Empirical tests of performance of some M – estimators

Abstract

The paper presents an empirical comparison of performance of three well known M -estimators (i.e. Huber, Tukey and Hampel's M -estimators) and also some new ones. The new M -estimators were motivated by weighting functions applied in orthogonal polynomials theory, kernel density estimation as well as one derived from Wigner semicircle probability distribution. M -estimators were used to detect outlying observations in contaminated datasets. Calculations were performed using iteratively reweighted least-squares (IRLS). Since the residual variance (used in covariance matrices construction) is not a robust measure of scale the tests employed also robust measures i.e. interquartile range and normalized median absolute deviation. The methods were tested on a simple leveling network in a large number of variants showing bad and good sides of M -estimation. The new M -estimators have been equipped with theoretical tuning constants to obtain 95% effi ciency with respect to the standard normal distribution. The need for data -dependent tuning constants rather than those established theoretically is also pointed out.

Multi split functional model of geodetic observations in deformation analyses of the Olsztyn castle

“…This refers to the observations not disturbed by outliers. For those check new methods of geodetic observations adjustments were developed, focused on their robustness to outliers (see, eg., Baselga, 2011;Kamiński, 2011;Banaś and Ligas, 2014;Štroner et al, 2014;Třasák and Štroner, 2014; The results of the I Scenario clearly indicate that the tested object did not deform. To verify whether the proposed strategy will give the correct results, the geodetic network should deform, the authors introduced artificial displacements of the selected points.…”

The Best Robust Estimation Method to Determine Local Surface

“…The estimation of polynomial coefficients was carried out with the use of the least squares method with iterative change of weights. In order to make it possible to compare the results, the control parameters were adopted in such a way as to obtain 95% efficiency in relation to the standard normal distribution (Banaś and Ligas, 2014).…”

M-Estimators in Business Statistics1

“…The weighting function i w , which is a ratio of the influence function and the residual, is the third function which characterizes M-estimation. The weighting function meets the following conditions (Banaś, Ligas, 2014):…”

“…The values of weights depend on what function  is chosen to correspond with function  . In the literature of the subject many variations of M-estimators are suggested, with different variants of function  (Fair, 1974), (Holland, Welsch 1977), (Huber, 1981), (Hampel et al, 1986), (Chen, Yin, 2002), (Banaś, Ligas, 2014). Curves of the weighting functions are different, but in M-estimation one always attempts to minimize or eliminate the influence of outliers.…”