Confidence sets for high-dimensional empirical linear prediction (HELP) models with dependent error structure

This paper is concerned with the exact joint tail distributions of order statistics for i.i.d. random variables with arbitrary common distribution function. It is pointed out that the left tail distributions can be expressed in terms of Abel–Gontcharoff polynomials, while the right tail distributions can be written using Appell polynomials. This property is exploited to obtain, in a simple and unified way, closed forms and recursions for the evaluation of the corresponding probabilities. The Kolmogorov–Smirnov goodness-of-fit test for discrete models is discussed as a special case.