Inference on model parameters with many L-moments
Luis Alvarez, Chang Chiann, Pedro Morettin
TL;DR
The paper develops a generalised method of L-moments (GML) estimator that allows the number of L-moments $R$ to grow with the sample size $T$, coupled with an optimal weighting scheme to harness overidentifying information. Under mild conditions, the authors prove consistency, an asymptotic linear representation, asymptotic normality, and asymptotic efficiency when $R$ grows with $T$, extending classical LM estimation beyond fixed-moment settings. Monte Carlo experiments with GEV and GPD distributions show finite-sample mean-squared-error improvements over the MLE in tail quantile estimation, with two-step weighting and a data-driven, RMSE-guided selection of $R$ providing robust gains across sample sizes. The paper also extends the approach to residual analysis in semi- and nonparametric models and to conditional quantile models, and demonstrates an empirical application to expenditure patterns on a Brazilian ridesharing platform, highlighting heavy-tailed behavior even after controlling for observed and unobserved heterogeneity. Overall, the work offers a computationally attractive, automatable alternative to MLE in small samples and a theoretically sound path to asymptotic efficiency as data grow, with broad applicability to tail risk and distributional inference.
Abstract
This paper studies parameter estimation using L-moments, an alternative to traditional moments with attractive statistical properties. The estimation of model parameters by matching sample L-moments is known to outperform maximum likelihood estimation (MLE) in small samples from popular distributions. The choice of the number of L-moments used in estimation remains ad-hoc, though: researchers typically set the number of L-moments equal to the number of parameters, which is inefficient in larger samples. In this paper, we show that, by properly choosing the number of L-moments and weighting these accordingly, one is able to construct an estimator that outperforms MLE in finite samples, and yet retains asymptotic efficiency. We do so by introducing a generalised method of L-moments estimator and deriving its properties in an asymptotic framework where the number of L-moments varies with sample size. We then propose methods to automatically select the number of L-moments in a sample. Monte Carlo evidence shows our approach can provide mean-squared-error improvements over MLE in smaller samples, whilst working as well as it in larger samples. We consider extensions of our approach to the estimation of conditional models and a class semiparametric models. We apply the latter to study expenditure patterns in a ridesharing platform in Brazil.