Uniform Consistency of Marked and Weighted Empirical Distributions of Residuals

Abstract

A uniform weak consistency theory is presented for the marked and weighted empirical distribution function of residuals. New and weaker su¢ cient conditions for uniform consistency are derived. The theory allows for a wide variety of regressors and error distributions. We apply the theory to 1-step Huber-skip estimators. These estimators describe the widespread practice of removing outlying observations from an intial estimation of the model of interest and updating the estimation in a second step by applying least squares to the selected observations. Two results are presented. First, we give new and weaker conditions for consistency of the estimators. Second, we analyze the gauge, which is the rate of false detection of outliers, and which can be used to decide the cut-o¤ in the rule for selecting outliers.

Heteroscedasticity testing after outlier removal

“…setting holds by, for instance, the least squares estimator, the least trimmed squares estimator of Rousseeuw (1984), or the MMestimator of Yohai (1987) -see Víšek (1999) for an asymptotic analysis of the least trimmed squares estimator and Marona, Martin and Yohai (2006) for the MM-estimator. In the second stage, the estimator^ de…ned in (2.2) will be p n-consistent under symmetry, if the …rst stage estimators are also p n-consistent (see Johansen and Nielsen, 2016a, 2016band Berenguer-Rico, Johansen and Nielsen, 2019a. In our simulations section, given its widespread use, special attention will be devoted to the robusti…ed least squares procedure, that is, the case in which~ ;~ are least squares estimates of and and^ is as de…ned in (2.2).…”

“…Under asymmetry, two issues arise in the context of the DAS. First, the second stage estimator will be inconsistent for the intercept -see Berenguer-Rico, Johansen and Nielsen (2019a). This inconsistency makes Assumption 2.3 invalid since n 1=2 (^ ) is not bounded in probability.…”

“…A question that applied researchers face when implementing the DAS is: what cut-o¤ value, c, to choose to detect outliers? Here, we make this choice by using the concept of gauge introduced by Hendry and Santos (2010) which, in turn, is based on an idea by Hoover and Perez (1999) in the context of variable selection -see also Johansen and Nielsen (2016b) and Berenguer-Rico, Johansen and Nielsen (2019a). The gauge is based on the number of falsely detected outliers, when there are none.…”

“…More precisely, the usual normalizing constants of the test statistic change and depend on the truncation imposed at the deselection stage and the estimation method being used. Johansen and Nielsen (2009), Chen and Bien (2017) or Berenguer-Rico, Johansen and Nielsen (2019a) show that conducting standard inferences on the parameters of the model after detecting and removing outliers can be misleading. These results are reminiscent of the phenomenon of post-selection inferencesee, for instance, Pöstcher (1991) or Leeb and Pöstcher (2005)-which is well-documented in the literature.…”

Models Where the Least Trimmed Squares and Least Median of Squares Estimators Are Maximum Likelihood

“…As maximum likelihood concept, we use pair-wise comparison of probability measures, as suggested by Kiefer and Wolfowitz (1956), and consider small hyper-cubes around the data point following Fisher (1922) and Scholz (1980). For the asymptotic analysis of the LTS model, we apply marked and weighted empirical processes of residuals (Johansen and Nielsen, 2016a;Berenguer-Rico, Johansen and Nielsen, 2019), quantile processes (Csörg½ o, 1983), and extreme value theory for the normal distribution (Leadbetter, Lindgren andRootzén, 1982, Watts 1980). For the asymptotic analysis of the LMS model, we apply results for uniform spacings (Pyke, 1965).…”