Links with Other Test Statistics and Measures of Divergence

Abstract

In the preceding chapters, we have concentrated on test statistics that measure the degree of divergence between the observed frequencies for a group of cells and the corresponding expected frequencies based on some null model. Throughout we have assumed that we are dealing with discrete counts. If the variable under observation has a continuous (in general, multivariate) distribution, we assume that the outcomes have been grouped into mutually exclusive intervals (in general, compact regions) or cells, whose union contains the support (i.e., the region where the density is positive) of the random variable. We can then use the frequencies with which observations from a sample fall in these cells to perform a goodness-of-fit test (as described in the previous chapters). Such grouping of continuous data necessarily results in some loss of information, and we may wish to use an alternative test statistic that is more efficient in these circumstances. The first two sections of this chapter are devoted to discussing test statistics that are designed for continuous data. In Section 7.3 we draw some comparisons between these statistics and the power-divergence statistic based on the grouping described earlier.