Robert Serfling

Quantile functions for multivariate analysis: Approaches and applications

Despite the absence of a natural ordering of Euclidean space for dimensions greater than one, the effort to define vector‐valued quantile functions for multivariate distributions has generated several approaches. To support greater discrimination in comparing, selecting and using such functions, we introduce relevant criteria, including a notion of “medianoriented quantile function”. On this basis we compare recent quantile approaches and several multivariate versions of trimmed mean and interquartile range. We also discuss a univariate “generalized quantile” approach that enables particular features of multivariate distributions, for example scale and kurtosis, to be studied by two‐dimensional plots. Methods based on statistical depth functions are found to be especially attractive for quantile‐based multivariate inference.

Some Elementary Results on Poisson Approximation in a Sequence of Bernoulli Trials

In a finite series of independent success-failure trials, the total number of successes has a binomial probability distribution. It is a classical result that this probability distribution is subje...

A Depth Function and a Scale Curve Based on Spatial Quantiles

Spatial quantiles, based on the L 1 norm in a certain sense, provide an appealing vector extension of univariate quantiles and generate a useful “volume” functional based on spatial “central regions” of increasing size. A plot of this functional as a “spatial scale curve” provides a convenient two-dimensional characterization of the spread of a multivariate distribution of any dimension. We discuss this curve and establish weak convergence of the empirical version. As a tool, we introduce and study a new statistical depth function which is naturally associated with spatial quantiles. Other depth functions that generate L 1-based multivariate quantiles are also noted.

Nonparametric Multivariate Descriptive Measures Based on Spatial Quantiles

Abstract An appealing way of working with probability distributions, especially in nonparametric inference, is through “descriptive measures” that characterize features of particular interest. One attractive approach is to base the measures on quantiles. Here we consider the multivariate context and utilize the “spatial quantiles”, a recent vector extension of univariate quantiles that is becoming increasingly popular. In terms of these quantiles, we introduce and study nonparametric measures of multivariate location, spread, skewness and kurtosis. In particular, we define a useful “location” functional which augments the well-known “spatial” median and a “volume” functional which plotted as a “spatial scale curve” yields a convenient two-dimensional characterization of the spread of a multivariate distribution of any dimension. These spatial location and volume functionals also play roles in the formulation of “spatial” skewness and kurtosis functionals which reduce to known versions in the univariate case. We also define corresponding spatial “asymmetry” and “kurtosis” curves which are new devices even in the univariate case. Tailweight and peakedness measures, as distinct from kurtosis, are also discussed. To aid better understanding of the spatial quantiles as a foundation for nonparametric multivariate inference and analysis, we also provide some basic perspective on them: their interpretations, properties, strengths and weaknesses.

Equivariance and invariance properties of multivariate quantile and related functions, and the role of standardisation

Equivariance and invariance issues arise as a fundamental but often problematic aspect of multivariate statistical analysis. For multivariate quantile and related functions, we provide coherent definitions of these properties. For standardisation of multivariate data to produce equivariance or invariance of procedures, three important types of matrix-valued functional are studied: ‘weak covariance’ (or ‘shape’), ‘transformation–retransformation’ (TR), and ‘strong invariant coordinate system’ (SICS). The clarification of TR affine equivariant versions of the sample spatial quantile function is obtained. It is seen that geometric artefacts of SICS-standardised data are invariant under affine transformation of the original data followed by standardisation using the same SICS functional, subject only to translation and homogeneous scale change. Some applications of SICS standardisation are described.

Robust and Efficient Estimation of the Tail Index of a Single-Parameter Pareto Distribution

Abstract Estimation of the tail index parameter of a single-parameter Pareto model has wide application in actuarial and other sciences. Here we examine various estimators from the standpoint of two competing criteria: efficiency and robustness against upper outliers. With the maximum likelihood estimator (MLE) being efficient but nonrobust, we desire alternative estimators that retain a relatively high degree of efficiency while also being adequately robust. A new generalized median type estimator is introduced and compared with the MLE and several well-established estimators associated with the methods of moments, trimming, least squares, quantiles, and percentile matching. The method of moments and least squares estimators are found to be relatively deficient with respect to both criteria and should become disfavored, while the trimmed mean and generalized median estimators tend to dominate the other competitors. The generalized median type performs best overall. These findings provide a basis for revision and updating of prevailing viewpoints. Other topics discussed are applications to robust estimation of upper quantiles, tail probabilities, and actuarial quantities, such as stop-loss and excess-of-loss reinsurance premiums that arise concerning solvency of portfolios. Robust parametric methods are compared with empirical nonparametric methods, which are typically nonrobust.

Glivenko-Cantelli properties of some generalized empirical DF's and strong convergence of generalized L-statistics

SummaryWe study a nonclassical form of empirical df Hnwhich is of U-statistic structure and extend to Hnthe classical exponential probability inequalities and Glivenko-Cantelli convergence properties known for the usual empirical df. An important class of statistics is given byT(Hn), where T(·) is a generalized form of L-functional. For such statisticswe prove almost sure convergence using an approach which separates the functional-analytic and stochastic components of the problem and handles the latter component by application of Glivenko-Cantelli type properties.Classical results for U-statistics and L-statistics are obtained as special cases without addition of unnecessary restrictions.Many important new types of statistics of current interest are covered as well by our result.

On the performance of some robust nonparametric location measures relative to a general notion of multivariate symmetry

Abstract Several robust nonparametric location estimators are examined with respect to several criteria, with emphasis on the criterion that they should agree with the point of symmetry in the case of a symmetric distribution. For this purpose, a broad version of multidimensional symmetry is introduced, namely “halfspace symmetry”, generalizing the well-known notions of “central” and “angular” symmetry. Characterizations of these symmetry notions are established, permitting their properties and interrelations to be illuminated. The particular location measures considered consist of several nonparametric notions of multidimensional median: The “L2” (or “spatial”), “Tukey/Donoho halfspace”, “projection”, and “Liu simplicial” medians, all of which are robust in the sense of nonzero breakdown point. It is established that the first three of these in general do identify the point of symmetry when it exists, whereas the latter, however, fails to do so in some circumstances. Combining this finding with consideration of other criteria such as affine equivariance, stochastic order preserving, and degree of robustness, we conclude that among these choices, the “halfspace” and “projection” medians, both of which are based on projection pursuit methodology, are the most attractive overall.

Generalized Order Statistics, Bahadur Representations, and Sequential Nonparametric Fixed-Width Confidence Intervals.

Abstract Let X 1 ,…, X n be an i.i.d. sample from df F , let H F be the df of h ( X 1 ,…, X m ), based on a given ‘kernel’ h ( x 1 ,…, x m ), and consider confidence interval estimation of a parameter of the form H -1 F ( p ). This paper introduces confidence intervals formed by a pair of ‘generalized order statistics’, develops Bahadur-type representation theory for these order statistics, and constructs corresponding sequential fixed-width confidence interval procedures. Previous work of Bahadur (1966) and Geertsema (1970) is sharpened and extended.

Robust Estimation of Tail Parameters for Two-Parameter Pareto and Exponential Models via Generalized Quantile Statistics

Robust estimation of tail index parameters is treated for (equivalent) two-parameter Pareto and exponential models. These distributions arise as parametric models in actuarial science, economics, telecommunications, and reliability, for example, as well as in semiparametric modeling of upper observations in samples from distributions which are regularly varying or in the domain of attraction of extreme value distributions. New estimators of “generalized quantile” type are introduced and compared with several well-established estimators, for the purpose of identifying which estimators provide favorable trade-offs between efficiency and robustness. Specifically, we examine asymptotic relative efficiency with respect to the (efficient but nonrobust) maximum likelihood estimator, and breakdown point. The new estimators, in particular the generalized median types, are found to dominate well-established and popular estimators corresponding to methods of trimming, least squares, and quantiles. Further, we establish that the least squares estimator is actually deficient with respect to both criteria and should become disfavored. The generalized median estimators manifest a general principle: “smoothing” followed by “medianing” produces a favorable trade-off between efficiency and robustness.