23 research outputs found
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 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 and . The proof of this four
moments theorem is based on a number of new estimates for contraction norms.
Applications concern homogeneous sums and -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
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 given that a reference position 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 , 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 . 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 based on an
information set is the conditional distribution of given
. In the context of point forecasts aiming to specify a functional
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 -measurable random variable. To that end,
the appropriate notion of measurability of is clarified and this
measurability is established for a large class of practically relevant
functionals, including elicitable ones. More generally, the measurability of
implies the measurability of any point forecast which arises by applying
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
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
Evaluating Range Value at Risk Forecasts
The debate of what quantitative risk measure to choose in practice has mainly
focused on the dichotomy between Value at Risk (VaR) -- a quantile -- and
Expected Shortfall (ES) -- a tail expectation. Range Value at Risk (RVaR) is a
natural interpolation between these two prominent risk measures, which
constitutes a tradeoff between the sensitivity of the latter and the robustness
of the former, turning it into a practically relevant risk measure on its own.
As such, there is a need to statistically validate RVaR forecasts and to
compare and rank the performance of different RVaR models, tasks subsumed under
the term 'backtesting' in finance. The predictive performance is best evaluated
and compared in terms of strictly consistent loss or scoring functions. That
is, functions which are minimised in expectation by the correct RVaR forecast.
Much like ES, it has been shown recently that RVaR does not admit strictly
consistent scoring functions, i.e., it is not elicitable. Mitigating this
negative result, this paper shows that a triplet of RVaR with two VaR
components at different levels is elicitable. We characterise the class of
strictly consistent scoring functions for this triplet. Additional properties
of these scoring functions are examined, including the diagnostic tool of
Murphy diagrams. The results are illustrated with a simulation study, and we
put our approach in perspective with respect to the classical approach of
trimmed least squares in robust regression.Comment: 25 pages, 2 figures An earlier version of this paper was circulated
under the name 'Elicitability of Range Value at Risk'. The presentation has
been made more concise and minor errors have been corrected. Statistics &
Risk Modeling, 202
Generalised Covariances and Correlations
The covariance of two random variables measures the average joint deviations
from their respective means. We generalise this well-known measure by replacing
the means with other statistical functionals such as quantiles, expectiles, or
thresholds. Deviations from these functionals are defined via generalised
errors, often induced by identification or moment functions. As a normalised
measure of dependence, a generalised correlation is constructed. Replacing the
common Cauchy-Schwarz normalisation by a novel Fr\'echet-Hoeffding
normalisation, we obtain attainability of the entire interval for any
given marginals. We uncover favourable properties of these new dependence
measures. The families of quantile and threshold correlations give rise to
function-valued distributional correlations, exhibiting the entire dependence
structure. They lead to tail correlations, which should arguably supersede the
coefficients of tail dependence. Finally, we construct summary covariances
(correlations), which arise as (normalised) weighted averages of distributional
covariances. We retrieve Pearson covariance and Spearman correlation as special
cases. The applicability and usefulness of our new dependence measures is
illustrated on demographic data from the Panel Study of Income Dynamics