The classical quantile approximation for the sample mean, based on the central limit theorem, has been proved to fail when the sample size is small and we approach the tail of the distribution. In this paper we will develop a second order approximation formula for the quantile which improves the classical one under heavy tails underlying distributions, and performs very accurately in the upper tail of the distribution even for relatively small samples.

Perez’s approximations of probability distributions by dependence structure simplification were introduced in 1970s, much earlier than graphical Markov models. In this paper we will recall these Perez’s models, formalize the notion of a compatible system of elementary simplifications and show the necessary and sufficient conditions a system must fulfill to be compatible. For this we will utilize the apparatus of compositional models.

Let (X,Y) be a random vector with joint probability measure σ and with margins μ and ν. Let ${\left(Pₙ\right)}_{n\in \mathbb{N}}$ and ${\left(Qₙ\right)}_{n\in \mathbb{N}}$ be two bases of complete orthonormal polynomials with respect to μ and ν, respectively. Under integrability conditions we have the following polynomial expansion: $\sigma (dx,dy)={\sum }_{n,k\in \mathbb{N}}{\varrho }_{n,k}Pₙ\left(x\right)Q_k\left(y\right)\mu \left(dx\right)\nu \left(dy\right)$. In this paper we consider the problem of changing the margin μ into μ̃ in this expansion. That is the case when μ is the true (or estimated) margin and μ̃ is its approximation. It is shown that a new joint probability with new margins...