Test for Parameter Change Based on the Estimator Minimizing Density-based Divergence Measures

In this paper we consider the problem of testing for a parameter change based on the cusum test proposed by Lee et al. The cusum test statistic is constructed via employing the estimator minimizing density-based divergence measures. It is shown that under regularity conditions, the test statistic has the limiting distribution of the sup of standard Brownian bridge. Simulation results demonstrate that the cusum test is robust when outliers exist. The problem of testing for parameter changes in statistical models has a long history. It originally started in the quality control context and then has been extended to various areas such as economics, finance, medicine, and seismic signal analysis. Since the paper of Page (1955), there have been published a vast amount of articles. For a general review of the change point problem, see CsSrg5 and Horv~th (1997) and the papers therein. In iid samples, the parametric approach based on the likelihood was taken by many authors (cf. Chan and Gupta (2000)). However, the parametric approach is not proper when no assumptions are imposed on the underlying distribution of observations. For instance, no parametric approaches are directly applicable to the test for changes in the autocorrelations of stationary time series. To overcome such a problem, Lee et al. (2003) devised a cusum test in the same spirit of Incls and Tiao (1994). The idea of the cusum test is the same as the one for the mean and variance change, but it includes a large number of other cases, such as the autoregressive coefficient in the random coefficient autoregressive models and the ARCH parameters. The cusum test has merit that it can test the existence of change points and, at the same time, allocate their locations. Furthermore, one can employ any estimators in construction of the cusum test as long as they satisfy certain regularity conditions. For instance, when there is a concern about outliers, a robust estimator can be utilized. Recently, Basu et al. (1998) (BHHJ in the sequel) introduced a new estimation procedure minimizing a density-based divergence measures, called density power divergences. Compared to other density-based methods, such as Beran (1977), Tamura and Boos (i 986) and Simpson (i 987), which use the Hellinger distance, and Basu and Lindsey (1994) and Cao et al. (1995), the new method has an advantage of requiring no smoothing methods. In this case, one can avoid the drawbacks and difficulties, like the …