Do mass extinctions grade continuously into the background extinctions occurring throughout the history of life, or are they a fundamentally distinct phenomenon that cannot be explained by processes responsible for background extinction? Various criteria have been proposed for addressing this question, including approaches based on physical mechanisms, ecological selectivity, and statistical characterizations of extinction intensities.

Here I propose a framework defining three types of continuity of mass and background extinctions—continuity of cause, continuity of effect, and continuity of magnitude. I test the third type of continuity with a statistical method based on kernel density estimation. Previous statistical approaches typically have examined quantitative characteristics of mass extinctions (such as metrics of extinction intensity) and compared them with the distribution of such characteristics associated with background extinctions. If mass extinctions are outliers, or are separated by a gap from background extinctions, the distinctness of mass extinctions is supported.

In this paper I apply Silverman's Critical Bandwidth Test to test for the continuity of mass extinctions by applying kernel density estimation and bootstrap modality testing. The method improves on existing work based on searching for gaps in histograms, in that it does not depend on arbitrary choices of parameters (such as bin widths for histograms), and provides a direct estimate of the significance of continuities or gaps in observed extinction intensities. I am thus able to test rigorously whether differences between mass extinctions and background extinctions are statistically significant.

I apply the methodology to Sepkoski's database of Phanerozoic marine genera. I conclude that mass and background extinctions appear to be continuous at this third level—continuity of magnitude—even though evidence suggests that they are discontinuous at the first and second levels.