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Modern methods for detecting changes in the scale or covariance of multivariate distributions rely primarily on testing for the constancy of the covariance matrix. These depend on higher‐order moment conditions, and also do not work well when the dimension of the data is large or even moderate relative to the sample size. In this paper, we propose a nonparametric change point test for multivariate data using rankings obtained from data depth measures. As the data depth of an observation measures its centrality relative to the sample, changes in data depth may signify a change of scale of the underlying distribution, and the proposed test is particularly responsive to detecting such changes. We provide a full asymptotic theory for the proposed test statistic under the null hypothesis that the observations are stable, and natural conditions under which the test is consistent. The finite sample properties are investigated by means of a Monte Carlo simulation, and these along with the theoretical results confirm that the test is robust to heavy tails, skewness and high dimensionality. The proposed methods are demonstrated with an application to structural break detection in the rate of change of pollutants linked to acid rain measured in Turkey lake, a lake in central Ontario, Canada. Our test suggests a change in the rate of acid rain in the late 1980s/early 1990s, which coincides with clean air legislation in Canada and the US. The Canadian Journal of Statistics 48: 417–446; 2020 © 2020 Statistical Society of Canada

a b s t r a c tThe main goal of the current study is to take advantage of advanced numerical and intelligent tools to predict the speed of a vehicle using time series. It is clear that the uncertainty caused by temporal behavior of the driver as well as various external disturbances on the road will affect the vehicle speed, and thus, the vehicle power demands. The prediction of upcoming power demands can be employed by the vehicle powertrain control systems to improve significantly the fuel economy and emission performance. Therefore, it is important to systems design engineers and automotive industrialists to develop efficient numerical tools to overcome the risk of unpredictability associated with the vehicle speed profile on roads. In this study, the authors propose an intelligent tool called evolutionary least learning machine (E-LLM) to forecast the vehicle speed sequence. To have a practical evaluation regarding the efficacy of E-LLM, the authors use the driving data collected on the San Francisco urban roads by a private Honda Insight vehicle. The concept of sliding window time series (SWTS) analysis is used to prepare the database for the speed forecasting process. To evaluate the performance of the proposed technique, a number of well-known approaches, such as auto regressive (AR) method, back-propagation neural network (BPNN), evolutionary extreme learning machine (E-ELM), extreme learning machine (ELM), and radial basis function neural network (RBFNN), are considered. The performances of the rival methods are then compared in terms of the mean square error (MSE), root mean square error (RMSE), mean absolute percentage error (MAPE), median absolute percentage error (MDAPE), and absolute fraction of variances (R 2 ) metrics. Through an exhaustive comparative study, the authors observed that E-LLM is a powerful tool for predicting the vehicle speed profiles. The outcomes of the current study can be of use for the engineers of automotive industry who have been seeking fast, accurate, and inexpensive tools capable of predicting vehicle speeds up to a given point ahead of time, known as prediction horizon (H P ), which can be used for designing efficient predictive powertrain controllers.

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