance matrix of a multivariate normal process without subgrouping. Using extensive Monte Carlo simulations, the length characteristics of these control charts are computed. It turns out that the ACCUSUM and AEWMA charts perform uniformly and substantially better than the CCUSUM and EWMA charts when detecting a range of shift sizes in the covariance matrix. Moreover, the AEWMA chart outperforms the ACCUSUM chart. A real dataset is used to explain the implementation of the proposed control charts.     

Monitoring means and covariances of multivariate non linear time series with heavy tails

In this paper, the focus is on sequential analysis of multivariate financial time series with heavy tails. The mean vector and the covariance matrix of multivariate non linear models are simultaneously monitored by modifying conventional control charts to identify structural changes in the data. The considered target process is a constant conditional correlation model (cf. Bollerslev, 1990 Bollerslev, T. (1990). Modeling the coherence in short-run nominal exchange rates: A multivariate generalized ARCH model. Rev. Econ. Stat. 72:498–505.[Crossref], [Web of Science ®] , [Google Scholar]), an extended constant conditional correlation model (cf. He and Teräsvirta, 2004 He, C., Teräsvirta, T. (2004). An extended constant conditional correlation GARCH model and its fourth-moment structure. Economet. Theory 20:904–926.[Crossref], [Web of Science ®] , [Google Scholar]), a dynamic conditional correlation model (cf. Engle, 2002 Engle, R.F. (2002). Dynamic conditional correlation: A simple class of multivariate GARCH models. J. Bus. Econ. Stat. 20(3):339–350.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]), or a generalized dynamic conditional correlation model (cf. Capiello et al., 2006 Capiello, L., Engle, R., Sheppard, K. (2006). Asymmetric correlations in the dynamics of global equity and bond returns. J. Financial Economet. 4(4):537–572.[Crossref] , [Google Scholar]). For statistical surveillance we use control charts based on residuals. Further, the procedures are constructed for t-distribution. The detection speed of these charts is compared via Monte Carlo simulation. In the empirical study, the procedure with the best performance is applied to log-returns of the stock market indices FTSE and CAC.     

The Continuous Run Sum Chart

The run sum chart is an effective two-sided chart that can be used to monitor for process changes. It is known that it is more sensible than the Shewhart chart with runs rules and its performance improves as the number of regions increases. However, as the number of regions increses the resulting chart has more parameters to be defined and its design becomes more involved. In this article, we introduce a one-parameter run sum chart. This chart accumulates scores equal to the subgroup means and signals when the cummulative sum exceeds a limit value. A fast initial response feature is proposed and its run length distribution function is found by a set of recursive relations. We compare this chart with other charts suggested in the literature and find it competitive with the CUSUM, the FIR CUSUM, and the combined Shewhart FIR CUSUM schemes.     

Dynamic sampling plans in on-line control charts

Three simple dynamic sampling plans for detecting the change point are investigated in the discrete-time case. The first is a two-rate sampling CUSUM procedure. The second is a two-rate sampling Shiryayev-Roberts procedure. The third is a periodic sequential testing procedure. Two problems are discussed. First, simple design methods are provided for practical use. Second, a comparison between the three plans is made in the continuous-time case, which shows that by properly choosing the design parameters, the three plans can be made equally efficient in certain senses.     

QUALITY CONTROL WITH MULTIVARIATE DATA

The paper examines statistical process control of bivariate and multivariate data, using in particular the multivariate equivalents of the univariate Shewhart chart, CUSUM chart and the Exponentially Weighted Moving Average chart. This illustrates the usefulness of Principal Component methods in statistical process control with multivariate data.     

Testing for abrupt breaks in variance structures with smooth changes

Abstract

We investigate the problem of testing for variance breaks in the case where the variance structure is assumed to be smoothly time-varying under the null. Since the classical tests are aimed to detect any change in the variance, they are not able to distinguish between smooth non constant variance and abrupt breaks. In this paper a new procedure for detecting variance breaks taking into account for smooth changes in the variance under the null is proposed. The finite sample properties of the test we introduce are investigated by Monte Carlo experiments. The theoretical outputs are illustrated using U.S. macroeconomic data.     

PROPERTIES OF THE MINIMUM POINT OF AN UNBALANCED TWO-SIDED RANDOM WALK

In this paper, properties of minimum point of a unbalanced two-sided random walk are investigated. Under the condition that the parameters at both sides tend to zero at the same order, probabilities that the minimum point is on which side, and the second order expansions for the first two moments of the minimum point are obtained. Applications of these results are very promising. First, they can be used to study the properties of the maximum likelihood estimator for the change point in the large sample case; second, they can be used to study inference problems after CUSUM test.     

Darling–Erdős limit results for change-point detection in panel data

We wish to test the null hypothesis if the means of N panels remain the same during the observation period of length T. A quasi-likelihood argument leads to self-normalized statistics whose limit distribution under the null hypothesis is double exponential. The main results are derived assuming that the each panel is based on independent observations and then extended to linear processes. The proofs are based on an approximation of the sum of squared CUSUM processes using the Skorokhod embedding scheme. A simulation study illustrates that our results can be used in case of small and moderate N and T. We apply our results to detect change in the “corruption index”.     

Two tests for sequential detection of a change-point in a nonlinear model

In this paper, two tests, based on weighted CUSUM of the least squares residuals, are studied to detect in real time a change-point in a nonlinear model. A first test statistic is proposed by extension of a method already used in the literature but for the linear models. It is tested under the null hypothesis, at each sequential observation, that there is no change in the model against a change presence. The asymptotic distribution of the test statistic under the null hypothesis is given and its convergence in probability to infinity is proved when a change occurs. These results will allow to build an asymptotic critical region. Next, in order to decrease the type I error probability, a bootstrapped critical value is proposed and a modified test is studied in a similar way. A generalization of the Hájek–Rényi inequality is established.