Elvezio Ronchetti

Robust Inference for Generalized Linear Models

By starting from a natural class of robust estimators for generalized linear models based on the notion of quasi-likelihood, we define robust deviances that can be used for stepwise model selection as in the classical framework. We derive the asymptotic distribution of tests based on robust deviances, and we investigate the stability of their asymptotic level under contamination. The binomial and Poisson models are treated in detail. Two applications to real data and a sensitivity analysis show that the inference obtained by means of the new techniques is more reliable than that obtained by classical estimation and testing procedures.

Robust inference with GMM estimators

The local robustness properties of generalized method of moments (GMM) estimators and of a broad class of GMM based tests are investigated in a unified framework. GMM statistics are shown to have bounded influence if and only if the function defining the orthogonality restrictions imposed on the underlying model is bounded. Since in many applications this function is unbounded, it is useful to have procedures that modify the starting orthogonality conditions in order to obtain a robust version of a GMM estimator or test. We show how this can be obtained when a reference model for the data distribution can be assumed. We develop a flexible algorithm for constructing a robust GMM (RGMM) estimator leading to stable GMM test statistics. The amount of robustness can be controlled by an appropriate tuning constant. We relate by an explicit formula the choice of this constant to the maximal admissible bias on the level or (and) the power of a GMM test and the amount of contamination that one can reasonably assume given some information on the data. Finally, we illustrate the RGMM methodology with some simulations of an application to RGMM testing for conditional heteroscedasticity in a simple linear autoregressive model. In this example we find a significant instability of the size and the power of a classical GMM testing procedure under a non-normal conditional error distribution. On the other side, the RGMM testing procedures can control the size and the power of the test under non-standard conditions while maintaining a satisfactory power under an approximatively normal conditional error distribution.

Robust model selection in regression

A robust version of Akaikes model selection procedure for regression models is introduced and its relationship with robust testing procedures is discussed.

Robust Linear Model Selection by Cross-Validation

Abstract This article gives a robust technique for model selection in regression models, an important aspect of any data analysis involving regression. There is a danger that outliers will have an undue influence on the model chosen and distort any subsequent analysis. We provide a robust algorithm for model selection using Shaos cross-validation methods for choice of variables as a starting point. Because Shaos techniques are based on least squares, they are sensitive to outliers. We develop our robust procedure using the same ideas of cross-validation as Shao but using estimators that are optimal bounded influence for prediction. We demonstrate the effectiveness of our robust procedure in providing protection against outliers both in a simulation study and in a real example. We contrast the results with those obtained by Shaos method, demonstrating a substantial improvement in choosing the correct model in the presence of outliers with little loss of efficiency at the normal model.

A Robust Version of Mallows's C P

Abstract We present a robust version of Mallowss C P for regression models. It is defined by RC P = W Pσ2 - (U P - V P), where W P = ω i ω2 i r 2 i is a weighted residual sum of squares computed from a robust fit of model P, σ2 is a robust and consistent estimator of σ2 in the full model, and U P and V P are constants depending on the weight function and the number of parameters in model P. Good subset models are those with RC P close to V P or smaller than V P. When the weights are identically 1, W P becomes the residual sum of squares of a least squares fit, and RC P reduces to Mallowss C P. The robust model selection procedure based on RC P allows us to choose the models that fit the majority of the data by taking into account the presence of outliers and possible departures from the normality assumption on the error distribution. Together with the classical C P, the robust version suggests several models from which we can choose.

General Saddlepoint Approximations with Applications to L Statistics

Abstract Saddlepoint approximations are extended to general statistics. The technique is applied to derive approximations to the density of linear combinations of order statistics, including trimmed means. A comparison with exact results shows the accuracy of these approximations even in very small sample sizes.

Resistant selection of the smoothing parameter for smoothing splines

Robust automatic selection techniques for the smoothing parameter of a smoothing spline are introduced. They are based on a robust predictive error criterion and can be viewed as robust versions of Cp and cross-validation. They lead to smoothing splines which are stable and reliable in terms of mean squared error over a large spectrum of model distributions.

General Saddlepoint Approximations of Marginal Densities and Tail Probabilities

Saddlepoint approximations of marginal densities and tail probabilities of general nonlinear statistics are derived. These are based on the expansion of the statistic up to the second order. Their accuracy is shown in a variety of examples, including logit and probit models and rank estimators for regression.

Robust Methods for Personal-Income Distribution Models

Statistical problems in modelling personal-income distributions include estimation procedures, testing, and model choice. Typically, the parameters of a given model are estimated by classical procedures such as maximum-likelihood and least-squares estimators. Unfortunately, the classical methods are very sensitive to model deviations such as gross errors in the data, grouping effects, or model misspecifications. These deviations can ruin the values of the estimators and inequality measures and can produce false information about the distribution of the personal income in a country. In this paper we discuss the use of robust techniques for the estimation of income distributions. These methods behave like the classical procedures at the model but are less influenced by model deviations and can be applied to general estimation problems.

The Change-of-Variance Curve and Optimal Redescending M-Estimators

Abstract We define the change-of-variance curve (CVC) of location M-estimators in order to investigate the infinitesimal stability of the asymptotic variance. We also construct the so-called hyperbolic tangent estimators, proving their existence and performing certain numerical computations of their defining constants. Their introduction is motivated by a theorem that shows they are the optimally robust redescending M-estimators in the sense of the CVC.