Nonparametric Location and Scatter Functionals

摘要：非参数位置和散射函数给定n个独立的观测值，每个观测值都具有分布P，通过取观测点质量的平均值，可以形成一个经验性的度量。对于P的泛函T，如果T在P处适当连续，则以经验度量计算T(P)的估计量，该估计量几乎必然收敛于T(P)。本项目将继续寻找在尽可能多的P处连续的泛函，这些泛函可能在一维上全为P，但在更高维上是开的稠密集。然后，如果T在P处也是充分意义上可微的，则估计将是渐近正态的，并且以n的平方根上的1的速率收敛到T(P)。在一个维度中，另一个位置泛函是中位数m，而尺度泛函包括标准差西格玛和中位数绝对偏差。经典的Mu和Sigma可以是未定义的，也可以是无限大的，因为定律在无限大时衰减得太慢。对于某些定律P，存在一个中值区间，其中点给出了m的唯一选择，但是m在这样的P处是不连续的。同时，自由度nu大于1的t分布的位置和尺度的最大似然估计扩展到在直线上的每个概率分布处定义和连续的位置和尺度泛函，并且在不具有如nu大的原子的分布上对Nu加一无限可微。在多维空间上，尺度参数的平方被类似于协方差矩阵的散布矩阵所代替，并且将在密集的开放分布集上寻求连续性和可微性，通过t泛函和其他方法。位置和尺度或散布泛函是统计学中最基本的一些泛函。但到目前为止，位置的两个最著名的泛函--平均值和中位数--都有严重的缺陷。使用平均值，人们实际上假设所有数据都是正确的。平均值可能会受到离群索居的极端观察结果的过度影响，例如粗略误差。另一方面，使用中位数可以防止近一半的数据可能是不正确的。如果分布在一个区间内的概率超过其一半，则中位数必须在该区间内，并且可能很差地代表该分布的其余部分。泛函结合了平均数和中位数的一些优点，同时避免了两者的最坏缺点，可以而且应该得到更广泛的研究和使用。然后，人们可以拥有各种平均值，以防止某些小部分数据可能是不正确的道德可能性。

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.