Before revision:  We also construct a novel approximation-based estimator for the sorted distribution with a near-optimal concentration property without any sample splitting, where as a byproduct we obtain tight trade-offs between the polynomial approximation error and the maximum magnitude of coefficients in the Poisson approximation of 1-Lipschitz functions.
After revision:  We also construct a novel approximation-based estimator for the sorted distribution with a near-optimal concentration property without any sample splitting, where as a byproduct we obtain better trade-offs between the polynomial approximation error and the maximum magnitude of coefficients in the Poisson approximation of 1-Lipschitz functions.