Robust estimation of scale of an exponential distribution

We consider the standardized median as an estimator of scale for exponential samples which is most B-robust in the sense of H ampel et al. (1986). This estimator is compared with two other estimators which were proposed to R ousseeuw and C roux (1993) but for a Gaussian model. All three estimators have the same breakdown point, but their bias curves are different. It is shown that under a gross error model the explosion bias curve of the most B-robust estimator performs better than the bias curves of the other estimators. But this estimator is worse than the two estimators proposed by R ousseeuw and C roux (1993) if the implosion bias curve is considered.     

The minimum weighted covariance determinant estimator

In this paper, we introduce weighted estimators of the location and dispersion of a multivariate data set with weights based on the ranks of the Mahalanobis distances. We discuss some properties of the estimators like the breakdown point, influence function and asymptotic variance. The outlier detection capacities of different weight functions are compared. A simulation study is given to investigate the finite-sample behavior of the estimators. The research of Stefan Van Aelst was supported by a grant of the Fund for Scientific Research-Flanders (FWO-Vlaanderen) and by IAP research network grant nr. P6/03 of the Belgian government (Belgian Science Policy).     

Jackknife estimators of a relative risk in 2×2 and 2×2×K contingency tables

Several jackknife estimators of a relative risk in a single 2×2 contingency table and of a common relative risk in a 2×2× K contingency table are presented. The estimators are based on the maximum likelihood estimator in a single table and on an estimator proposed by Tarone (1981) for stratified samples, respectively. For the stratified case, a sampling scheme is assumed where the number of observations within each table tends to infinity but the number of tables remains fixed. The asymptotic properties of the above estimators are derived. Especially, we present two general results which under certain regularity conditions yield consistency and asymptotic normality of every jackknife estimator of a bunch of functions of binomial probabilities.     

Maxbias curves of robust scale estimators based on subranges

A maxbias curve is a powerful tool to describe the robustness of an estimator. It is an asymptotic concept which tells how much an estimator can change due to a given fraction of contamination. In this paper, maxbias curves are computed for some univariate scale estimators based on subranges: trimmed standard deviations, interquantile ranges and the univariate Minimum Volume Ellipsoid (MVE) and Minimum Covariance Determinant (MCD) scale estimators. These estimators are intuitively appealing and easy to calculate. Since the bias behavior of scale estimators may differ depending on the type of contamination (outliers or inliers), expressions for both explosion and implosion maxbias curves are given. On the basis of robustness and efficiency arguments, the MCD scale estimator with 25% breakdown point can be recommended for practical use. Received: February 2000     

Robust explicit estimators of Weibull parameters

The Weibull distribution plays a central role in modeling duration data. Its maximum likelihood estimator is very sensitive to outliers. We propose three robust and explicit Weibull parameter estimators: the quantile least squares, the repeated median and the median/Q _n estimator. We derive their breakdown point, influence function, asymptotic variance and study their finite sample properties in a Monte Carlo study. The methods are illustrated on real lifetime data affected by a recording error.     

The identification of outliers in exponential samples

In this paper, the task of identifying outliers in exponential samples is treated conceptionally in the sense of DAVIES and GATHER (1989, 1993) by means of a so-called outlier region. In case of an exponential distribution, an empirical version of such a region – also called an outlier identifier – is mainly dependent on some estimator of the unknown scale parameter. The worst-case behaviour of several reasonable outlier identifiers is investigated thoroughly and it is shown that only robust estimators of scale should be used to construct reliable identifiers. These findings lead to the recommendation of an outlier identifier that is based on a standardized version of the sample median.     

Estimation in Surveys Using Conditional Inclusion Probabilities: Simple Random Sampling

In survey sampling, auxiliary information on the population is often available. The aim of this paper is to develop a method which allows one to take into account such auxiliary information at the estimation stage by means of conditional bias adjustment. The basic idea is to attempt to construct a conditionally unbiased estimator. Four estimators that have a small conditional bias with respect to a statistic are proposed. It is shown that many of the estimators used in the literature in the case of simple random sampling can be obtained by using this estimation principle. The problem of simple random sampling with replacement, poststratification, and adjustment of a 2 x 2 dimensional contingency table to marginal totals are discussed in the conditional framework. Finally it is shown that the regression estimator can be viewed as an approximation of an application of the conditional principle.     

Exponential deconvolution: two asymptotically equivalent estimators

Two isotonic estimators for the distribution function in a specific deconvolution model, the exponential deconvolution model, are considered. The first estimator is a least squares projection of a naive estimator for the distribution function on the set of distribution functions. The second estimator is the well known maximum likelihood estimator. The two estimators are shown to be first order asymptotically equivalent at a fixed point.     

Combining kernel estimators in the uniform deconvolution problem

We construct a density estimator and an estimator of the distribution function in the uniform deconvolution model. The estimators are based on inversion formulas and kernel estimators of the density of the observations and its derivative. Initially the inversions yield two different estimators of the density and two estimators of the distribution function. We construct asymptotically optimal convex combinations of these two estimators. We also derive pointwise asymptotic normality of the resulting estimators, the pointwise asymptotic biases and an expansion of the mean integrated squared error of the density estimator. It turns out that the pointwise limit distribution of the density estimator is the same as the pointwise limit distribution of the density estimator introduced by Groeneboom and Jongbloed (Neerlandica, 57, 2003, 136), a kernel smoothed nonparametric maximum likelihood estimator of the distribution function.     

Likelihood estimation and inference in threshold regression

This paper studies likelihood-based estimation and inference in parametric discontinuous threshold regression models with i.i.d. data. The setup allows heteroskedasticity and threshold effects in both mean and variance. By interpreting the threshold point as a “middle” boundary of the threshold variable, we find that the Bayes estimator is asymptotically efficient among all estimators in the locally asymptotically minimax sense. In particular, the Bayes estimator of the threshold point is asymptotically strictly more efficient than the left-endpoint maximum likelihood estimator and the newly proposed middle-point maximum likelihood estimator. Algorithms are developed to calculate asymptotic distributions and risk for the estimators of the threshold point. The posterior interval is proved to be an asymptotically valid confidence interval and is attractive in both length and coverage in finite samples.     

A robust scale estimator based on the shortest half

A new robust estimator of scale is considered, which is proportional to the length of the shortest half of the sample. The estimator is compared to the interquartile range and the median absolute deviation, that are also based on order statistics. All three estimators have the same influence function, but their breakdown points differ. It also turns out that one needs a finite-sample correction factor which depends on mod(sample size, 4) to achieve approximate unbiasedness at normal distributions.     

Bayesian point estimation of the cointegration space

A neglected aspect of the otherwise fairly well developed Bayesian analysis of cointegration is point estimation of the cointegration space. It is pointed out here that, due to the well known non-identification of the cointegration vectors, the parameter space is not Euclidean and the loss functions underlying the conventional Bayes estimators are therefore questionable. We present a Bayes estimator of the cointegration space which takes the curved geometry of the parameter space into account. This estimate has the interpretation of being the posterior mean cointegration space and is invariant to the order of the time series, a property not shared with many of the Bayes estimators in the cointegration literature. An overall measure of cointegration space uncertainty is also proposed. Australian interest rate data are used for illustration. A small simulation study shows that the new Bayes estimator compares favorably to the maximum likelihood estimator.     

Mean square error estimation in multi-stage sampling

Summary: Suppose for a homogeneous linear unbiased function of the sampled first stage unit (fsu)-values taken as an estimator of a survey population total, the sampling variance is expressed as a homogeneous quadratic function of the fsu-values. When the fsu-values are not ascertainable but unbiased estimators for them are separately available through sampling in later stages and substituted into the estimator, Raj (1968) gave a simple variance estimator formula for this multi-stage estimator of the population total. He requires that the variances of the estimated fsu-values in sampling at later stages and their unbiased estimators are available in certain `simple forms'. For the same set-up Rao (1975) derived an alternative variance estimator when the later stage sampling variances have more ‘complex forms’. Here we pursue with Raj's (1968) simple forms to derive a few alternative variance and mean square error estimators when the condition of homogeneity or unbiasedness in the original estimator of the total is relaxed and the variance of the original estimator is not expressed as a quadratic form.  We illustrate a particular three-stage sampling strategy and present a simulation-based numerical exercise showing the relative efficacies of two alternative variance estimators. Received: 19 February 1999     

Mostly harmless simulations? Using Monte Carlo studies for estimator selection

We consider two recent suggestions for how to perform an empirically motivated Monte Carlo study to help select a treatment effect estimator under unconfoundedness. We show theoretically that neither is likely to be informative except under restrictive conditions that are unlikely to be satisfied in many contexts. To test empirical relevance, we also apply the approaches to a real‐world setting where estimator performance is known. Both approaches are worse than random at selecting estimators that minimize absolute bias. They are better when selecting estimators that minimize mean squared error. However, using a simple bootstrap is at least as good and often better. For now, researchers would be best advised to use a range of estimators and compare estimates for robustness.     

Cross-validated SNP density estimates

We consider cross-validation strategies for the seminonparametric (SNP) density estimator, which is a truncation (or sieve) estimator based upon a Hermite series expansion with coefficients determined by quasi-maximum likelihood. Our main focus is on the use of SNP density estimators as an adjunct to efficient method of moments (EMM) structural estimation. It is known that for this purpose a desirable truncation point occurs at the last point at which the integrated squared error (ISE) curve of the SNP density estimate declines abruptly. We study the determination of the ISE curve for iid data by means of leave-one-out cross-validation and hold-out-sample cross-validation through an examination of their performance over the Marron–Wand test suite and models related to asset pricing and auction applications. We find that both methods are informative as to the location of abrupt drops, but that neither can reliably determine the minimum of the ISE curve. We validate these findings with a Monte Carlo study. The hold-out-sample method is cheaper to compute because it requires fewer nonlinear optimizations. We consider the asymptotic justification of hold-out-sample cross-validation. For this purpose, we establish rates of convergence of the SNP estimator under the Hellinger norm that are of interest in their own right.     

Reliable Robust Regression Diagnostics

Motivated by the requirement of controlling the number of false discoveries that arises in several application fields, we study the behaviour of diagnostic procedures obtained from popular high‐breakdown regression estimators when no outlier is present in the data. We find that the empirical error rates for many of the available techniques are surprisingly far from the prescribed nominal level. Therefore, we propose a simulation‐based approach to correct the liberal diagnostics and reach reliable inferences. We provide evidence that our approach performs well in a wide range of settings of practical interest and for a variety of robust regression techniques, thus showing general appeal. We also evaluate the loss of power that can be expected from our corrections under different contamination schemes and show that this loss is often not dramatic. Finally, we detail some possible extensions that may further enhance the applicability of the method.     

Robust estimators for estimating discontinuous functions

We study the asymptotic behavior of a wide class of kernel estimators for estimating an unknown regression function. In particular we derive the asymptotic behavior at discontinuity points of the regression function. It turns out that some kernel estimators based on outlier robust estimators are consistent at jumps.     

Semiparametric robust estimation of truncated and censored regression models

Many estimation methods of truncated and censored regression models such as the maximum likelihood and symmetrically censored least squares (SCLS) are sensitive to outliers and data contamination as we document. Therefore, we propose a semiparametric general trimmed estimator (GTE) of truncated and censored regression, which is highly robust but relatively imprecise. To improve its performance, we also propose data-adaptive and one-step trimmed estimators. We derive the robust and asymptotic properties of all proposed estimators and show that the one-step estimators (e.g., one-step SCLS) are as robust as GTE and are asymptotically equivalent to the original estimator (e.g., SCLS). The finite-sample properties of existing and proposed estimators are studied by means of Monte Carlo simulations.     

Discussion: Efficiency and Self-efficiency With Multiple Imputation Inference

By closely examining the examples provided in Nielsen (2003), this paper further explores the relationship between self-efficiency (Meng, 1994) and the validity of Rubin's multiple imputation (RMI) variance combining rule. The RMI variance combining rule is based on the common assumption/intuition that the efficiency of our estimators decreases when we have less data. However, there are estimation procedures that will do the opposite, that is, they can produce more efficient estimators with less data. Self-efficiency is a theoretical formulation for excluding such procedures. When a user, typically unaware of the hidden self-inefficiency of his choice, adopts a self-inefficient complete-data estimation procedure to conduct an RMI inference, the theoretical validity of his inference becomes a complex issue, as we demonstrate. We also propose a diagnostic tool for assessing potential self-inefficiency and the bias in the RMI variance estimator, at the outset of RMI inference, by constructing a convenient proxy to the RMI point estimator.     

Yet another breakdown point notion: EFSBP

The breakdown point in its different variants is one of the central notions to quantify the global robustness of a procedure. We propose a simple supplementary variant which is useful in situations where we have no obvious or only partial equivariance: Extending the Donoho and Huber (The notion of breakdown point, Wadsworth, Belmont, 1983) Finite Sample Breakdown Point?, we propose the Expected Finite Sample Breakdown Point to produce less configuration-dependent values while still preserving the finite sample aspect of the former definition. We apply this notion for joint estimation of scale and shape (with only scale-equivariance available), exemplified for generalized Pareto, generalized extreme value, Weibull, and Gamma distributions. In these settings, we are interested in highly-robust, easy-to-compute initial estimators; to this end we study Pickands-type and Location-Dispersion-type estimators and compute their respective breakdown points.