272 research outputs found
A journey in single steps: robust one-step M-estimation
No full textWe present a unified treatment of different types of one-step M-estimation in regression models which incorporates the Newton–Raphson, method of scoring and iteratively reweighted least squares forms of one-step estimator. We use higher order expansions to distinguish between the different forms of estimator and the effects of different initial estimators. We show that the Newton–Raphson form has better properties than the method of scoring form which, in turn, has better properties than the iteratively reweighted least squares form. We also show that the best choice of initial estimator is a smooth, robust estimator which converges at the rate n?1/2. These results have important consequences for the common data-analytic strategy of using a least squares analysis on "clean" data obtained by deleting observations with extreme residuals from an initial least squares fit. It is shown that the resulting estimator is an iteratively reweighted least squares one-step estimator with least squares as the initial estimator, giving it the worst performance of the one-step estimators we consider: inferences resulting from this strategy are neither valid nor robust
Composite Likelihood Inference by Nonparametric Saddlepoint Tests
No full textThe class of composite likelihood functions provides a flexible and powerful toolkit to
carry out approximate inference for complex statistical models when the full likelihood
is either impossible to specify or unfeasible to compute. However, the strenght of the
composite likelihood approach is dimmed when considering hypothesis testing about a
multidimensional parameter because the finite sample behavior of likelihood ratio, Wald,
and score-type test statistics is tied to the Godambe information matrix. Consequently
inaccurate estimates of the Godambe information translate in inaccurate p-values. In this
paper it is shown how accurate inference can be obtained by using a fully nonparametric
saddlepoint test statistic derived from the composite score functions. The proposed statis-
tic is asymptotically chi-square distributed up to a relative error of second order and does
not depend on the Godambe information. The validity of the method is demonstrated
through simulation studies
Robust methods for personal‐income distribution models
No full textStatistical 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
Longitudinal variable selection by cross-validation in the case of many covariates
No full textLongitudinal models are commonly used for studying data collected on individuals repeatedly through time. While there are now a variety of such models available (Marginal Models, Mixed Effects Models, etc.), far fewer options appear to exist for the closely related issue of variable selection. In addition, longitudinal data typically derive from medical or other large-scale studies where often large numbers of potential explanatory variables and hence even larger numbers of candidate models must be considered. Cross-validation is a popular method for variable selection based on the predictive ability of the model. Here, we propose a cross-validation Markov Chain Monte Carlo procedure as a general variable selection tool which avoids the need to visit all candidate models. Inclusion of a “one-standard error” rule provides users with a collection of good models as is often desired. We demonstrate the effectiveness of our procedure both in a simulation setting and in a real application.
Contributions to robustness theory
No full textThe goal of this PhD Thesis is the definition of new robust estimators, thereby extending the available theory and exploring new directions for applications in finance. The Thesis contains three papers, which analyse three different types of estimators: M-, Minimum Distance- and R-estimators. The focus is manly of their infinitesimal robustness, but global robustness properties are also considered. The first paper (“Higher-order infinitesimal robustness”) studies M-estimators and it is a joint work with Elvezio Ronchetti and Fabio Trojani. Using the higher-order von Mises expansion, we go beyond the Influence Function and we extend Hampel's paradigm of robust ness, introducing higher-order infinitesimally robust M-estimators. We show that a bounded estimating function having also bounded gradient with respect to the parameter ensures, at the same time, the stability of the: (i) second-order approximated bias (B-robustness); (ii) asymptotic variance (V-robustness) and (iii) saddle point density approximation. An application in finance (risk management) concludes the paper. The second paper (“On robust estimation via pseudo-additive information measures”) is jointly written with Davide Ferrari and it studies a new class of Minimum Divergence (in the following, MD) estimators. The theoretical contribution of the paper is to show that robustness is dual to information theory. Information theory plays a crucial role in statistical inference : Maximum Likelihood estimators are related to it through the minimization of Shannon entropy (namely, minimization of the Kullback-Leibler divergence). The fundamental axiom characterizing Shannon entropy is additivity. Relaxing this assumption, we obtain a generalized entropy (called q-entropy) which exploits the link between information theory and infinitesimal robustness. Minimizing the q-entropy, we define a new class of MD robust re-descending estimators, featuring B-, V-robustness and that have also good global robustness properties in terms of high-breakdown. The third paper (“Semi-parametric rank-based tests and estimators for Markov processes”) contains the preliminary results of a working paper that I have started in Princeton, working with Marc Hallin. The paper deals with R-estimators and rank-based tests. Precisely, combining the flexibility of the semi-parametric approach with the distribution-freeness of rank statistics, we define R-estimators and tests for stationary Markov processes. An application for inference and testing in stochastic volatility (SV) models concludes the paper
Robust and accurate inference for generalized linear models
No full textIn the framework of generalized linear models, the nonrobustness of classical estimators and tests for the parameters is a well known problem, and alternative methods have been proposed in the literature. These methods are robust and can cope with deviations from the assumed distribution. However, they are based on first order asymptotic theory, and their accuracy in moderate to small samples is still an open question. In this paper, we propose a test statistic which combines robustness and good accuracy for moderate to small sample sizes. We combine results from Cantoni and Ronchetti [E. Cantoni, E. Ronchetti, Robust inference for generalized linear models, Journal of the American Statistical Association 96 (2001) 1022-1030] and Robinson, Ronchetti and Young [J. Robinson, E. Ronchetti, G.A. Young, Saddlepoint approximations and tests based on multivariate M-estimators, The Annals of Statistics 31 (2003) 1154-1169] to obtain a robust test statistic for hypothesis testing and variable selection, which is asymptotically [chi]2-distributed as the three classical tests but with a relative error of order O(n-1). This leads to reliable inference in the presence of small deviations from the assumed model distribution, and to accurate testing and variable selection, even in moderate to small samples.M-estimators Monte Carlo Robust inference Robust variable selection Saddlepoint techniques Saddlepoint test
Estimation of generalized linear latent variable models
No full textGeneralized linear latent variable models (GLLVMs), as defined by Bartholomew and Knott, enable modelling of relationships between manifest and latent variables. They extend structural equation modelling techniques, which are powerful tools in the social sciences. However, because of the complexity of the log-likelihood function of a GLLVM, an approximation such as numerical integration must be used for inference. This can limit drastically the number of variables in the model and can lead to biased estimators. We propose a new estimator for the parameters of a GLLVM, based on a Laplace approximation to the likelihood function and which can be computed even for models with a large number of variables. The new estimator can be viewed as an "M"-estimator, leading to readily available asymptotic properties and correct inference. A simulation study shows its excellent finite sample properties, in particular when compared with a well-established approach such as LISREL. A real data example on the measurement of wealth for the computation of multidimensional inequality is analysed to highlight the importance of the methodology. Copyright 2004 Royal Statistical Society.
Recently, there has been growing evidence of stock and bond return predictability ac...
No full textwww.elsevier.com/locate/csda Stock and bond return predictability: the discrimination power of model selection criteria Rosario Dell’Aquila a, ∗ , Elvezio Ronchetti
Robust and accurate inference for generalized linear models
No full textIn the framework of generalized linear models, the nonrobustness of classical estimators and tests for the parameters is a well known problem and alternative methods have been proposed in the literature. These methods are robust and can cope with deviations from the assumed distribution. However, they are based on ¯rst order asymptotic theory and their accuracy in moderate to small samples is still an open question. In this paper we propose a test statistic which combines robustness and good accuracy for moderate to small sample sizes. We combine results from Cantoni and Ronchetti (2001) and Robinson, Ronchetti and Young (2003) to obtain a robust test statistic for hypothesis testing and variable selection which is asymptotically Â2¡distributed as the three classical tests but with a relative error of order O(n¡1). This leads to reliable inference in the presence of small deviations from the assumed model distribution and to accurate testing and variable selection even in moderate to small samples
Sequential automatic search of subsets of classifiers in multi class classification
No full textMulticlass Learning (ML) requires a classifier to discriminate instances (objects) among several classes of an outcome (response) variable. Most of the proposed methods for ML do not consider that analyzing complex datasets requires the results to be easily interpretable. We refer to the Sequential Automatic Search of Subset of Classifiers (SASSC) algorithm as an approach able to find the right compromise between knowledge extraction and good prediction. SASSC is an iterative algorithm that works by building a taxonomy of classes in an ascendant manner: this is done by the solution of a multiclass problem obtained by decomposing it into several r-nary problems (r >> 2) in an agglomerative way. We consider the use of different classification methods as base classifiers and evaluate the performance of SASSC with respect to alternative classes aggregation criteria which allow us to compare either the predictive performance or the interpretation issues related to the use of each set of classifiers
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