Quantile Based Estimation of Scale and Dependence

Abstract

QUANTILE BASED ESTIMATION OF SCALE AND DEPENDENCE Garth Tarr 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. Chapter 1 addresses the need for reliable confidence intervals for quantile regression coefficients particularly in small samples. We demonstrate the competitive performance of the xy-pair quantile bootstrap approach in a broad range of model designs with a focus on small and moderate sample sizes. Chapter 2 forms the core of this thesis with its investigation into robust estimation of scale. Common robust estimators of scale such as the interquartile range and the median absolute deviation from the median are inefficient when the observations come from a Gaussian distribution. We present a new robust scale estimator, Pn, which is proportional to the interquartile range of the pairwise means. When the underlying distribution is Gaussian, Pn trades some robustness for high Gaussian efficiency. Chapter 3 extends our robust scale estimator to the bivariate setting. We show that the resulting covariance estimator inherits the robustness and efficiency properties of the underlying scale estimator. We also consider the problem of estimating scale and autocovariance in dependent processes. We establish the asymptotic normality of Pn under short and mildly long range dependent Gaussian processes. In the case of extreme long range dependence, we prove a non-normal limit result for the interquartile range. Simulation suggests that an equivalent result holds for Pn. Chapter 4 looks at the problem of estimating covariance and precision matrices under cellwise contamination. A pairwise approach is shown to perform well under much higher levels of contamination than standard robust techniques would allow. Our approach works well with high levels of scattered contamination and has the advantage of being able to impose sparsity on the resulting precision matrix.