The limiting distribution of a test for multivariate structure

We define a chi-squared statistic for p-dimensional data as follows. First, we transform the data to remove the correlations between the p variables. Then, we discretize each variable into groups of equal size and compute the cell counts in the resulting p-way contingency table. Our statistic is just the usual chi-squared statistic for testing independence in a contingency table. Because the cells have been chosen in a data-dependent manner, this statistic does not have the usual limiting distribution. We derive the limiting joint distribution of the cell counts and the limiting distribution of the chi-squared statistic when the data is sampled from a multivariate normal distribution. The chi-squared statistic is useful in detecting hidden structure in raw data or residuals. It can also be used as a test for multivariate normality.