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The power of monitoring: how to make the most of a contaminated multivariate sample. Statistical Methods and Applications.

The Box-Cox power transformation family for non-negative responses in linear models has a long and interesting history in both statistical practice and theory, which we summarize. The relationship between generalized linear models and log transformed data is illustrated. Extensions investigated include the transform both sides model and the Yeo-Johnson transformation for observations that can be positive or negative. The paper also describes an extended Yeo-Johnson transformation that allows positive and negative responses to have different power transformations. Analyses of data show this to be necessary. Robustness enters in the fan plot for which the forward search provides an ordering of the data. Plausible transformations are checked with an extended fan plot. These procedures are used to compare parametric power transformations with nonparametric transformations produced by smoothing.

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Summary We analyse data on the performance of investment funds, 99 out of 309 of which report a loss, and on the profitability of 1405 firms, 407 of which report losses. The problem in both cases is to use regression to predict performance from sets of explanatory variables. In one case, it is clear from scatter plots of the data that the negative responses have a lower variance than the positive responses and a different relationship with the explanatory variables. Because the data include negative responses, the Box–Cox transformation cannot be used. We develop a robust version of an extension to the Yeo–Johnson transformation which allows different transformations for positive and negative responses. Tests and graphical methods from our robust analysis enable the detection of outliers, the assessment of values of the two transformation parameters and the building of simple regression models. Performance comparisons are made with non‐parametric transformations.