Sequential monitoring for change in scale

We propose two sequential monitoring schemes for detecting a change in scale. We consider a stable historical period of length m. The goal is to propose test with asymptotically small probability of false alarm and power 1 as the length of the historical period tends to infinity. The distribution under the null hypothesis and also under the alternative hypothesis is shown. A small simulation study illustrates the finite sample performance of both monitoring schemes. Introduction The paper concerns the question of structural stability of a model. Such problems occur in number of applications, such as in economics and finance, statistical quality control, medical care etc. Precisely speaking the paper is devoted to the detection of a change in scale in location model when the data arrive sequentially and training (historical) data with no change are available. We assume that the observations Yi follow the location model Yi = μ + σi ei, 1 ≤ i < ∞, (1) where μ is unknown location parameter, {ei, 1 ≤ i < ∞} are independent identically distributed (i.i.d.) random errors satisfying further conditions specified below and {σi, 1 ≤ i < ∞} are constants determining the variance of the observations {Yi, 1 ≤ i < ∞}. Our goal is to monitor the change in variance of the observation, i.e. the change in σ i . The observations Y1, . . . , Ym are assumed to represent the training data for which the variance is constant, i.e. σ 1 = · · · = σ m = σ 0 , where σ 0 is unknown. Our problem of detection of a change can be formulated as a sequential hypothesis testing problem, where the null hypothesis corresponds to the model without any change: H0 : σ i = σ 2 0 , 1 ≤ i < ∞ (2) against the alternative that the model changes in some unknown time point m + k∗: H1 : there exists k∗ ≥ 1 such that σ i = σ 2 0 , 1 ≤ i < m + k∗, σ i = σ ∗, m + k∗ ≤ i < ∞, σ 0 6= σ ∗. (3) The k∗, σ ∗ and σ 2 0 are unknown. This represents the so called online monitoring and was originated in Chu et al. [1996] where two types of such monitoring procedures in the linear regression settings were studied. The first procedure was based on CUSUM type test statistics calculated from recursive residuals and the second one was the fluctuation test based on differences between estimates of the regression parameters. This test was generalized in Leisch et al. [2000] to the so called generalized fluctuation test. Similarly to the cumulative sum of residuals one can consider a moving sum. These MOSUM type test statistics were suggested in Zeileis et al. [2005]. All the three described monitoring procedures were compared there through a simulation study which showed that the MOSUM type statistics behave better when the change occurs later in the monitoring period. CUSUM type statistics based on ordinary residuals and on recursive residuals are studied in Horváth et al. [2004] and also in Koubková [2006]. A practical side of this testing approach was described in Zeileis et al. [2003], where tests were conducted using the R package strucchange. The method used in this paper is related to those introduced in Horváth et al. [2004] and in Koubková [2006], however it is adapted to the change in scale. 74 WDS'08 Proceedings of Contributed Papers, Part I, 74–79, 2008. ISBN 978-80-7378-065-4 © MATFYZPRESS