Assessing the effect of kurtosis deviations from Gaussianity on conditional distributions

• We study the impact of changes in kurtosis on marginal and conditional distributions.
• A relative sensitivity measure of the conditional distributions is proposed.
• For large dimensions the effect of non-normality depends only on the variables size.

The multivariate exponential power family is considered for n  -dimensional random variables, ZZ, with a known partition Z≡(Y,X)Z≡(Y,X) of dimensions p   and n-pn-p, respectively, with interest focusing on the conditional distribution Y|XY|X. An infinitesimal variation of any parameter of the joint distribution produces perturbations in both the conditional and marginal distributions. The aim of the study was to determine the local effect of kurtosis deviations using the Kullback–Leibler divergence measure between probability distributions. The additive decomposition of this measure in terms of the conditional and marginal distributions, Y|XY|X and XX, is used to define a relative sensitivity measure of the conditional distribution family {Y|X=x}{Y|X=x}. Finally, simulated results suggest that for large dimensions, the measure is approximately equal to the ratio p/np/n, and then the effect of non-normality with respect to kurtosis depends only on the relative size of the variables considered in the partition of the random vector.