Nonparametric estimation of distributions of order statistics with application to nuclear engineering

In this paper we focus on the modelling of d -dimensional random vectors with fixed marginals and components which verify an ordering constraint almost surely. We aim to calculate the distribution which maximizes the Shannon-entropy under these constraints. We provide the solution with explicit formulas to the maximum entropy problem, which has a particular form. Namely, the density of the optimal joint distribution becomes a product of univariate functions on the support of the random vector. To exploit the special structure of such distributions, we propose a nonparametric approach to estimate the density of the joint distribution based on an available sample. We propose an exponential model based on series of quasi-orthogonal polynomials specially designed to suit this particular structure. We show that by exploiting the special features of the model, we achieve a fast convergence rate which depends only linearly on the dimension of our random vector, thus does not suffer from the curse of dimensionality which arises when dealing with high-dimensional density estimation problems. We apply the proposed method to an industrial application case involving the estimation of mechanical flaw dimensions in a component of a power plant, with experimental data available. We compare the results obtained with the maximum entropy approach to previously considered modelling schemes.