Chapter 4 - Local Gaussian correlation and dependence

Abstract

In the present chapter, we introduce and described the basic concepts of the book. The chapter starts with some introductory material on local dependence measures and local parametric models, and we explain how this leads to the introduction of the local Gaussian correlation. We explain this concept and its properties in some detail, including description of heavy tails. We also evaluate it in the context of the so-called Rényi criteria for a dependence measure. We show how it can be estimated via the local log likelihood and derive asymptotic results both for a fixed bandwidth and for the case where the bandwidth tends to zero. In particular, we derive the convergence rates for these two cases. We give a normalized version of the local Gaussian correlation by transforming the marginals to standard normals or pseudo-standard normals. We point out the problem of the curse of dimensionality and suggest a pairwise simplification as a possible approximate solution. We illustrate the concepts on simulated and real data examples.