Quantile Based Estimation of Scale and Dependence

Abstract

The sample quantile has a long history in statistics. The aim of this thesis is to explore some further applications of quantiles as simple, convenient and robust alternatives to classical procedures. The first application we consider is estimating confidence intervals for quantile regression coefficients, however, the core of this thesis is the development of a new, quantile based, robust scale estimator and its extension to autocovariance estimation in the time series setting and precision matrix estimation in the multivariate setting.Chapter 1 addresses the need for reliable confidence intervals for quantile regression coefficients, particularly in small samples. The existing methods for constructing confidence intervals tend to be based on complex asymptotic arguments and little is known about their finite sample performance. We consider taking xy-pair bootstrap samples and calculating the corresponding quantile regression coefficient estimates for each sample. Instead of estimating a covariance matrix based on these bootstrap samples, our approach is to take the appropriate upper and lower quantiles of the bootstrap sample estimates as the bounds of the confidence interval. The resulting confidence interval estimate is not necessarily symmetric, only covers admissible parameter values and is shown to have good coverage properties. This work demonstrates the competitive performance of our quantile based approach in a broad range of model designs with a focus on small and moderate sample sizes. These results were published in [5].A reliable estimate of the scale of the residuals from a regression model is often of interest, whether it be parametrically estimating confidence intervals, determining a goodness-of-fit measure, performing model selection, or identifying

Modern Nonparametric, Robust and Multivariate Methods

“…Our approach is similar in spirit as Tarr et al (2015) (see also Tarr 2014), but we emphasize the difference in Sect. Our approach is similar in spirit as Tarr et al (2015) (see also Tarr 2014), but we emphasize the difference in Sect.…”

Robust high-dimensional precision matrix estimation

“…In this paper, we consider different high-dimensional precision matrix estimators robust to cellwise contamination. Our approach is similar in spirit as Tarr et al [2015] [see also Tarr, 2014], but we emphasize the difference in Section 3. We start with pairwise robust correlation estimates from which we then estimate a covariance matrix by multiplication with robust standard deviations.…”

Robust High-Dimensional Precision Matrix Estimation

“…Afterwards the OGK estimate is applied to obtain a positive semidefinite covariance estimate. This method has been fine tuned by Tarr et al [2015] who use pairwise covariances instead of correlations [see also Tarr, 2014]. This matrix is then plugged into the graphical lasso (and similar techniques) instead of the sample covariance matrix, resulting in a sparse precision matrix estimate.…”

“…Our approach is similar in spirit as Tarr et al [2015] [see also Tarr, 2014], but we emphasize the difference in Section 3. We start with pairwise robust correlation estimates from which we then estimate a covariance matrix by multiplication with robust standard deviations.…”