Robustness in Asymmetric Departures from the Model Normal Distribution

Alistair Martin

Maximum likelihood and least squares methods lead to ideal estimates when parametric requirements are met. When dealing with real data, such requirements are almost never satisfied. Early attempts to remedy this problem focussed on tedious outlier identification and treatment. Robustness theory emerged in the second half of the twentieth century, with tools and theory to formulate and compare estimators tolerant of deviations from distribution assumptions. M-estimates, a result of the generalisation of maximum likelihood estimation, became the dominant class of robust estimators defined by this body of work. Historically, asymptotic assessment of robust location and scale estimates has seemingly been predicated on symmetric contamination. In this work, asymptotic assessment for the simultaneous estimates under an asymmetric contaminated mixture model framework is provided. Optimal estimators are compared, including Hampel’s three-part estimator and the tanh-estimator. The influence and change-of-variance functions are compared, along with derived measures of robustness such as gross error sensitivity, local-shift sensitivity and change-of-variance sensitivity. Hampel and tanh estimators are both optimal in some senses. They are weakly continuous and Fréchet differentiable at the normal distribution, though it is not known if these estimators retain these properties in either small Kolmogorov-Smirnov or even Prohorov neighbourhoods of the normal distribution. Problems may exist due to existence of sharp corners of the M-estimators defining continuous but not smooth ψ functions. Smoothing Hampel’s redescender yields simple, piecewise polynomial, weakly continuous and Fréchet differentiable functionals. These qualities guarantee consistency, asymptotic normality, and qualitative robustness, particularly in small neighbourhoods of the model distribution. Asymptotic calculus for piecewise polynomial M-estimates is introduced, for fast computation of asymptotic values and variance. The smoothed estimator is shown to achieve asymptotic values consistent with the aforementioned optimal estimators, and improved asymptotic variance and change-of variance sensitivity; particularly for scale, and when contamination is present in the tails of the normal distribution. The three-part Hampel, Bachmaier, tanh and Tukey estimators are applied to real world and simulated data for location and scale estimation. Regression estimates and model selection are considered with application to the robust adaptive lasso.

Robustness in Asymmetric Departures from the Model Normal Distribution

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