Joshua D. Naranjo

High-Breakdown Rank Regression

Abstract A weighted rank estimate is proposed that has 50% breakdown and is asymptotically normal at rate √n. Based on this theory, inferential procedures, including asymptotic confidence and tests, and diagnostic procedures, such as studentized residuals, are developed. The influence function of the estimate is derived and shown to be continuous and bounded everywhere in (x, Y) space. Examples show that robustness against outlying high-leverage clusters may approach that of the least median of squares, while retaining more stability against inliers. The estimator uses weights that correct for both factor and response spaces. A Monte Carlo study shows that the estimate is more efficient than the generalized rank estimates, which are generalized R estimates with weights that only correct for factor space. When weights are constant, the estimate reduces to the regular Wilcoxon rank estimate.

A pretest for choosing between logrank and wilcoxon tests in the two-sample problem

SummaryTwo commonly used tests for comparison of survival curves are the generalized Wilcoxon procedure of Gehan (1965) and Breslow (1970) and the logrank test proposed by Mantel (1966) and Cox (1972). In applications, the logrank test is often used after checking for validity of the proportional hazards (PH) assumption, with Wilcoxon being the fallback method when the PH assumption fails.However, the relative performance of the two procedures depend not just on the PH assumption but also on the pattern of differences between the two survival curves. We show that the crucial factor is whether the differences tend to occur early or late in time. We propose diagnostics to measure early-or-late differences between two survival curves. We propose a pretest that will help the user choose the more efficient test under various patterns of treatment differences.

GR-estimates for an autoregressive time series

A weighted rank-based (GR) estimate for estimating the parameter vector of an autoregressive time series is considered. When the weights are constant, the estimate is equivalent to using Jaeckels estimate with Wilcoxon scores. Asymptotic linearity properties are derived for the GR-estimate. Based on these properties, the GR-estimate is shown to be asymptotically normal at rate n1/2.

The use and interpretation of rank-based residuals

The diagnostic properties of residuals based on generalized rank-based (GR) estimates are investigated. These are bounded-influence estimates which are resistant to the effects of outliers in either the y- or x-space. Using first order approximations based on asymptotic linearity results, the interpretation of residuals based on GR-fits is discussed for true and misspecified models. A measure of departure from othogonality of both R- and GR-residuals and their corresponding fitted values is also proposed. This measure and the interpretation of GR-residuals prove insightful in an understanding of GR-residual plots. Proper standardization of GR-residuals is also discussed. This is effective in declaration of potential outliers. A small Monte Carlo study along with examples verify the asymptotic theory .

A ROBUST METHOD FOR THE ANALYSIS OF EXPERIMENTS WITH ORDERED TREATMENT LEVELS

A robust approach for the analysis of experiments with ordered treatment levels is presented as an alternative to existing approaches such as the parametric Abelson-Tukey test for monotone alternatives and the nonparametric Terpstra-Jonckheere test. The method integrates the familiar Spearman rank-order correlation with bootstrap routines to provide magnitude of association measures, p values, and confidence intervals for magnitude of association measures. The advantages of this method relative to five alternative approaches are pointed out.

Highly efficient weighted for autoregression wilcoxon estimes for autoregression

In this paper we explore the use of Schweppe-type weights for a class of weighted Wiicoxon estimates and apply the corresponding estimates to an autoregressive time series model This special class of estimates is essentially the autoregressive analog of the HBR-estimates proposed by Chang et al. (1999) in the linear regression context. Assuming a stationary finite second moment autoregressive model of order p, asymptotic linearity properties are derived for the HBR-estimate. Based on these properties, the HBR-estimate is shown to be asymptotically normal at rate nl/2. Tests of general linear hypotheses as well as standard errors for confidence interval procedures can be based on such results. In a linear regression setting, the HBR-estimate is highly efficient and inherits a totally bounded influence function and a 50percent breakdown point. Examples and a Monte Carlo study over innovated and additive outlier models indicate that these properties of the HBR-estimate are preserved in an autoregressive time series context, Thus, the HBR-estimate provides a highly efficient and robust alternative for autoregressive time series estimation.

Weighted Wilcoxon estimates for autoregression

This paper explores the class of weighted Wilcoxon (WW) estimates in the context of autoregressive parameter estimation, giving special attention to three sub-classes of so-called WW-estimates. When the weights are constant, the estimate is equivalent to using Jaeckel’s estimate with Wilcoxon scores. The paper presents asymptotic linearity properties for the three sub-classes of WW-estimates. These properties imply that the estimates are asymptotically normal at rate n½. Tests of hypotheses as well as standard errors for confidence interval procedures can be based on such results. Furthermore, the estimates can be computed with an L1 regression routine once the weights have been calculated. Examples and a Monte Carlo study over innovation and additive outlier models suggest that WW-estimates can be both robust and highly efficient.

Rank regression with estimated scores

Rank-based estimates are asymptotically efficient when optimal scores are used. This paper describes a method for estimating the optimal score function based on residuals from an initial fit. The resulting adaptive estimate is shown to be asymptotically efficient.

Diagnostics for comparing robust and least squares fits

Is a simple least squares (LS) fit appropriate for the data at hand? How different would a more robust estimate be from LS? Is a high breakdown estimator necessary, or is a highly efficient robust estimator sufficient? We propose diagnostics which help answer these questions by measuring the difference in fits between least squares and, successively, a highly efficient robust estimate and a bounded influence robust estimate. Our diagnostic TDBETAS measures the overall change in parameter estimates among these three fits, while the casewise diagnostic CFITS measures change in individual fitted values. We also propose a plot based on CFITS which provides an effective graphical summary of underlying data structure.

An efficient and high breakdown procedure for model criticism

Part of a linear model analysis is the examination of the appropriateness of the chosen model. We propose an exploratory model criticism procedure that exposes hidden outliers, clusters of outliers, or underlying curvature by using diagnostics that exploit the differences between an efficient robust fit and a high breakdown fit. Examples illustrate the procedure.