Research team

Fundamental Mathematics

Expertise

Applications of mathematical techniques in probability theory and (medical) statistics.

A study of the impact of stopping rules on conventional estimation with probabilistic and approach theoretic techniques 01/10/2017 - 31/08/2020

Abstract

In a sequential trial, the data collector is allowed to take intermediate looks at the data. After each intermediate look, he can decide, based on the data observed so far and a prescribed stopping rule, if the trial is stopped or continued. Stopping a trial early is potentially beneficial for economic and ethical reasons in e.g. a clinical study. The existing literature on sequential trials has reported that simple conventional estimators, such as the sample mean, become biased in the presence of observation based stopping rules. However, very recently, Molenberghs and colleagues have taken away some of this concern by showing that in many cases the bias, caused by stopping rules, vanishes quickly as the number of collected data increases. Building upon the insights obtained by Molenberghs and colleagues, we will provide a theoretical underpinning of the fact that conventional estimation remains in many important cases legitimate in a sequential trial. More precisely, we seek to establish Berry-Esseen type inequalities in sequential analysis that justify the use of confidence intervals based on conventional estimators. Also, the implications for hypothesis testing will be investigated. To this end, we will use mathematical techniques from probability, e.g. Stein's method, and approach theory, a topological theory pioneered by R. Lowen, which has already been useful in the theory of probability metrics, central limit theory, and estimation with contaminated data.

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Approach theory meets likelihood theory. 01/10/2014 - 30/09/2017

Abstract

Let the density of an unknown probability distribution belong to a family of densities which are determined up to an unknown parameter vector. Based on a sample of observations, the maximum likelihood estimator (MLE) then provides an estimate for the unknown parameter vector by picking the vector under which the chance of observing the particular given sample is maximal. Statisticians value the MLE because it behaves well asymptotically. This roughly means that the estimates given by the MLE will converge to the true value of the unknown parameter vector as the sample size grows to infinity. However, doubts about the applicability of the MLE have emerged as `misspecified models', i.e. models in which the density of the unknown probability distribution fails to belong to the family of densities producing the MLE, are common in realistic settings. Many researchers have investigated under which additional regularity conditions the MLE in a misspecified model continues to behave well asymptotically. Here we want to follow a different route. More precisely, instead of adding extra regularity conditions on the model (which may again destroy the applicability), we will try to establish results in which we measure how irregular a model is and then use this information to assess how asymptotically well the MLE behaves. To this end, we will use approach theory, a mathematical theory which is designed to cope rigorously with notions such as `almost behaving well asymptotically'

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Approach structures in probability theory. 01/10/2012 - 30/09/2014

Abstract

In this project we aim to develop a comprehensive and universally applicable theory of quantitative analysis of hitherto only topological structures (e.g. weak convergence, finite dimensional convergence, convergence in probability and in law) on spaces of probability measures and random variables (in particular continuous and cadlag stochastic processes) by replacing the topologies by canonical and intrinsically richer isometric counterparts, eventually aiming to prove quantitative versions of the fundamental results of stochastic analysis such as e.g. Prohorov's theorem and various important limit theorems.

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Approach structures in probability theory. 01/10/2010 - 30/09/2012

Abstract

In this project we aim to develop a comprehensive and universally applicable theory of quantitative analysis of hitherto only topological structures (e.g. weak convergence, finite dimensional convergence, convergence in probability and in law) on spaces of probability measures and random variables (in particular continuous and cadlag stochastic processes) by replacing the topologies by canonical and intrinsically richer isometric counterparts, eventually aiming to prove quantitative versions of the fundamental results of stochastic analysis such as e.g. Prohorov's theorem and various important limit theorems.

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A study of new quantitative convergence structures in probability theory and their application in stochastic analysis and parametric and non-parametric statistics. 01/10/2009 - 30/09/2010

Abstract

The first cornerstone is the study of new quantitative convergence structures in measure theoretic context, in particular on spaces of random variables and probability measures. We will mainly be concerned with structures strongly related to the p-Wasserstein distance, a topic popular for both applications and theoretical aspects. The second cornerstone is the application of the theory to stochastic analysis (convergence of Feller processes, martingales and solutions of stochastic differential equations) and statistics (convergence of estimators in parametric and non-parametric models).

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Research team(s)