L-moments under Nuisance Regression

The L-moments are analogues of the conventional moments and have similar interpretations. They are calculated using linear combinations of the expectation of ordered data. In practice, L-moments must usually be estimated from a random sample drawn from an unknown distribution as a linear combination of ordered statistics. Jureckova and Picek (2014) showed that averaged regression quantile is asymptotically equivalent to the location quantile. We therefore propose a generalization of L-moments in the model with nuisance regression using the averaged regression quantiles.