Change detection in hydrological records—a review of the methodology / Revue méthodologique de la détection de changements dans les chroniques hydrologiques <s> Cumulative deviations and other CUSUM tests <s> Abstract An alternative to the hypothesis that the sequence X 1 ···, X n are independent and identically distributed normal random variables, with mean μ and variance σ2, is that the location parameter μ shifts at some unknown instant. The null distributions of likelihood ratio test statistics are given by Hawkins (1977) for the two cases of known and unknown σ2. Unfortunately, the null distribution for unknown σ2 obtained in that article is incorrect. In this article the correct null distribution is found and a numerical integration technique is used to obtain standard percentage points for n = 3(1)10. A Monte Carlo method is used to obtain additional standard percentage points for n = 15(5)50. <s> BIB001 </s> Change detection in hydrological records—a review of the methodology / Revue méthodologique de la détection de changements dans les chroniques hydrologiques <s> Cumulative deviations and other CUSUM tests <s> Abstract Cumulative deviations from the mean are often used in the analysis of homogeneity. Features of five tests on the cumulative deviations are discussed. Some of these tests have optimal properties in testing the null hypothesis of homogeneity against a shift in the mean at an unknown point. Together with the classical von Neumann ratio the tests were applied to the annual amounts of 30-yr. rainfall records in The Netherlands. For a large number of records strong indications for a change in the mean were found. There were only small differences between the various test-statistics with respect to the number of records for which the null hypothesis was rejected. <s> BIB002 </s> Change detection in hydrological records—a review of the methodology / Revue méthodologique de la détection de changements dans les chroniques hydrologiques <s> Cumulative deviations and other CUSUM tests <s> A practical problem in time-series analysis of hydrological and meteorological data is to find statistical techniques for testing for an abrupt change in the mean at an unknown time. Suitable techniques have been developed in the situation of a single time series, {yi}. Attention is paid to the likelihood ratio statistic V and to a Bayesian statistic U. ::: ::: Complications arise if we want to test for a shift in the mean using a regression on a second correlated sequence, {xi}, because the critical values of the test statistic in general depend on the configuration of the xi's. For the statistic U, this problem can be solved using techniques similar to those for testing for serial correlation in least-squares regression. A so-called bounds test can be performed on the least-squares residuals. Unfortunately, there is quite a large possibility that this test is inconclusive. As an alternative to the bounds test, the statistic U can be applied to transformed residuals. The tests are illustrated with runoff and precipitation data for the Colorado River Basin, U.S.A., and the River Thames at Teddington, U.K. <s> BIB003
The cumulative deviation test BIB002 is based on the rescaled cumulative sum of the deviations from the mean. The test is relatively powerful in comparison with other tests (e.g. Worsley likelihood ratio test; BIB002 for a change-point that occurs towards the centre of the time series. The basic test assumes normally distributed data. Other CUSUM based tests (using Bayesian and likelihood methods) are described in BIB003 . 6. Student's t test This is a standard parametric test for testing whether two samples have different means. In its basic form it assumes normally distributed data and a known change-point time. BIB001 is similar to Student's t test but is suitable for use when the change-point time is unknown. It assumes normality.