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 ε-contamination at oo, are presented for the special subclass arising from Huber's 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.     

Multivariate τ-estimators for location and scatter

We discuss the robustness and asymptotic behaviour of τ-estimators for multivariate location and scatter. We show that τ-estimators correspond to multivariate M-estimators defined by a weighted average of redescending ψ-functions, where the weights are adaptive. We prove consistency and asymptotic normality under weak assumptions on the underlying distribution, show that τ-estimators have a high breakdown point, and obtain the influence function at general distributions. In the special case of a location-scatter family, τ-estimators are asymptotically equivalent to multivariate S-estimators defined by means of a weighted ψ-function. This enables us to combine a high breakdown point and bounded influence with good asymptotic efficiency for the location and covariance estimator.     

Lp-estimators in ARCH models

For ergodic ARCH processes, we introduce a one-parameter family of L_p-estimators. The construction is based on the concept of weighted M-estimators. Under weak assumptions on the error distribution, the consistency is established. The asymptotic normality is proved for the special cases p=1 and 2. To prove the asymptotic normality of the L₁-estimator, one needs the existence of a density of the squares of the errors, whereas for the L₂-estimator the existence of fourth moments is assumed. The asymptotic covariance matrix of the estimator depends on the unknown parameter which can be substituted by consistent estimators. For the L₁-estimator we construct a kernel estimator for the unknown density of the square of the errors.     

The influence function of penalized regression estimators

To perform regression analysis in high dimensions, lasso or ridge estimation are a common choice. However, it has been shown that these methods are not robust to outliers. Therefore, alternatives as penalized M-estimation or the sparse least trimmed squares (LTS) estimator have been proposed. The robustness of these regression methods can be measured with the influence function. It quantifies the effect of infinitesimal perturbations in the data. Furthermore, it can be used to compute the asymptotic variance and the mean-squared error (MSE). In this paper we compute the influence function, the asymptotic variance and the MSE for penalized M-estimators and the sparse LTS estimator. The asymptotic biasedness of the estimators make the calculations non-standard. We show that only M-estimators with a loss function with a bounded derivative are robust against regression outliers. In particular, the lasso has an unbounded influence function.     

Asymptotic properties of one-step M-estimators

We study the asymptotic behavior of one-step M-estimators based on not necessarily independent identically distributed observations. In particular, we find conditions for asymptotic normality of these estimators. Asymptotic normality of one-step M-estimators is proven under a wide spectrum of constraints on the exactness of initial estimators. We discuss the question of minimal restrictions on the exactness of initial estimators. We also discuss the asymptotic behavior of the solution to an M-equation closest to the parameter under consideration. As an application, we consider some examples of one-step approximation of quasi-likelihood estimators in nonlinear regression.     

Robust M-estimators of multivariate location and scatter in the presence of asymmetry

Robust estimation of location vectors and scatter matrices is studied under the assumption that the unknown error distribution is spherically symmetric in a central region and completely unknown in the tail region. A precise formulation of the model is given, an analysis of the identifiable parameters in the model is presented, and consistent initial estimators of the identifiable parameters are constructed. Consistent and asymptotically normal M-estimators are constructed (solved iteratively beginning with the initial estimates) based on “influence functions” which vanish outside specified compact sets. Finally M-estimators which are asymptotically minimax (in the sense of Huber) are derived.     

M-Estimators Based on Inverse Probability Weighted Estimating Equations with Response Missing at Random

Asymptotic properties of M-estimators with complete data are investigated extensively. In the presence of missing data, however, the standard inference procedures for complete data cannot be applied directly. In this article, the inverse probability weighted method is applied to missing response problem to define M-estimators. The existence of M-estimators is established under very general regularity conditions. Consistency and asymptotic normality of the M-estimators are proved, respectively. An iterative algorithm is applied to calculating the M-estimators. It is shown that one step iteration suffices and the resulting one-step M-estimate has the same limit distribution as in the fully iterated M-estimators.     

Minimax Asymptotic Mean-Squared-Error of L-Estimators of Scale Parameter

We derive the AMSE (maximal asymptotic mean-squared-error) of the general class of L-estimators of scale that are location-scale equivariant and Fisher consistent. For non-normal error distributions, we determined estimators that have minimum AMSE over the subclass of (i) α-interquantile ranges and (ii) mixtures of at most two α-interquantile ranges. Finally, the L-estimators of scale symmetrized about the median were found to have the same AMSE as their nonsymmetrized counterparts, thus yielding the same results as in the symmetrized case.     

On the trimmed mean and minimax-variance L-estimation in Kolmogorov neighbourhoods

We consider the properties of the trimmed mean, as regards minimax-variance L-estimation of a location parameter in a Kolmogorov neighbourhood K() of the normal distribution: We first review some results on the search for an L-minimax estimator in this neighbourhood, i.e. a linear combination of order statistics whose maximum variance in K_t() is a minimum in the class of L-estimators. The natural candidate – the L-estimate which is efficient for that member of K_t,() with minimum Fisher information – is known not to be a saddlepoint solution to the minimax problem. We show here that it is not a solution at all. We do this by showing that a smaller maximum variance is attained by an appropriately trimmed mean. We argue that this trimmed mean, as well as being computationally simple – much simpler than the efficient L-estimate referred to above, and simpler than the minimax M- and R-estimators – is at least “nearly” minimax.     

Ordres stochastiques et efficacité asymptotique des L-estimateurs

The efficiency of an estimator depends heavily on the tails of the distribution of the observations. Several partial orders have been defined to compare probability distributions according to their tails. In this paper we show that the asymptotic relative efficiency of two L-estimators with monotone weight functions is isotonic with respect to the partial orders defined by van Zwet (1964) and Lawrence (1975). We also give results concerning trimmed means.     

M-Estimates of regression when the scale is unknown and the error distribution is possibly asymmetric: A minimax result

Huber (1964) found the minimax-variance M-estimate of location under the assumption that the scale parameter is known; Li and Zamar (1991) extended this result to the case when the scale is unknown. We consider the robust estimation of the regression coefficients (β₁,…,β_p) when the scale and the intercept parameters are unknown. The minimax-variance estimates of (β₁,…,β_p) with respect to the trace of their asymptotic covariance matrix are derived. The maximum is taken over ?-contamination neighbourhoods of a central regression model with Gaussian errors (asymmetric contamination is allowed), and the minimum is taken over a large class of generalized M-estimates of regression of the Mallow type. The optimal choice of estimates for the nuisance parameters (scale and intercept) is also considered.     

Simultaneous estimation of several CDF’s: homogeneity constraint

Let {x_ij(1 ? j ? n_i)|i = 1, 2, …, k} be k independent samples of size n_j from respective distributions of functions F_j(x)(1 ? j ? k). A classical statistical problem is to test whether these k samples came from a common distribution function, F(x) whose form may or may not be known. In this paper, we consider the complementary problem of estimating the distribution functions suspected to be homogeneous in order to improve the basic estimator known as “empirical distribution function” (edf), in an asymptotic setup. Accordingly, we consider four additional estimators, namely, the restricted estimator (RE), the preliminary test estimator (PTE), the shrinkage estimator (SE), and the positive rule shrinkage estimator (PRSE) and study their characteristic properties based on the mean squared error (MSE) and relative risk efficiency (RRE) with tables and graphs. We observed that for k ? 4, the positive rule SE performs uniformly better than both shrinkage and the unrestricted estimator, while PTEs works reasonably well for k < 4.     

Optimal Nonparametric Quantile Estimators. Towards a General Theory. A Survey

“Nonparametric” in the title is used to say that observations X ₁,…,X _n come from an unknown distribution F ∈ ? with ? being the class of all continuous and strictly increasing distribution functions. The problem is to estimate the quantile of a given order q ∈ (0,1) of the distribution F. The class ? of distributions is very large; it is so large that even X _nq:n , where nq is an integer, may be very poor estimator of the qth quantile. To assess the performance of estimators no properties based on moments may be used: expected values of estimators should be replaced by their medians, their variances—by some characteristics of concentration of distributions around the median. If an estimator is median-biased for one of distributions, the bias of the estimator may be infinitely large for other distributions. In the note optimal estimators with respect to various criteria of optimality are presented. The pivotal function F(T) of the estimator T is introduced which enables us to apply the classical statistical approach.     

Asymptotics for L1-estimators of regression parameters under heteroscedasticityY

We consider the asymptotic behaviour of L₁ -estimators in a linear regression under a very general form of heteroscedasticity. The limiting distributions of the estimators are derived under standard conditions on the design. We also consider the asymptotic behaviour of the bootstrap in the heteroscedastic model and show that it is consistent to first order only if the limiting distribution is normal.     

A note on the asymptotic distribution of the process capability index C pmk

Process capability indices have been widely used to evaluate the process performance to the continuous improvement of quality and productivity. The distribution of the estimator of the process capability index C _pmk is very complicated and the asymptotic distribution is proposed by Chen and Hsu [The asymptotic distribution of the processes capability index C _pmk , Comm. Statist. Theory Methods 24(5) (1995), pp. 1279–1291]. However, we found a critical error for the asymptotic distribution when the population mean is not equal to the midpoint of the specification limits. In this paper, a correct version of the asymptotic distribution is given. An asymptotic confidence interval of C _pmk by using the correct version of asymptotic distribution is proposed and the lower bound can be used to test if the process is capable. A simulation study of the coverage probability of the proposed confidence interval is shown to be satisfactory. The relation of six sigma technique and the index C _pmk is also discussed in this paper. An asymptotic testing procedure to determine if a process is capable based on the index of C _pmk is also given in this paper.     

Redescending M-estimators

In finite sample studies redescending M-estimators outperform bounded M-estimators (see for example, Andrews et al. [1972. Robust Estimates of Location. Princeton University Press, Princeton]). Even though redescenders arise naturally out of the maximum likelihood approach if one uses very heavy-tailed models, the commonly used redescenders have been derived from purely heuristic considerations. Using a recent approach proposed by Shurygin, we study the optimality of redescending M-estimators. We show that redescending M-estimator can be designed by applying a global minimax criterion to locally robust estimators, namely maximizing over a class of densities the minimum variance sensitivity over a class of estimators. As a particular result, we prove that Smith's estimator, which is a compromise between Huber's skipped mean and Tukey's biweight, provides a guaranteed level of an estimator's variance sensitivity over the class of densities with a bounded variance.     

On shrinking minimax convergence in nonparametric statistics

‘?…?if we are prepared to assume that the unknown density has k derivatives, then?…?the optimal mean integrated squared error is of order n^{?2 k/(2 k+1)}?…?’ The citation is from Silverman [(1986), Density Estimation for Statistics and Data Analysis, London: Chapman &; Hall] and its assertion is based on a classical minimax lower bound which is the pillar of the modern nonparametric statistics. This paper proposes a new minimax methodology that implies a faster decreasing minimax lower bound that is attainable by a data-driven estimator, and the same estimator is also minimax under the classical approach. The recommendation is to test performance of estimators via the new and classical minimax approaches.     

The Recursive Kernel Distribution Function Estimator Based on Negatively and Positively Associated Sequences

Let {X _j , j ≥ 1} be a strictly stationary negatively or positively associated sequence of real valued random variables with unknown distribution function F(x). On the basis of the random variables {X _j , j ≥ 1}, we propose a smooth recursive kernel-type estimate of F(x), and study asymptotic bias, quadratic-mean consistency and asymptotic normality of the recursive kernel-type estimator under suitable conditions.