Frontiers in Nonparametric Statistics

The goal of this workshop was to discuss recent developments of nonparametric statistical inference. A particular focus was on high dimensional statistics, semiparametrics, adaptation, nonparametric bayesian statistics, shape constraint estimation and statistical inverse problems. The close interaction of these issues with optimization, machine learning and inverse problems has been addressed as well

Approximations of Semicontinuous Functions with Applications to Stochastic Optimization and Statistical Estimation

Upper semicontinuous (usc) functions arise in the analysis of maximization problems, distributionally robust optimization, and function identification, which includes many problems of nonparametric statistics. We establish that every usc function is the limit of a hypo-converging sequence of piecewise affine functions of the difference-of-max type and illustrate resulting algorithmic possibilities in the context of approximate solution of infinite-dimensional optimization problems. In an effort to quantify the ease with which classes of usc functions can be approximated by finite collections, we provide upper and lower bounds on covering numbers for bounded sets of usc functions under the Attouch-Wets distance. The result is applied in the context of stochastic optimization problems defined over spaces of usc functions. We establish confidence regions for optimal solutions based on sample average approximations and examine the accompanying rates of convergence. Examples from nonparametric statistics illustrate the results

Symbolic computation and exact distributions of nonparametric test statistics

We show how to use computer algebra for computing exact distributions on nonparametric statistics. We give several examples of nonparametric statistics with explicit probability generating functions that can be handled this way. In particular, we give a new table of critical values of the Jonckheere-Terpstra test that extends tables known in the literature

Nonparametric statistics

1 online resource (PDF, 28 pages

The Mellin transform in Nonparametric Statistics

This thesis deals with the nonparametric estimation for a special class of ill-posed inverse prob- lems, the so-called multiplicative measurement error models. In these models, the observations of the unknown, to be estimated quantity is only accessible with a multiplicative measurement error. As a consequence, the instability of the reconstruction depends on the distribution of the error by effecting the ill-posedness of the underlying inverse problem. The theory of Mellin transform al- lows to express the influence of the error distribution on the instability of the reconstruction and to reduce the estimation of the unknown quantity to a regularized estimation of its unknown Mellin transform. The proposed estimation strategies will be evaluated in terms of a mean weighted(- integrated) squared risk. Aside from being an introduction to the theory of Mellin transforms and multiplicative convolu- tions, this thesis is structuered in three topics. In the first part, we consider global density estimation under multiplicative measurement error. After a comparison between direct and noisy observations, we study several families of error dis- tributions, the multivariate case and the influence of dependence structures in the data. Here in each case we will propose an estimation strategy, discuss its minimax-optimality and consider data- driven choices of smoothing parameters. The theoritcal expected behavior of the estimators are illustrated through Monte-Carlo simulations. In the second part, we study global survival function estimation, which is, alongside the density of a distribution, a frequently considered characterization of a distribution. We once again propose an estimation method, prove its minimax-optimality and discuss data-driven choices of smoothing parameters. Furthermore, we analyse the stability of the estimator for Bernoulli-shift processes and visualize it using a Monte-Carlo simulation. The third part considers the estimation of the evaluation of an linear functional under multiplicative measurement errors. The point evaluation of the density, the survival function and the cumulative distribution function, to mention only a few, can be intrepreted as an evaluation of a linear func- tional. This allows the simultaneous analysis of these different estimation problems and the com- parison of the ill-posedness of the underlying inverse problems. A minimax-optimal estimation strategy as well as a data-driven choice of the smoothing parameters are presented and analyzed

Nonparametric statistics. Part 3. Correlation coefficients

A measure of correlation or strength of association between random variables is the correlation coefficient. In scientific research, correlation analysis is most often carried out using various correlation coefficients without explaining why this particular coefficient was chosen and what the resulting value of this coefficient means. The article discusses Spearman correlation coefficient, Kendall correlation coefficient, phi (Yule) correlation coefficient, Cramér’s correlation coefficient, Matthews correlation coefficient, Fechner correlation coefficient, Tschuprow correlation coefficient, rank-biserial correlation coefficient, point-biserial correlation coefficient, as well as association coefficient and contingency coefficient. The criteria for applying each of the coefficients are given. It is shown how to establish the significance (insignificance) of the resulting correlation coefficient. The scales in which the correlated variables should be located for the coefficients under consideration are presented. Spearman rank correlation coefficient and other nonparametric indicators are independent of the distribution law, and that is why they are very useful. They make it possible to measure the contingency between such attributes that cannot be directly measured, but can be expressed by points or other conventional units that allow ranking the sample. The benefit of rank correlation coefficient also lies in the fact that it allows to quickly assess the relationship between attributes regardless of the distribution law. Examples are given and step-by-step application of each coefficient is described. When analyzing scientific research and evaluating the results obtained, the strength of association is most commonly assessed by the correlation coefficient. In this regard, a number of scales are given (Chaddock scale, Cohen scale, Rosenthal scale, Hinkle scale, Evans scale) grading the strength of association for correlation coefficient, both widely recognized and not so well known