Robust statistics: a brief introduction and overview

Abstract

The paper gives a highly condensed first introduction into robust statistics and some guidance for the interpretation of the literature, with some consideration for the uses in geodetics in the background.Robust statistics is the stability theory of statistical procedures.It systematically investigates the effects of deviations from modelling assumptions on known procedures and, if necessary, develops new, better procedures.Common modelling assumptions are those of normality and of independence of the random errors.For both exist sophisticated theories with important practical consequences, and specifically normality can be replaced by any other reasonable parametric model (cf.Huber 1981, Hampel et al. 1986, and also Beran 1994).As a simple example, let us consider n "independent" measurements of the same quantity, for example a distance.In general they will differ, namely by what we now call the "random error", and the question arises which value we should take as the "best estimate" of the unknown "true value".This question was already considered by Gauss (1821; cf. also Huber 1972), and he, noticing that he needed the unknown error distribution to answer this question, turned the problem upside down and asked for that error distribution which made a rule "generally accepted as a good one", namely the arithmetic mean, optimal (in location or shift models).This led to the normal distribution as assumed error distribution, and to a deeper justification (other than by simplicity) of the method of least squares.Less than a hundred years later, almost everybody believed in the "dogma of normality", the mathematicians because they believed it to be an empirical fact, and the users of statistics because they believed it to be a mathematical theorem.But the central limit theorem, being a limit theorem, only suggests approximate normality under well-specified conditions in real situations; and empirical investigations, already by Bessel (1818) and later by Newcomb (1886), Jeffreys (1939) and others, show that typical error distributions of high-quality data are slightly but clearly longertailed (i.e., with higher kurtosis or standardized 4th moment) than the normal.Gauss (1821) had been careful to talk about "observations of equal accuracy", but obviously real data have different accuracy, as modeled by Newcomb (1886).The implicit or explicit hope that under approximate (instead of exact) normality least squares would still be approximately optimal -a belief still held by many statisticians today -was thwarted by Tukey (1960; cf. also Huber 1981or Hampel et al. 1986) who showed among other things that already under tiny deviations from normality the mean deviation -formerly much used as scale measure until the dispute between Fisher (1920) and Eddington -was better than the standard deviation, despite its efficiency loss of 12 percent under strict normality.In fact, the (avoidable) efficiency losses of least squares under high-quality data are more typically between 10 percent and 50 percent or even