Isotonic Distributional Regression

Distributional regression estimates the probability distribution of a response variable conditional on covariates. The estimated conditional distribution comprehensively summarizes the available information on the response variable, and allows to derive all statistical quantities of interest, such as the conditional mean, threshold exceedance probabilities, or quantiles. This thesis develops isotonic distributional regression, a method for estimating conditional distributions under the assumption of a monotone relationship between covariates and a response variable. The response variable is univariate and real-valued, and the covariates lie in a partially ordered set. The monotone relationship is formulated in terms of stochastic order constraints, that is, the response variable increases in a stochastic sense as the covariates increase in the partial order. This assumption alone yields a shape-constrained non-parametric estimator, which does not involve any tuning parameters. The estimation of distributions under stochastic order restrictions has already been studied for various stochastic orders, but so far only with totally ordered covariates. Apart from considering more general partially ordered covariates, the first main contribution of this thesis lies in a shift of focus from estimation to prediction. Distributional regression is the backbone of probabilistic forecasting, which aims at quantifying the uncertainty about a future quantity of interest comprehensively in the form of probability distributions. When analyzed with respect to predominant criteria for probabilistic forecast quality, isotonic distributional regression is shown to have desirable properties. In addition, this thesis develops an efficient algorithm for the computation of isotonic distributional regression, and proposes an estimator under a weaker, previously not thoroughly studied stochastic order constraint. A main application of isotonic distributional regression is the uncertainty quantification for point forecasts. Such point forecasts sometimes stem from external sources, like physical models or expert surveys, but often they are generated with statistical models. The second contribution of this thesis is the extension of isotonic distributional regression to allow covariates that are point predictions from a regression model, which may be trained on the same data to which isotonic distributional regression is to be applied. This combination yields a so-called distributional index model. Asymptotic consistency is proved under suitable assumptions, and real data applications demonstrate the usefulness of the method. Isotonic distributional regression provides a benchmark in forecasting problems, as it allows to quantify the merits of a specific, tailored model for the application at hand over a generic method which only relies on monotonicity. In such comparisons it is vital to assess the significance of forecast superiority or of forecast misspecification. The third contribution of this thesis is the development of new, safe methods for forecast evaluation, which require no or minimal assumptions on the data generating processes

Nonparametric Statistical Inference with an Emphasis on Information-Theoretic Methods

This book addresses contemporary statistical inference issues when no or minimal assumptions on the nature of studied phenomenon are imposed. Information theory methods play an important role in such scenarios. The approaches discussed include various high-dimensional regression problems, time series and dependence analyses

Handbook of Mathematical Geosciences

This Open Access handbook published at the IAMG's 50th anniversary, presents a compilation of invited path-breaking research contributions by award-winning geoscientists who have been instrumental in shaping the IAMG. It contains 45 chapters that are categorized broadly into five parts (i) theory, (ii) general applications, (iii) exploration and resource estimation, (iv) reviews, and (v) reminiscences covering related topics like mathematical geosciences, mathematical morphology, geostatistics, fractals and multifractals, spatial statistics, multipoint geostatistics, compositional data analysis, informatics, geocomputation, numerical methods, and chaos theory in the geosciences

Statistical analysis of child growth data

The study of child growth is complex. There are many clinical questions to answer but not necessarily the statistical methodology to deal with these questions. Human growth begins at conception and continues into adult life. In chapter 1 we discuss the characteristics of the growth process from conception to maturity and the purpose of growth monitoring. In chapter 2 we summarise the mathematical approaches to growth data. In chapter 3 we summarise the approaches that have been used to detect growth faltering. In this chapter we introduce the conditional gain Z-score. The data set analysed within this thesis is from the Newcastle growth and development study. In infancy we have routine weights of 3415 term infants. A sub-sample of these infants were followed-up at 7-9 years as part of a research study. These children belonged to three subgroups: cases were children that were defined as failing to thrive in infancy, controls were matched to cases and a 20% systematic sample. The school entry data of the sub-sample followed at 7-9 years were retrieved from school health records. In chapter 4 we carry out a preliminary analysis of the routine infancy weight Z-scores. The infancy data provided the opportunity to generate the correlation structure of routine weight Z-scores in infancy. In chapter 5 we develop a model for this correlation structure. In chapter 7 we explore patterns in the conditional weight gain Z-scores and also suggest some alternative criteria for identifying growth faltering in infancy. In chapters 6, 8 and 9 we analyse the anthropometric data obtained at follow-up and school entry. In childhood, the conditional gain Z-score is used to contrast height with mid-parental height and height at follow-up with height at school entry. The anthropometric data of the case and control children will be compared

Proceedings of the NASA Workshop on Density Estimation and Function Smoothing

Statistical model identification techniques being developed to provide workable solutions to problems in density estimation and function smoothing are examined

Proceedings of the 36th International Workshop Statistical Modelling July 18-22, 2022 - Trieste, Italy

The 36th International Workshop on Statistical Modelling (IWSM) is the first one held in presence after a two year hiatus due to the COVID-19 pandemic. This edition was quite lively, with 60 oral presentations and 53 posters, covering a vast variety of topics. As usual, the extended abstracts of the papers are collected in the IWSM proceedings, but unlike the previous workshops, this year the proceedings will be not printed on paper, but it is only online. The workshop proudly maintains its almost unique feature of scheduling one plenary session for the whole week. This choice has always contributed to the stimulating atmosphere of the conference, combined with its informal character, encouraging the exchange of ideas and cross-fertilization among different areas as a distinguished tradition of the workshop, student participation has been strongly encouraged. This IWSM edition is particularly successful in this respect, as testified by the large number of students included in the program