Universal Residuals: A Multivariate Transformation

Abstract

Rosenblatt's transformation has been used extensively for evaluation of model goodness-of-fit, but it only applies to models whose joint distribution is continuous. In this paper we generalize the transformation so that it applies to arbitrary probability models. The transformation is simple, but has a wide range of possible applications, providing a tool for exploratory data analysis and formal goodness-of-fit testing for a very general class of probability models. The method is demonstrated with specific examples.

Figure 1

Test data generated from the generalized linear model (3), along with different types of residuals. Top left: simulated observations {y_i, i = 1, … , 5000}. Top right: residuals, under correct model specification, obtained using T₂ and applying the inverse cumulative distribution Φ⁻¹ of a standard normal. Bottom left: Anscombe residuals, under the correct model specification. Bottom right: residuals obtained using T₂ and applying Φ⁻¹, under a constant rate (incorrect) model specification.