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Statistical science --- 519.233 --- #WWIS:d.d. Prof. Teugels --- Parametric methods --- 519.233 Parametric methods

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In this thesis we propose some robust estimators and techniques for outlier detection for skewed, Pareto-type and multivariate distributions.First we explain how to improve the estimation of quantile-based measures for a univariate, possibly skewed, distribution based on a small number of observations. It has been theoretically pointed out that smoothing the empirical distribution function with an appropriate kernel and bandwidth can reduce the variance and mean squared error (MSE) of some quantile-based estimators in small data sets. We apply this idea on several quantile-based estimators of location, scale and skewness and propose a robust bandwidth selection and bias reduction procedure. We show that the use of this smoothing method indeed leads to smaller MSEs, also at contaminated data sets. In particular, we obtain better performances for the medcouple which is a robust measure of skewness often used for outlier detection in skewed distributions, as shown through a simulation study. We use smoothed quantile-based estimators to improve the outlier detection performance of the adjusted boxplot (which computes whiskers that are adjusted to the estimated skewness of the underlying distribution of the data set) when the data set is small.Next, we propose an outlier detection tool for Pareto-type distributions. Pareto-type distributions are univariate extreme value distributions for which the extreme value index is strictly positive. Classical estimators for the extreme value index, like the Hill estimator, tend to overestimate this parameter in the presence of outliers in the data. In order to measure the influence every (potentially outlying) data point has on the Hill estimator, the empirical influence function plot, which displays the influence that each data point has on the Hill estimator, is introduced. To avoid a masking effect, the empirical influence function is based on a new Robust Generalized Linear Model (RGLM) estimator for the extreme value index. This RGLM estimator is also used to determine high quantiles of the data generating distribution, allowing to flag data points as unusually large if they exceed this high quantile.The final contribution of this thesis consists of the proposal of a deterministic algorithm to compute multivariate S- and MM-estimators of location and scatter. Multivariate S- and MM-estimators are robust location- and scale estimators that allow to detect potential outliers in a multivariate data set. The current fast random algorithm to compute an S-estimator, which we refer to as the FastS estimator, applies improvement steps to decrease the objective function in each step, similar to the FAST-MCD algorithm to compute the Minimum Covariance Determinant (MCD) estimator. Recently a deterministic algorithm for the MCD-estimator has been proposed, and we combine ideas from this method with the improvement steps from the FastS algorithm in order to construct a deterministic algorithm for multivariate S-estimators of location and scatter, which we name DetS. We also explain how DetS can be used as an initial scale estimator for a deterministic MM-estimator, called DetMM. The performance of DetS is compared to FastS using simulated and real data. We show through a simulation study that our proposed S- and MM-estimators are very close to affine equivariant, and that they are permutation invariant.

academic collection --- 519.233 <043.3> --- Parametric methods--dissertaties --- Theses

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The first systematic, book-length treatment of the subject. Begins with a general introduction and the formal mathematical background behind qualitative and quantitative robustness. Stresses concepts. Provides selected numerical algorithms for computing robust estimates, as well as convergence proofs. Tables contain quantitative robustness information for a variety of estimates.

Mathematical statistics --- 519.233 --- Robust statistics --- Statistics, Robust --- Distribution (Probability theory) --- Parametric methods --- Robust statistics. --- 519.233 Parametric methods

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Mathematical statistics --- Statistique mathématique --- 519.233 --- Mathematics --- Statistical inference --- Statistics, Mathematical --- Statistics --- Probabilities --- Sampling (Statistics) --- Parametric methods --- Statistical methods --- 519.233 Parametric methods --- Statistique mathématique

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Sciences appliquées --- Toegepaste wetenschappen --- 534 --- 519.233 --- Academic collection --- #TELE:ELEN --- Vibrations. Acoustics --- Parametric methods --- 519.233 Parametric methods --- 534 Vibrations. Acoustics