27,766 research outputs found
Robust estimation of risks from small samples
Data-driven risk analysis involves the inference of probability distributions
from measured or simulated data. In the case of a highly reliable system, such
as the electricity grid, the amount of relevant data is often exceedingly
limited, but the impact of estimation errors may be very large. This paper
presents a robust nonparametric Bayesian method to infer possible underlying
distributions. The method obtains rigorous error bounds even for small samples
taken from ill-behaved distributions. The approach taken has a natural
interpretation in terms of the intervals between ordered observations, where
allocation of probability mass across intervals is well-specified, but the
location of that mass within each interval is unconstrained. This formulation
gives rise to a straightforward computational resampling method: Bayesian
Interval Sampling. In a comparison with common alternative approaches, it is
shown to satisfy strict error bounds even for ill-behaved distributions.Comment: 13 pages, 3 figures; supplementary information provided. A revised
version of this manuscript has been accepted for publication in Philosophical
Transactions of the Royal Society A: Mathematical, Physical and Engineering
Science
A General Framework for Updating Belief Distributions
We propose a framework for general Bayesian inference. We argue that a valid
update of a prior belief distribution to a posterior can be made for parameters
which are connected to observations through a loss function rather than the
traditional likelihood function, which is recovered under the special case of
using self information loss. Modern application areas make it is increasingly
challenging for Bayesians to attempt to model the true data generating
mechanism. Moreover, when the object of interest is low dimensional, such as a
mean or median, it is cumbersome to have to achieve this via a complete model
for the whole data distribution. More importantly, there are settings where the
parameter of interest does not directly index a family of density functions and
thus the Bayesian approach to learning about such parameters is currently
regarded as problematic. Our proposed framework uses loss-functions to connect
information in the data to functionals of interest. The updating of beliefs
then follows from a decision theoretic approach involving cumulative loss
functions. Importantly, the procedure coincides with Bayesian updating when a
true likelihood is known, yet provides coherent subjective inference in much
more general settings. Connections to other inference frameworks are
highlighted.Comment: This is the pre-peer reviewed version of the article "A General
Framework for Updating Belief Distributions", which has been accepted for
publication in the Journal of Statistical Society - Series B. This article
may be used for non-commercial purposes in accordance with Wiley Terms and
Conditions for Self-Archivin
Data and uncertainty in extreme risks - a nonlinear expectations approach
Estimation of tail quantities, such as expected shortfall or Value at Risk,
is a difficult problem. We show how the theory of nonlinear expectations, in
particular the Data-robust expectation introduced in [5], can assist in the
quantification of statistical uncertainty for these problems. However, when we
are in a heavy-tailed context (in particular when our data are described by a
Pareto distribution, as is common in much of extreme value theory), the theory
of [5] is insufficient, and requires an additional regularization step which we
introduce. By asking whether this regularization is possible, we obtain a
qualitative requirement for reliable estimation of tail quantities and risk
measures, in a Pareto setting
Estimating Financial Risk Measures for Futures Positions:A Non-Parametric Approach
This paper presents non-parametric estimates of spectral risk measures applied to long and short positions in 5 prominent equity futures contracts. It also compares these to estimates of two popular alternative measures, the Value-at-Risk (VaR) and Expected Shortfall (ES). The spectral risk measures are conditioned on the coefficient of absolute risk aversion, and the latter two are conditioned on the confidence level. Our findings indicate that all risk measures increase dramatically and their estimators deteriorate in precision when their respective conditioning parameter increases. Results also suggest that estimates of spectral risk measures and their precision levels are of comparable orders of magnitude as those of more conventional risk measures. Running head: financial risk measures for futures positions.
Regularizing Portfolio Optimization
The optimization of large portfolios displays an inherent instability to
estimation error. This poses a fundamental problem, because solutions that are
not stable under sample fluctuations may look optimal for a given sample, but
are, in effect, very far from optimal with respect to the average risk. In this
paper, we approach the problem from the point of view of statistical learning
theory. The occurrence of the instability is intimately related to over-fitting
which can be avoided using known regularization methods. We show how
regularized portfolio optimization with the expected shortfall as a risk
measure is related to support vector regression. The budget constraint dictates
a modification. We present the resulting optimization problem and discuss the
solution. The L2 norm of the weight vector is used as a regularizer, which
corresponds to a diversification "pressure". This means that diversification,
besides counteracting downward fluctuations in some assets by upward
fluctuations in others, is also crucial because it improves the stability of
the solution. The approach we provide here allows for the simultaneous
treatment of optimization and diversification in one framework that enables the
investor to trade-off between the two, depending on the size of the available
data set
Extreme Measures of Agricultural Financial Risk
This paper is from the Centre for Financial Markets (CFM) Working Paper series at University College Dublin.Agricultural financial risk, Spectral risk measures, Expected Shortfall, Value at Risk, Extreme Value Theory., Agricultural Finance, Risk and Uncertainty, E17, G19, N52,
There is a VaR beyond usual approximations
Basel II and Solvency 2 both use the Value-at-Risk (VaR) as the risk measure
to compute the Capital Requirements. In practice, to calibrate the VaR, a
normal approximation is often chosen for the unknown distribution of the yearly
log returns of financial assets. This is usually justified by the use of the
Central Limit Theorem (CLT), when assuming aggregation of independent and
identically distributed (iid) observations in the portfolio model. Such a
choice of modeling, in particular using light tail distributions, has proven
during the crisis of 2008/2009 to be an inadequate approximation when dealing
with the presence of extreme returns; as a consequence, it leads to a gross
underestimation of the risks. The main objective of our study is to obtain the
most accurate evaluations of the aggregated risks distribution and risk
measures when working on financial or insurance data under the presence of
heavy tail and to provide practical solutions for accurately estimating high
quantiles of aggregated risks. We explore a new method, called Normex, to
handle this problem numerically as well as theoretically, based on properties
of upper order statistics. Normex provides accurate results, only weakly
dependent upon the sample size and the tail index. We compare it with existing
methods.Comment: 33 pages, 5 figure
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