Study of partial and average conditional Kendall's tau

When the interest is in studying conditional dependencies, and more precisely the strength of conditional dependencies, some kind of averaging over the conditioning random vector may be needed. Examples of average measures that can serve in this context are the average conditional Kendall's tau and partial Kendall's tau. It is known that these measures differ in general. Some statistical tests are based on these average measures, and a better knowledge of them is of importance. The aim of this paper is to provide a quantitative study of the possible differences of these two average measures, and to establish sufficient conditions under which they coincide. Both measures are studied in two fairly general settings. In each setting theoretical results are established as well as several illustrative examples given.