Authors: Nicolás Hernández Pérez, Santiago Carrillo Menéndez and Luis Seco.
The classical problem in statistics of estimating an unknown distribution from a given a series of observations is approached from the point of view of interpolating primary features of the shape of any distribution. Unlike traditional approaches that aim at matching descriptive measures based on algebraic moments, we choose to match more robust statistics: the L-moments. Our main contribution is to present a new system of parametric families that are capable of interpolating an arbitrary ﬁnite set of L-moments. Our methodology is based on the representation of certain subsets of quantile functions by means of positive measures and the concept of entropy for making up for missing information. This approach also allows to incorporate additional constraints of a more qualitative nature such as unimodality.
The calibration of these families is accomplished by simply matching sample L-moments. We justify the feasibility of this method almost surely, and discuss a numerically tractable algorithm for its implementation.