Minimax rates of convergence for nonparametric location-scale models

This paper studies minimax rates of convergence for nonparametric location-scale models, which include mean, quantile and expectile regression settings. Under Hellinger differentiability on the error distribution and other mild conditions, we show that the minimax rate of convergence for estimating the regression function under the squared $L_2$ loss is determined by the metric entropy of the nonparametric function class. Different error distributions, including asymmetric Laplace distribution, asymmetric connected double truncated gamma distribution, connected normal-Laplace distribution, Cauchy distribution and asymmetric normal distribution are studied as examples. Applications on low order interaction models and multiple index models are also given

Bayesian Semi-parametric Expected Shortfall Forecasting in Financial Markets

Bayesian semi-parametric estimation has proven effective for quantile estimation in general and specifically in financial Value at Risk forecasting. Expected short-fall is a competing tail risk measure, involving a conditional expectation beyond a quantile, that has recently been semi-parametrically estimated via asymmetric least squares and so-called expectiles. An asymmetric Gaussian density is proposed allowing a likelihood to be developed that leads to Bayesian semi-parametric estimation and forecasts of expectiles and expected shortfall. Further, the conditional autoregressive expectile class of model is generalised to two fully nonlinear families. Adaptive Markov chain Monte Carlo sampling schemes are employed for estimation in these families. The proposed models are clearly favoured in an empirical study forecasting eleven financial return series: clear evidence of more accurate expected shortfall forecasting, compared to a range of competing methods is found. Further, the most favoured models are those estimated by Bayesian methods

Kernel-based expectile regression

Conditional expectiles are becoming an increasingly important tool in finance as well as in other areas of application such as demography when the goal is to explore the conditional distribution beyond the conditional mean. In this thesis, we consider a support vector machine (SVM) type approach with the asymmetric least squares loss for estimating conditional expectiles. Firstly, we establish learning rates for this approach that are minimax optimal modulo a logarithmic factor if Gaussian RBF kernels are used and the desired expectile is smooth in a Besov sense. It turns out that our learning rates, as a special case, improve the best known rates for kernel-based least squares regression in aforementioned scenario. As key ingredients of our statistical analysis, we establish a general calibration inequality for the asymmetric least squares loss, a corresponding variance bound as well as an improved entropy number bound for Gaussian RBF kernels. Furthermore, we establish optimal learning rates in the case of a generic kernel under the assumption that the target function is in a real interpolation space. Secondly, we complement the theoretical results of our SVM approach with the empirical findings. For this purpose we use a sequential minimal optimization method and design an SVM-like solver for expectile regression considering Gaussian RBF kernels. We conduct various experiments in order to investigate the behavior of the designed solver with respect to different combinations of initialization strategies, working set selection strategies, stopping criteria and number of nearest neighbors, and then look for the best combination of them. We further compare the results of our solver to the recent R-package ER-Boost and find that our solver exhibits a better test performance. In terms of training time, our solver is found to be more sensitive to the training set size and less sensitive to the dimensions of the data set, whereas, ER-Boost behaves the other way around. In addition, our solver is found to be faster than a similarly implemented solver for the quantile regression. Finally, we show the convergence of our designed solver

Mathematical Statistics of Partially Identified Objects

The workshop brought together leading experts in mathematical statistics, theoretical econometrics and bio-mathematics interested in mathematical objects occurring in the analysis of partially identified structures. The mathematical core of these ubiquitous structures has an impact on all three research areas and is expected to lead to the development of new algorithms for solving such problems

What is Essential for Unseen Goal Generalization of Offline Goal-conditioned RL?

Offline goal-conditioned RL (GCRL) offers a way to train general-purpose agents from fully offline datasets. In addition to being conservative within the dataset, the generalization ability to achieve unseen goals is another fundamental challenge for offline GCRL. However, to the best of our knowledge, this problem has not been well studied yet. In this paper, we study out-of-distribution (OOD) generalization of offline GCRL both theoretically and empirically to identify factors that are important. In a number of experiments, we observe that weighted imitation learning enjoys better generalization than pessimism-based offline RL method. Based on this insight, we derive a theory for OOD generalization, which characterizes several important design choices. We then propose a new offline GCRL method, Generalizable Offline goAl-condiTioned RL (GOAT), by combining the findings from our theoretical and empirical studies. On a new benchmark containing 9 independent identically distributed (IID) tasks and 17 OOD tasks, GOAT outperforms current state-of-the-art methods by a large margin.Comment: Accepted by Proceedings of the 40th International Conference on Machine Learning, 202