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.

翻译：本文重新审视了位置-尺度族的稳健估计与验证问题。所提出的方法利用独立同分布随机变量样本分位数的联合渐近正态性，构建了位置参数和尺度参数的普通最小二乘估计量及广义最小二乘估计量。这些分位数最小二乘估计量具有显式表达式，易于计算；其稳健性通过从最小二乘估计中排除极端分位数实现，而效率则通过使用尽可能多的非极端实际相关分位数得到提升。针对多个位置-尺度族，明确了分位数最小二乘估计量的影响函数并绘制了图形。这些影响函数与一些经典影响函数的形状高度相似，但前者是自动生成的（即无需预先指定）。建立了所提出估计量的联合渐近正态性，并通过模拟实验探究了其有限样本性质。此外，还评估了这些估计量以及极大似然估计在样本量n=10^6、10^7、10^8、10^9下的计算成本。在模型验证方面，构建了两个拟合优度检验，并通过模拟和真实数据研究了其性能。特别地，针对谷歌过去四年的日股票收益率数据，这两个检验均强烈支持逻辑分布假设，并拒绝了其他钟形分布竞争模型。