Quantile Least Squares: A Flexible Approach for Robust Estimation and Validation of Location-Scale Families

Abstract: In this paper, the problem of robust estimation and validation of location-scale families is revisited. The proposed methods exploit the joint asymptotic normality of sample quantiles (of i.i.d random variables) to construct the ordinary and generalized least squares estimators of location and scale parameters. These quantile least squares (QLS) estimators are easy to compute because they have explicit expressions, their robustness is achieved by excluding extreme quantiles from the least-squares estimation, and efficiency is boosted by using as many non-extreme quantiles as practically relevant. The influence functions of the QLS estimators are specified and plotted for several location-scale families. They closely resemble the shapes of some well-known influence functions yet those shapes emerge automatically (i.e., do not need to be specified). The joint asymptotic normality of the proposed estimators is established, and their finite-sample properties are explored using simulations. Also, computational costs of these estimators, as well as those of MLE, are evaluated for sample sizes n = 10^6, 10^7, 10^8, 10^9. For model validation, two goodness-of-fit tests are constructed and their performance is studied using simulations and real data. In particular, for the daily stock returns of Google over the last four years, both tests strongly support the logistic distribution assumption and reject other bell-shaped competitors.