Ivan Mizera

QUANTILE TOMOGRAPHY: USING QUANTILES WITH MULTIVARIATE DATA

The use of quantiles to obtain insights about multivariate data is ad- dressed. It is argued that incisive insights can be obtained by considering direc- tional quantiles, the quantiles of projections. Directional quantile envelopes are proposed as a way to condense this kind of information; it is demonstrated that they are essentially halfspace (Tukey) depth levels sets, coinciding for elliptic distri- butions (in particular multivariate normal) with density contours. Relevant ques- tions concerning their indexing, the possibility of the reverse retrieval of directional quantile information, invariance with respect to affine transformations, and approx- imation/asymptotic properties are studied. It is argued that analysis in terms of directional quantiles and their envelopes offers a straightforward probabilistic inter- pretation and thus conveys a concrete quantitative meaning; the directional defini- tion can be adapted to elaborate frameworks, like estimation of extreme quantiles and directional quantile regression, the regression of depth contours on covariates. The latter facilitates the construction of multivariate growth charts—the question that motivated this development.

Convex Optimization, Shape Constraints, Compound Decisions, and Empirical Bayes Rules

Estimation of mixture densities for the classical Gaussian compound decision problem and their associated (empirical) Bayes rules is considered from two new perspectives. The first, motivated by Brown and Greenshtein, introduces a nonparametric maximum likelihood estimator of the mixture density subject to a monotonicity constraint on the resulting Bayes rule. The second, motivated by Jiang and Zhang, proposes a new approach to computing the Kiefer–Wolfowitz nonparametric maximum likelihood estimator for mixtures. In contrast to prior methods for these problems, our new approaches are cast as convex optimization problems that can be efficiently solved by modern interior point methods. In particular, we show that the reformulation of the Kiefer–Wolfowitz estimator as a convex optimization problem reduces the computational effort by several orders of magnitude for typical problems, by comparison to prior EM-algorithm based methods, and thus greatly expands the practical applicability of the resulting methods. Our new procedures are compared with several existing empirical Bayes methods in simulations employing the well-established design of Johnstone and Silverman. Some further comparisons are made based on prediction of baseball batting averages. A Bernoulli mixture application is briefly considered in the penultimate section.

Quasi-concave density estimation

Maximum likelihood estimation of a log-concave probability density is formulated as a convex optimization problem and shown to have an equivalent dual formulation as a constrained maximum Shannon entropy problem. Closely related maximum Renyi entropy estimators that impose weaker concavity restrictions on the fitted density are also considered, notably a minimum Hellinger discrepancy estimator that constrains the reciprocal of the square-root of the density to be concave. A limiting form of these estimators constrains solutions to the class of quasi-concave densities.

Location-Scale Depth

This article introduces a halfspace depth in the location–scale model that is along the lines of the general theory given by Mizera, based on the idea of Rousseeuw and Hubert, and is complemented by a new likelihood-based principle for designing criterial functions. The most tractable version of the proposed depth—the Student depth—turns out to be nothing but the bivariate halfspace depth interpreted in the Poincaré plane model of the Lobachevski geometry. This fact implies many fortuitous theoretical and computational properties, in particular equivariance with respect to the Möbius group and favorable time complexities of algorithms. It also opens a way to introduce some other depth notions in the location–scale context, for instance, location–scale simplicial depth. A maximum depth estimator of location and scale—the Student median—is introduced. Possible applications of the proposed concepts are investigated on data examples.

Breakdown points of Cauchy regression-scale estimators

The lower bounds for the explosion and implosion breakdown points of the simultaneous Cauchy M-estimator (Cauchy MLE) of the regression and scale parameters are derived. For appropriate tuning constants, the breakdown point attains the maximum possible value.

Generalized runs tests for heteroscedastic time series

The problem of testing for nonhomogeneous white noise (i.e. independently but possibly nonidentically distributed observations, with a common, specified or unspecified, median) against alternatives of serial dependence is considered. This problem includes as a particular case the important problem of testing for heteroscedastic white noise. When the value of the common median is specified, invariance arguments suggest basing this test on a generalized version of classical runs: the generalized runs statistics. These statistics yield a run-based correlogram concept with exact (under the hypothesis of nonhomogeneous white noise) p-values. A run-based portmanteau test is also provided. The local powers and asymptotic relative efficiencies (AREs) of run-based correlograms and the corresponding run-based tests with respect to their traditional parametric counterparts (based on classical correlograms) are investigated and explicitly computed. In practice, however, the value of the exact median of the observations is seldom specified. For such situations, we propose two different solutions. The first solution is based on the classical idea of replacing the unknown median by its empirical counterpart, yielding aligned runs statistics. The asymptotic equivalence between exact and aligned runs statistics is established under extremely mild assumptions. These assumptions do not require that the empirical median consistently estimates the exact one, so that the continuity properties usually invoked in this context are totally helpless. The proofs we are giving are of a combinatorial nature, and related to the so-called Banach match box problem. The second solution is a finite-sample, nonasymptotic one, yielding (for fixed n) strictly conservative testing procedures, irrespectively of the underlying densities. Instead of the empirical median, a nonparametric confidence interval for the unknown median is considered. Run-based correlograms can be expected to play the same role in the statistical analysis of time series with nonhomogeneous innovation process as classical correlograms in the traditional context of second-order stationary ARMA series.

Partial functional linear quantile regression for neuroimaging data analysis

We propose a prediction procedure for the functional linear quantile regression model by using partial quantile covariance techniques and develop a simple partial quantile regression (SIMPQR) algorithm to efficiently extract partial quantile regression (PQR) basis for estimating functional coefficients. We further extend our partial quantile covariance techniques to functional composite quantile regression (CQR) defining partial composite quantile covariance. There are three major contributions. (1) We define partial quantile covariance between two scalar variables through linear quantile regression. We compute PQR basis by sequentially maximizing the partial quantile covariance between the response and projections of functional covariates. (2) In order to efficiently extract PQR basis, we develop a SIMPQR algorithm analog to simple partial least squares (SIMPLS). (3) Under the homoscedasticity assumption, we extend our techniques to partial composite quantile covariance and use it to find the partial composite quantile regression (PCQR) basis. The SIMPQR algorithm is then modified to obtain the SIMPCQR algorithm. Two simulation studies show the superiority of our proposed methods. Two real data from ADHD-200 sample and ADNI are analyzed using our proposed methods.

Continuous chaotic functions of an interval have generically small scrambled sets

It is shown that continuous self-mappings of a compact interval, chaotic in the sense of Li and Yorke, have generically, in the uniform topology, only scrambled sets which are nowhere dense and of zero Lebesgue measure.

Tail Behavior and Breakdown Properties of Equivariant Estimators of Location

For translation and scale equivariant estimators of location, inequalities connecting tail behavior and the finite-sample breakdown point are proved, analogous to those established by He et al. (1990, Econometrika, 58, 1195–1214) for monotone and translation equivariant estimators. Some other inequalities are given as well, enabling to establish refined bounds and in some cases exact values for the tail behavior under heavy- and light-tailed distributions. The inequalities cover translation and scale equivariant estimators in great generality, and they involve new breakdown-related quantities, whose relations to the breakdown point are discussed. The worth of tail-behavior considerations in robustness theory is demonstrated on examples, showing the impact of the basic two techniques in robust estimation: trimming and averaging. The mathematical language employs notions from regular variation theory.

Elastic and Plastic Splines: Some Experimental Comparisons

We give some empirical comparisons between two nonparametric regression methods based on regularization: the elastic or thin-plate splines, and plastic splines based on total variation penalties.