John R. Collins

Maximal asymptotic biases and variances of symmetrized interquantile ranges under asymmetric contamination

In the location-scale model where random observations come from the distribution we study asymptotic robustness properties of two estimators of the unknown scale parameter ?. The two estimators studied are: (i) the interquantile range, given by the functional where [Ftilde] is a normalizing constant; and (ii) the symmetrized inter-quantile range where F is obtained by symmetrizing F about the median Maximal asymptotic biases and variances of S?(F) and S? ?(F) are found under the e-contamination model where F0 is a fixed strongly unimodal symmetric distribution and G is an unknown and possibly asymmetric distribution. Asymptotic variances are sometimes (depending on e and ?) maximized when G places all its mass in a neighborhood of ? but in some cases the asymptotic variance can be increased by shifting a small amount of mass to a neighborhood of the quantile Numerical comparisons are given for the case when F0 is the standard normal distribution

Robust estimation in the linear model with asymmetric error distributions

In the linear model Xn - 1 = Cn - p[theta]p - 1 + En - 1, Hubers theory of robust estimation of the regression vector [theta]p - 1 is adapted for two models for the partially specified common distribution F of the i.i.d. components of the error vector En - 1. In the first model considered, the restriction of F to a set [-a0, b0] is a standard normal distribution contaminated, with probability [var epsilon], by an unknown distribution symmetric about 0. In the second model, the restriction of F to [-a0, b0] is completely specified (and perhaps asymmetrical). In both models, the distribution of F outside the set [-a0, b0] is completely unspecified. For both models, consistent and asymptotically normal M-estimators of [theta]p - 1 are constructed, under mild regularity conditions on the sequence of design matrices {Cn - p}. Also, in both models, M-estimators are found which minimize the maximal mean-squared error. The optimal M-estimators have influence curves which vanish off compact sets.

Robustness Comparisons of Some Classes of Location Parameter Estimators

Asymptotic biases and variances of M-, L- and R-estimators of a location parameter are compared under ε-contamination of the known error distribution F0 by an unknown (and possibly asymmetric) distribution. For each ε-contamination neighborhood of F0, the corresponding M-, L- and R-estimators which are asymptotically efficient at the least informative distribution are compared under asymmetric ε-contamination. Three scale-invariant versions of the M-estimator are studied: (i) one using the interquartile range as a preliminary estimator of scale: (ii) another using the median absolute deviation as a preliminary estimator of scale; and (iii) simultaneous M-estimation of location and scale by Hubers Proposal 2. A question considered for each case is: when are the maximal asymptotic biases and variances under asymmetric ε-contamination attained by unit point mass contamination at ∞? Numerical results for the case of the ε-contaminated normal distribution show that the L-estimators have generally better performance (for small to moderate values of ε) than all three of the scale-invariant M-estimators studied.

Maximal Asymptotic Biases of M-Estimators of Location with Preliminary Scale Estimates

Abstract We consider the problem of finding the maximal asymptotic bias of an M-estimator of a location parameter, using a preliminary estimate of the unknown scale parameter, when the error distribution is assumed to lie in an ϵ-contamination neighborhood of a fixed symmetric unimodal distribution. The least favorable contaminating distribution is shown to put all its mass at ∞ under some fairly general conditions. A particular case considered is that of the auxiliary scale estimator being a location-invariant and scale-equivariant version of a trimmed variance.

Bias-robust L-estimators of a scale parameter

We derive optimal bias-robust L-estimators of a scale parameter σ based on random samples from F(( ·−θ/σ), where θ and σ are unknown and F is an unknown member of a ε-contaminated neighborhood of a fixed symmetric error distribution F 0. Within a very general class S of L-estimators which are Fisher-consistent at F, we solve for: (i) the estimator with minimax asymptotic bias over the ε-contamination neighborhood; and (ii) the estimator with minimum gross error sensitivity at F 0 [the limiting case of (i) as ε → 0]. The solutions to problems (i) and (ii) are shown, using a generalized method of moment spaces, to be mixtures of at most two interquantile ranges. A graphical method is presented for finding the optimal bias-robust solutions, and examples are given.

Comparisons of asymptotic biases and variances of m-estimators of scale under asymmetric contamination

We study robustness properties of two types of M-estimators of scale when both location and scale parameters are unknown: (i) the scale estimator arising from simultaneous M-estimation of location and scale; and (ii) its symmetrization about the sample median. The robustness criteria considered are maximal asymptotic bias and maximal asymptotic variance when the known symmetric unimodal error distribution is subject to unknown, possibly asymmetric, £-con-tamination. Influence functions and asymptotic variance functionals are derived, and computations of asymptotic biases and variances, under the normal distribution with e-contamination at oo, are presented for the special subclass arising from Hubers Proposal 2 and its symmetrized version. Symmetrization is seen to reduce both asymptotic bias and variance. Some complementary theoretical results are obtained, and the tradeoff between asymptotic bias and variance is discussed.

M-Estimators of Scale with Minimum Gross Errors Sensitivity

The median absolute deviation (MAD) is known to be the M-estimator of scale with minimum gross errors sensitivity (GES) when the error distribution is known to be symmetric and strongly unimodal. The problem considered here is to find the Fisher consistent M-estimator with minimum GES when the error distribution is symmetric but not necessarily unimodal. Under some general conditions, the score function χ corresponding to the minimizing M-estimator has the form χ(x) = −1 when |x| < a; χ(x) = c when a < |x| < b; χ(x) = 1 when |x| > b. An example is given in which the M-estimator with minimum GES does not correspond to the MAD.

Asymptotic relative efficiency comparisions for some robust location estimators

Following results previous obtained by DasGupta (1994), we study the problem of finding the upper and lower bounds on asymptotic relative efficiencies (ARE) among three robust estimators of location when the error distribution F lies in the class Here H is an absolutely continuous distribution; the contamination parameter ∈ is fixed, 0 < ∈ < ½; and G, the mixing distribution on the scale parameter, has fixed support [s 1,s 2], where 1 ≤ s 1 < s 2 < ∞ but is otherwise arbitrary. Our modification of DasGuptas formulation was to replace his side conditions,G[s 1, ∞) = f sdG(s) by the single side condition G[s 1,s 2]=1 The three estimators for which bounds on pairwise AREs are found are the sample mean [Xbar], the α—trimmed mean [Xbar]α , and the sample median M. In some of the cases, the extremal problems have a simple structure and are solved explicitly by moment space methods. Numerical tabulations of the bounds on AREs are presented.