Empirical tests of performance of some M – estimators

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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% efficiency with respect to the standard normal distribution. The need for data - dependent tuning constants rather than those established theoretically is also pointed out.

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Geodesy and Cartography

The Journal of Committee on Geodesy of Polish Academy of Sciences

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