Professor Jana Jurečková has extensively contributed to many areas in statistics and probability theory. Her contributions are highlighted here with reverent appreciation and admiration from all her colleagues, advisees as well as research collaborators.

The problem of testing equivalence of two ROC curves is addressed. A transformation of corresponding ROC curves, which motivates a test statistic based on a distance of two empirical quantile processes, is suggested, its asymptotic distribution found and a simulation scheme proposed that enables us to find critical values.

Charles Stein [10] discovered that, under quadratic loss, the usual unbiased estimator for the mean vector of a multivariate normal distribution is inadmissible if the dimension n of the mean vector exceeds two. On the way, he constructed shrinkage estimators that dominate the usual estimator asymptotically in n. It has since been claimed that Stein’s results and the subsequent James–Stein estimator are counter-intuitive, even paradoxical, and not very useful. In response to such doubts, various authors have presented alternative derivations of Stein shrinkage estimators. Surely Stein himself did not find his results paradoxical. This paper argues that assertions of “paradoxical" or “counter-intuitive" or “not practical" have overlooked essential arguments and remarks in Stein’s beautifully written paper [10]. Among these overlooked aspects are the asymptotic geometry of quadratic loss in high dimensions that makes Stein estimation transparent; the asymptotic optimality results that can be associate with Stein estimation; the explicit mention of practical multiple shrinkage estimators; and the foreshadowing of Stein confidence balls. These ideas are fundamental for studies of modern regularization estimators that rely on multiple shrinkage, whether implicitly or overtly.

Rank-based inference and, in particular, R-estimation, is a red thread running through Jana Jurečková’s entire scientific career, starting with her dissertation in 1967, where she laid the foundations of an extension to linear regression of the R-estimation methods that had recently been proposed by Hodges and Lehmann [13]. Cross-information quantities in that context play an essential role. In location/regression problems, these quantities take the form ∫₀¹φ(u)φ_g(u) du where φ is a score function and φ_g(u):=g'(G⁻¹(u))/g(G⁻¹(u)) is the log-derivative of the unknown actual underlying density g computed at the quantile G⁻¹(u); in other models, they involve more general scores. Such quantities appear in the local powers of rank tests and the asymptotic variance of R-estimators. Estimating them consistently is a delicate problem that has been extensively considered in the literature. We provide here a new, flexible, and very general method for that problem, which furthermore applies well beyond the traditional case of regression models.

This paper deals with nonparametric estimation of an unknown density function which possibly is discontinuous or non-differentiable in an unknown finite number of points. Estimation of such irregular densities is accomplished by viewing the problem as a regression problem and applying recent techniques for estimation of irregular regression curves. Moreover, the method can deal with estimation of densities that have an irregularity at the endpoint(s) of their support. A simulation study compares the performance of the proposed method with those of other methods available in the literature. A further illustration on real data is provided.

In this article we propose new methods for finding the direction of dependency between two random variables which are related by a linear function.

The two-sample Kolmogorov–Smirnov test can lose power as the size of one sample grows while the size of the other sample remains constant. In this case, a paradoxical situation takes place: the use of additional observations weakens the ability of the test to reject the null hypothesis when it is false.

Consider the task of estimating a regression function for describing the relationship between a response and a vector of p predictors. Often only a small subset of all given candidate predictors actually effects the response, while the rest might inhibit the analysis. Procedures for variable selection aim to identify the true predictors. A method for variable selection when the dimension p of the regressor space is much larger than the sample size n is Sure Independence Screening (SIS). The number of predictors is to be reduced to a value less than the number of observations before conducting the regression analysis. As SIS is based on nonrobust estimators, outliers in the data might lead to the elimination of true predictors. Hence, a robustified version of SIS called RoSIS was proposed which is based on robust estimators. Here, we give a modification of RoSIS by using the MCD estimator in the new algorithm. The new procedure MCD-RoSIS leads to better results, especially under collinearity. In a simulation study we compare the performance of SIS, RoSIS and MCD-RoSIS w.r.t. their robustness against different types of data contamination as well as different degrees of collinearity.

We introduce a conceptual framework within which the problem of setting reference intervals is one of estimating population parameters. The framework enables us to broaden the possibilities for inference by showing how to create confidence intervals for population intervals. We propose a new kind of interval (the γ-mode interval) as the population parameter of interest and show how to estimate and make optimal inference about this interval. Finally, we clarify the relationship between our reference intervals and other types of intervals.

A simple sequential procedure is proposed for detection of a change in distribution when a training sample with no change is available. Its properties under both null and alternative hypothesis are studied and possible modifications are discussed. Theoretical results are accompanied by a simulation study.

A class of random vectors (X,Y),X∈ℝ^j,Y∈ℝ^k with characteristic functions of the form

h(s,t)=f(s)g(t)exp{s'Ct}

where C is a (j×k)-matrix and prime stands for transposition is introduced and studied. The class contains all Gaussian vectors and possesses some of their properties. A relation of the class to random vectors with Gaussian components is of a particular interest. The problem of describing all pairs of characteristic functions f(s),g(t) such that h(s,t) is a characteristic function is open.

This paper examines different approaches to classification and discrimination applied to two-dimensional (2D) grey-scale images of faces. The database containing 212 standardized images is divided to a training set and a validation set. The aim is the automatic localization of the mouth. We focus on template matching and compare the results with standard classification methods. We discuss the choice of a suitable template and inspect its robustness aspects.

While methods of image analysis are well-established, there exists a popular belief that statistical methods cannot handle this task. We ascertain that simple methods are successful even without a prior reduction of dimension and feature extraction. Template matching and linear discriminant analysis turn out to give very reliable results.

The analytic center estimator is defined as the analytic center of the so-called membership set. In this paper, we consider the asymptotics of this estimator under fairly general assumptions on the noise distribution.

Employing the regression rankscore approach of Gutenbrunner and Jurečková [2] we consider rank tests designed to detect heterogeneous treatment effects concentrated in the upper tail of the conditional response distribution given other covariates.

In this note we establish asymptotic normality of a class of minimum distance estimators of autoregressive parameters when error variance is infinite, thereby extending the domain of their applications to a larger class of error distributions that includes a class of stable symmetric distributions having Pareto-like tails. These estimators are based on certain symmetrized randomly weighted residual empirical processes. In particular they include analogs of robustly weighted least absolute deviation and Hodges–Lehmann type estimators.

We show how a simple argument based on an inequality of McDiarmid yields strong consistency and central limit results for plug-in estimators of integral functionals of the density.

We formulate versions of Hampel’s theorem and its converse, establishing the connection between qualitative robustness and weak continuity in full generality and under minimal assumptions.

In this paper we review, prove and collect the results on the limiting behavior of the regular spatial median and its affine equivariant modification, the transformation retransformation spatial median. Estimation of the limiting covariance matrix of the spatial median is discussed as well. Some algorithms for the computation of the regular spatial median and its different modifications are described. The theory is illustrated with two examples.

The asymptotic properties of fixed-scale as well as studentized M-estimators in linear models with fixed carriers are studied. A two term von Mises expansion (second order asymptotic representation) is derived and verified. Possible applications of this result are shortly discussed.

The article deals with estimators of extreme value index based on two-step regression quantiles in the linear regression model. Two-step regression quantiles can be seen as a possible generalization of the quantile idea and as an alternative to regression quantiles. We derive the approximation of the tail quantile function of errors. Following Drees (1998) we consider a class of smooth functionals of the tail quantile function as a tool for the construction of estimators in the linear regression context. Pickands, maximum likelihood and probability weighted moments estimators are illustrated on simulated data.

While censored data is sufficiently common to have generated an enormous field of applied statistical research, the basic model for such data is also sufficiently non-standard to provide ample surprises to the statistical theorist, especially one who is too quick to assume regularity conditions. Here we show that estimators of the survival quantile function based on assuming additional information about the censoring distribution behave more poorly than estimators (like the inverse of Kaplan–Meier) that discard this information. This phenomenon will be explored with special emphasis on the Powell estimator, which assumes that all censoring times are observed.

In a simple measurement error regression model, the classical least squares estimator of the slope parameter consistently estimates a discounted slope, though sans normality, some other properties may not hold. It is shown that for a broader class of error distributions, the Theil–Sen estimator, albeit nonlinear, is a median-unbiased, consistent and robust estimator of the same discounted parameter. For a general class of nonlinear (including R–, M– and L– estimators), study of asymptotic properties is greatly facilitated by using some uniform asymptotic linearity results, which are, in turn, based on contiguity of probability measures. This contiguity is established in a measurement error model under broader distributional assumptions. Some asymptotic properties of the Theil–Sen estimator are studied under slightly different regularity conditions in a direct way bypassing the contiguity approach.

We study the group Lasso, where the number of groups is very large, and the sizes of the groups is large as well. We assume there is within group structure, in the sense that the ordering of the variables within groups in some loose sense expresses their relevance. We propose a within group weighting of the variables, and show that with this structure, the group Lasso satisfies a sparsity oracle inequality.

This paper discusses nonparametric estimation of quantiles of the residual lifetime distribution. The underlying model is a generalized Koziol–Green model for censored data, which accomodates both dependent censoring and covariate information.

A scale-equivariant and regression-invariant estimator of the variance of error terms in the linear regression model is proposed and its consistency proved. The estimator is based on (down)weighting the order statistics of the squared residuals which corresponds to the consistent and scale- and regression-equivariant estimator of the regression coefficients. A small numerical study demonstrating the behaviour of the estimator under the various types of contamination is included.