Robust tobit regression for censored observations using extended Box–Cox transformations

Truncated regression data often occur in measuring consumer behaviour involving infrequent purchases. We consider data with truncation of the upper and lower tails of the response distribution at arbitrary values. If, in addition, the distribution of responses is also skewed, we use robust transformations of the response with the parametric Yeo–Johnson transformation to provide approximate normality for both positive and negative responses. Tests for the value of the transformation parameter use the signed square root of the loglikelihood ratio test. To achieve robustness we use the Forward Search which fits the model to data subsets of increasing size and so orders the observations by closeness to the fitted model. Monitoring the statistic for transformation during the Forward Search indicates an appropriate transformation. We initially exhibit the properties of our procedure on simulated data. Our practical regression analysis is of 493 observations derived from loyalty card data. 100 of the responses are censored at zero and there are ten explanatory variables.