Advancements in Distribution Sampling for Statistical Inference

The process of inference in a parametric statistical model involves assessing the uncertainty of the surrounding parameters based on an observed sample. Often, the lack of analytical solutions prohibits a direct precise quantification. Monte Carlo (MC) simulations play a central role in understanding this uncertainty by reproducing samples that mimic data-dependent probability distributions or replicating data-generating mechanisms to estimate functionals and specific quantities of interest. This dissertation is dedicated to the advancement of methods for sampling algorithms for statistical inference in different paradigms. Chapter 2 focuses on general simulation-based strategies for obtaining confidence distributions, confidence curves and confidence densities in non non regular settings, where for instance Bootstrap methods are not directly applicable. Special attention is paid to the treatment of parameter vectors and nuisance parameters, to ensure invariance of the procedures under reparametrizations. The developed techniques are investigated in the context of robust methods and estimating equations. Some extensions are considered for inference with non-parametric tests, with probability semi-metrics. Possible applications can be found in non-inferiority tests and in the realm of likelihood-free inference. In Chapter 3, we study a method in the context of Approximate Bayesian Computation which is free from the choice of the tuning parameter. The resulting approximation implicitly uses a pseudo-likelihood that exhibits some consistency properties, and is linked to Confidence Distributions and data depth functions. In Chapter 4, we derive and discuss coupling techniques for Markov Chain Monte Carlo (MCMC) algorithms on submanifolds. They form the basis for the generation of convergence diagnoses and principal possibilities for the execution of parallel chains. In particular, we describe probabilistic reflection-contract couplings and meeting-inducing couplings, placing the latter in the context of couplings for a broader class of MCMC algorithms with complex proposal mechanisms. Chapter 5 presents two novel MCMC strategies developed for sampling generic target distributions on $\mathrm{R}^d$. These algorithms utilise ideas derived from MCMC algorithms on manifolds, incorporating geometric information from the target distribution in the problem at hand. In particular, equations specifically relevant to the sampling problem are used to define an artificial submanifold, such as the graph of the target distribution and the contour set. Finally, Chapter 6 deals with the problem of performing Bayesian inference in the presence of an intractable matching prior distribution. This involves transitioning to a manifold characterized by an estimating equation that encompasses the derivatives of the said intractable matching prior distribution. An application in the context of Bayesian testing with $e$-values is presented, where the default prior guarantees the invariance properties of the procedure.