Summary Multivariate control charts are used to monitor stochastic processes for changes and unusual observations. Hotelling's T2 statistic is calculated for each new observation and an out‐of‐control signal is issued if it goes beyond the control limits. However, this classical approach becomes unreliable as the number of variables p approaches the number of observations n, and impossible when p exceeds n. In this paper, we devise an improvement to the monitoring procedure in high‐dimensional settings. We regularise the covariance matrix to estimate the baseline parameter and incorporate a leave‐one‐out re‐sampling approach to estimate the empirical distribution of future observations. An extensive simulation study demonstrates that the new method outperforms the classical Hotelling T2 approach in power, and maintains appropriate false positive rates. We demonstrate the utility of the method using a set of quality control samples collected to monitor a gas chromatography–mass spectrometry apparatus over a period of 67 days.

The traditional and readily available multivariate analysis of variance (MANOVA) tests such as Wilks' Lambda and the Pillai-Bartlett trace start to suffer from low power as the number of variables approaches the sample size. Moreover, when the number of variables exceeds the number of available observations, these statistics are not available for use. Ridge regularisation of the covariance matrix has been proposed to allow the use of MANOVA in high-dimensional situations and to increase its power when the sample size approaches the number of variables. In this paper two forms of ridge regression are compared to each other and to a novel approach based on lasso regularisation, as well as to more traditional approaches based on principal components and the Moore-Penrose generalised inverse. The performance of the different methods is explored via an extensive simulation study. All the regularised methods perform well; the best method varies across the different scenarios, with margins of victory being relatively modest. We examine a data set of soil compaction profiles at various positions relative to a ridgetop, and illustrate how our results can be used to inform the selection of a regularisation method.

In this paper we present a new redescending M-estimator "Insha's estimator" for robust regression and outliers detection that overcomes some drawbacks of other M-estimators for robust regression and outliers detection, such as destruction of the good observations and lack of simplicity in applications. The Ψ-function associated with the proposed estimator attains more linearity in the central section before it redescends, resulting in enhanced efficiency. Moreover the estimator is continuous everywhere and can be written in closed form without the use of an indictor function. The estimator is also applied to a real world example taken from the literature. For the purpose of comparison with other well-known redescending M-estimators extensive simulation study has been carried out. The example and simulation study show that using this estimator all the outliers can be successfully detected and is not affected by outliers.

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