Lhasa, China, 2019

Axiomatic theory of risk measures | Risk sharing, allocation, and equilibria |

Joint mixability | Robust risk aggregation |

Risk management with risk measures | E-values: Theory and methods |

The series of working papers contains recent advances on the theory and methods of e-values. The recently introduced notion of e-values have many advantages over p-values. In particular, they are convenient for optional stopping in data collection and experiment design, arbitrary dependence in multiple testing, and high-dimensional composite hypotheses. E-values are intimately linked to martingales, likelihood ratios, and betting scores (Glenn Shafer); see details in WP01. This working paper series has an overlap with Hypothesis testing with e-values maintained by Vladimir Vovk. See also the work of Peter Grünwald and Aaditya Ramdas. Some basic terminology and their references are listed below. A brief description of each working paper is provided below to explain its main results and the logical structure across papers. | |||

WP12 |
E-backtesting (by Qiuqi Wang, Ruodu Wang, Johanna Ziegel) | ||

One of the most challenging tasks in risk modeling practice is to backtest ES (see the working paper series on Axiomatic theory of risk measures for a background on risk measures) forecasts provided by financial institutions. We use e-values and e-processes to construct a model-free backtesting procedure for ES using a concept of universal e-statistics, which can be naturally generalized to many other risk measures and statistical quantities. | |||

WP11 |
Efficiency of nonparametric e-tests (by Vladimir Vovk, Ruodu Wang) | ||

In this short note, we introduce a simple analogue for e-values of Pitmanâ€™s asymptotic relative efficiency and apply it to three popular nonparametric tests. | |||

WP10 |
E-values as unnormalized weights in multiple testing (by Ruodu Wang, Aaditya Ramdas) | ||

We study procedures based on both e-values and p-values for each hypothesis to show the advantages of using e-values as random weights. We also collect several other results, such as a tiny but uniform improvement of e-BH, a soft-rank permutation e-value, and the use of e-values as masks in interactive multiple testing. | |||

WP09 |
Post-selection inference for e-value based confidence intervals
(by Ziyu Xu, Ruodu Wang, Aaditya Ramdas) | ||

This is a follow-up of WP06. Inspired by the Benjamini-Yekutieli (BY) procedure for reporting confidence intervals, we design the e-BY procedure which uses e-CIs instead of p-CIs. The e-BY procedure has a desirable false coverage rate control under arbitrary dependence and stopping rules, in contrast to the BY procedure. | |||

WP08 |
A unified framework for bandit multiple testing
(by Ziyu Xu, Ruodu Wang, Aaditya Ramdas) | ||

This is a follow-up of
WP06. We apply e-values and the e-BH procedure to multi-armed bandit testing problems. We propose a unified, modular framework for FDR control that emphasizes the decoupling of exploration and summarization of evidence. We utilize the powerful e-processes to ensure FDR control for arbitrary composite nulls, exploration rules and stopping times in generic problem settings.
The conference version is published in | |||

WP07 |
Testing with p*-values: Between p-values, mid p-values, and e-values
(by Ruodu Wang) | ||

We introduce the notion of p*-values (p*-variables), which generalizes p-values (p-variables) in several senses. The new notion has four natural interpretations: operational, probabilistic, Bayesian, and frequentist. A main example of a p*-value is a mid p-value. A unified stochastic representation (Theorem 3.1) for p-values, mid p-values, and p*-values is obtained. The notion of p*-values becomes useful in many situations even if one is only interested in p-values, mid p-values, or e-value; it is connected to e-values in a simple way: 1/(2e) is a p*-value for an e-value e. | |||

WP06 |
False discovery rate control with e-values (by Ruodu Wang, Aaditya Ramdas) | ||

We design a natural analog of the Benjamini-Hochberg (BH) procedure for false discovery control (FDR) control that utilizes e-values (e-BH) and compare it with the standard procedure for p-values. The e-BH procedure includes the BH procedure as a special case through calibration between p-values and e-values.
The journal version is published in | |||

WP05 |
Admissible ways of merging p-values under arbitrary dependence
(by Vladimir Vovk, Bin Wang, Ruodu Wang) | ||

This paper is a follow-up of WP12 of Joint mixability. We study the admissibility in Wald's sense of p-merging functions and their domination structure under arbitrary dependence. As a technical tool e-values become essential in the main results (Theorems 5.1 and 5.2).
Another result (Theorem 6.2) showing the admissibility of some p-merging methods uses results on Joint mixability.
The journal version is published in | |||

WP04 |
Merging sequential e-values via martingales
(by Vladimir Vovk, Ruodu Wang) | ||

This is a follow-up of WP01. We study the problem of merging sequential or independent e-values into one e-value for statistical decision making. We describe a class of e-value merging functions via martingale merging functions, and show that all merging methods for sequential e-values are dominated by such a class. In case of merging independent e-values, the situation becomes much more sophisticated, and we provide a general class of such merging functions based on reordered test martingales. | |||

WP03 |
True and false discoveries with independent e-values
(by Vladimir Vovk, Ruodu Wang) | ||

This is a follow-up of WP02. We use e-discovery matrices in the context of multiple hypothesis testing assuming that the base tests produce independent e-values, and compare those with procedures based on p-values. | |||

WP02 |
Confidence and discoveries with e-values (by Vladimir Vovk, Ruodu Wang) | ||

We discuss systematically two versions of confidence regions: those based on p-values and those based on e-values, the latter are e-confidence intervals (e-CIs). Both versions are applied to multiple hypothesis testing to construct procedures that control the number of false discoveries under arbitrary dependence between the base p- or e-values. We introduce the e-discovery matrices and show that it is efficient both computationally and statistically using simulated and real-world datasets. | |||

WP01 |
E-values: Calibration, combination, and applications
(by Vladimir Vovk, Ruodu Wang) | ||

We introduce e-values in their pure form to the statistical community. In particular, we demonstrate that e-values are often mathematically more tractable and develop procedures using e-values for multiple testing of a single hypothesis and testing multiple hypotheses.
Related concepts introduced in this paper are e-variables, sequential e-values, and e-merging functions.
The journal version is published in |