SOLVING NOT COMPLETELY INTEGRABLE QUANTILE PFAFFIAN DIFFERENTIAL EQUATIONS

The present work deals with quantile Pfaffian differential equations which are constructed using two-dimensional conditional quantiles of multidimensional probability distributions. As it was shown in [3] in case when the initial probability distributions have reproducible conditional quantiles this kind of Pfaan equations is completely integrable and the integral manifold is the conditional quantile of maximum dimension. In this paper we discuss properties of integral manifolds of maximum possible dimension for quantile Pfaffian equations which are not completely integrable. Manifolds of this type are described in terms of conditional quantiles of intermediate dimensions.

multidimensional probability distributions, conditional quantile reproducibility, quantile Pfaffian equations, Darboux class of differential 1-forms, first integrals, manifolds of maximum dimensions, mixture distributions