A four moments theorem for Gamma limits on a Poisson chaos

This paper deals with sequences of random variables belonging to a fixed chaos of order $q$ generated by a Poisson random measure on a Polish space. The problem is investigated whether convergence of the third and fourth moment of such a suitably normalized sequence to the third and fourth moment of a centred Gamma law implies convergence in distribution of the involved random variables. A positive answer is obtained for $q=2$ and $q=4$. The proof of this four moments theorem is based on a number of new estimates for contraction norms. Applications concern homogeneous sums and $U$-statistics on the Poisson space

Supplement to "Erratum: Higher Order Elicitability and Osband's Principle"

This note corrects conditions in Proposition 3.4 and Theorem 5.2(ii) and comments on imprecisions in Propositions 4.2 and 4.4 in Fissler and Ziegel (2016).Comment: 12 pages, 1 figure, to appear as a supplement in the Annals of Statistic

Higher order elicitability and Osband's principle

A statistical functional, such as the mean or the median, is called elicitable if there is a scoring function or loss function such that the correct forecast of the functional is the unique minimizer of the expected score. Such scoring functions are called strictly consistent for the functional. The elicitability of a functional opens the possibility to compare competing forecasts and to rank them in terms of their realized scores. In this paper, we explore the notion of elicitability for multi-dimensional functionals and give both necessary and sufficient conditions for strictly consistent scoring functions. We cover the case of functionals with elicitable components, but we also show that one-dimensional functionals that are not elicitable can be a component of a higher order elicitable functional. In the case of the variance this is a known result. However, an important result of this paper is that spectral risk measures with a spectral measure with finite support are jointly elicitable if one adds the `correct' quantiles. A direct consequence of applied interest is that the pair (Value at Risk, Expected Shortfall) is jointly elicitable under mild conditions that are usually fulfilled in risk management applications.Comment: 32 page

On Higher Order Elicitability and Some Limit Theorems on the Poisson and Wiener Space

This PhD thesis consists of two independent parts. The first one is dedicated to a thorough study of higher order elicitability whereas the second part is concerned with qualitative and quantitative limit theorems for Poisson and Gaussian functionals. It comprises a total number of four articles, three of them already published in peer-reviewed journals (Annals of Statistics, Risk Magazine, and ALEA), the fourth one in a preprint version. The articles are accompanied by detailed additional material, primarily concerning questions of order-sensitivity, order-preservingness and convexity of strictly consistent scoring functions

Expected Shortfall is jointly elicitable with Value at Risk - Implications for backtesting

In this note, we comment on the relevance of elicitability for backtesting risk measure estimates. In particular, we propose the use of Diebold-Mariano tests, and show how they can be implemented for Expected Shortfall (ES), based on the recent result of Fissler and Ziegel (2015) that ES is jointly elicitable with Value at Risk

Erratum: Higher order elicitability and Osband's principle.

This note corrects conditions in Proposition 3.4 and Theorem 5.2(ii) and comments on imprecisions in Propositions 4.2 and 4.4 in Fissler and Ziegel (2016)

Backtesting Systemic Risk Forecasts using Multi-Objective Elicitability

Backtesting risk measure forecasts requires identifiability (for model validation) and elicitability (for model comparison). The systemic risk measures CoVaR (conditional value-at-risk), CoES (conditional expected shortfall) and MES (marginal expected shortfall), measuring the risk of a position $Y$ given that a reference position $X$ is in distress, fail to be identifiable and elicitable. We establish the joint identifiability of CoVaR, MES and (CoVaR, CoES) together with the value-at-risk (VaR) of the reference position $X$, but show that an analogue result for elicitability fails. The novel notion of multi-objective elicitability however, relying on multivariate scores equipped with an order, leads to a positive result when using the lexicographic order on $\mathbb{R}^2$. We establish comparative backtests of Diebold--Mariano type for superior systemic risk forecasts and comparable VaR forecasts, accompanied by a traffic-light approach. We demonstrate the viability of these backtesting approaches in simulations and in an empirical application to DAX 30 and S&P 500 returns.Comment: 43 pages, 8 figure

Measurability of functionals and of ideal point forecasts

The ideal probabilistic forecast for a random variable $Y$ based on an information set $\mathcal{F}$ is the conditional distribution of $Y$ given $\mathcal{F}$. In the context of point forecasts aiming to specify a functional $T$ such as the mean, a quantile or a risk measure, the ideal point forecast is the respective functional applied to the conditional distribution. This paper provides a theoretical justification why this ideal forecast is actually a forecast, that is, an $\mathcal{F}$-measurable random variable. To that end, the appropriate notion of measurability of $T$ is clarified and this measurability is established for a large class of practically relevant functionals, including elicitable ones. More generally, the measurability of $T$ implies the measurability of any point forecast which arises by applying $T$ to a probabilistic forecast. Similar measurability results are established for proper scoring rules, the main tool to evaluate the predictive accuracy of probabilistic forecasts.Comment: 13 page