Nonparametric Location and Scatter Functionals

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Abstract: Nonparametric location and scatter functionalsGiven n independent observations, each with distribution P, onecan form an empirical measure by taking the average of point massesat the observations. For a functional T of P, evaluating T at theempirical measure gives an estimator of T(P) which will convergeto it almost surely if T is suitably continuous at P. The project will continue a search for functionals continuous at as many P's as possible, which may be all P in one dimension, but in higher dimensions an open dense set. Then if T is also differentiable at P in a sufficient sense, the estimators will be asymptoticallynormal and converge to T(P) at a rate of 1 over square root of n.Functionals of location include the mean mu when it is finite. In one dimension, another location functional is the median m, and scale functionals include the standard deviation sigma and median absolute deviation. The classical mu and sigma can be undefined or infinite at laws decreasing too slowly at infinity. For some laws P there is an interval of medians, whose midpoint gives a unique choice of m, but m is discontinuous at such P. Simultaneous maximum likelihood estimation of location and scale for t distributions with degrees of freedom nu larger than 1 extends to location and scale functionals defined and continuous at every probability distribution on the line, and infinitely differentiable at distributions having no atom as large as nu over nu plus one.On multidimensional spaces, the square of a scale parameteris replaced by a scatter matrix analogous to a covariancematrix, and the continuity and differentiability will be sought ona dense open set of distributions, via t functionals and by othermethods.Location and scale or scatter functionals are some of the most basic in statistics. But by far the two best known functionals of location, the mean and median, each have serious drawbacks. Using the mean, oneassumes in effect that all data are correct. The mean can be overly influenced by outlying, extreme observations such as gross errors. Using the median, on the other hand, guards against the possibilitythat nearly half the data may be incorrect. If a distribution has more than half its probability in an interval, the median must be in that interval and may very poorly represent the rest of the distribution. Functionals combining some of the advantages of the mean and median whileavoiding the worst drawbacks of either can and should be more widely studied and used. Then one can have kinds of averages that guard against a morerealistic possibility that some small fraction of the data may beincorrect.