19,908 research outputs found
Implementing Loss Distribution Approach for Operational Risk
To quantify the operational risk capital charge under the current regulatory
framework for banking supervision, referred to as Basel II, many banks adopt
the Loss Distribution Approach. There are many modeling issues that should be
resolved to use the approach in practice. In this paper we review the
quantitative methods suggested in literature for implementation of the
approach. In particular, the use of the Bayesian inference method that allows
to take expert judgement and parameter uncertainty into account, modeling
dependence and inclusion of insurance are discussed
Informative Data Projections: A Framework and Two Examples
Methods for Projection Pursuit aim to facilitate the visual exploration of
high-dimensional data by identifying interesting low-dimensional projections. A
major challenge is the design of a suitable quality metric of projections,
commonly referred to as the projection index, to be maximized by the Projection
Pursuit algorithm. In this paper, we introduce a new information-theoretic
strategy for tackling this problem, based on quantifying the amount of
information the projection conveys to a user given their prior beliefs about
the data. The resulting projection index is a subjective quantity, explicitly
dependent on the intended user. As a useful illustration, we developed this
idea for two particular kinds of prior beliefs. The first kind leads to PCA
(Principal Component Analysis), shining new light on when PCA is (not)
appropriate. The second kind leads to a novel projection index, the
maximization of which can be regarded as a robust variant of PCA. We show how
this projection index, though non-convex, can be effectively maximized using a
modified power method as well as using a semidefinite programming relaxation.
The usefulness of this new projection index is demonstrated in comparative
empirical experiments against PCA and a popular Projection Pursuit method
Detecting spatial patterns with the cumulant function. Part I: The theory
In climate studies, detecting spatial patterns that largely deviate from the
sample mean still remains a statistical challenge. Although a Principal
Component Analysis (PCA), or equivalently a Empirical Orthogonal Functions
(EOF) decomposition, is often applied on this purpose, it can only provide
meaningful results if the underlying multivariate distribution is Gaussian.
Indeed, PCA is based on optimizing second order moments quantities and the
covariance matrix can only capture the full dependence structure for
multivariate Gaussian vectors. Whenever the application at hand can not satisfy
this normality hypothesis (e.g. precipitation data), alternatives and/or
improvements to PCA have to be developed and studied. To go beyond this second
order statistics constraint that limits the applicability of the PCA, we take
advantage of the cumulant function that can produce higher order moments
information. This cumulant function, well-known in the statistical literature,
allows us to propose a new, simple and fast procedure to identify spatial
patterns for non-Gaussian data. Our algorithm consists in maximizing the
cumulant function. To illustrate our approach, its implementation for which
explicit computations are obtained is performed on three family of of
multivariate random vectors. In addition, we show that our algorithm
corresponds to selecting the directions along which projected data display the
largest spread over the marginal probability density tails.Comment: 9 pages, 3 figure
Among-site variability in the stochastic dynamics of East African coral reefs
Coral reefs are dynamic systems whose composition is highly influenced by
unpredictable biotic and abiotic factors. Understanding the spatial scale at
which long-term predictions of reef composition can be made will be crucial for
guiding conservation efforts. Using a 22-year time series of benthic
composition data from 20 reefs on the Kenyan and Tanzanian coast, we studied
the long-term behaviour of Bayesian vector autoregressive state-space models
for reef dynamics, incorporating among-site variability. We estimate that if
there were no among-site variability, the total long-term variability would be
approximately one third of its current value. Thus among-site variability
contributes more to long-term variability in reef composition than does
temporal variability. Individual sites are more predictable than previously
thought, and predictions based on current snapshots are informative about
long-term properties. Our approach allowed us to identify a subset of possible
climate refugia sites with high conservation value, where the long-term
probability of coral cover <= 0.1 was very low. Analytical results show that
this probability is most strongly influenced by among-site variability and by
interactions among benthic components within sites. These findings suggest that
conservation initiatives might be successful at the site scale as well as the
regional scale.Comment: 97 pages, 49 figure
Shrinkage Estimation in Multilevel Normal Models
This review traces the evolution of theory that started when Charles Stein in
1955 [In Proc. 3rd Berkeley Sympos. Math. Statist. Probab. I (1956) 197--206,
Univ. California Press] showed that using each separate sample mean from
Normal populations to estimate its own population mean can be
improved upon uniformly for every possible . The
dominating estimators, referred to here as being "Model-I minimax," can be
found by shrinking the sample means toward any constant vector. Admissible
minimax shrinkage estimators were derived by Stein and others as posterior
means based on a random effects model, "Model-II" here, wherein the
values have their own distributions. Section 2 centers on Figure 2, which
organizes a wide class of priors on the unknown Level-II hyperparameters that
have been proved to yield admissible Model-I minimax shrinkage estimators in
the "equal variance case." Putting a flat prior on the Level-II variance is
unique in this class for its scale-invariance and for its conjugacy, and it
induces Stein's harmonic prior (SHP) on .Comment: Published in at http://dx.doi.org/10.1214/11-STS363 the Statistical
Science (http://www.imstat.org/sts/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Selection of proposal distributions for generalized importance sampling estimators
The standard importance sampling (IS) estimator, generally does not work well
in examples involving simultaneous inference on several targets as the
importance weights can take arbitrarily large values making the estimator
highly unstable. In such situations, alternative generalized IS estimators
involving samples from multiple proposal distributions are preferred. Just like
the standard IS, the success of these multiple IS estimators crucially depends
on the choice of the proposal distributions. The selection of these proposal
distributions is the focus of this article. We propose three methods based on
(i) a geometric space filling coverage criterion, (ii) a minimax variance
approach, and (iii) a maximum entropy approach. The first two methods are
applicable to any multi-proposal IS estimator, whereas the third approach is
described in the context of Doss's (2010) two-stage IS estimator. For the first
method we propose a suitable measure of coverage based on the symmetric
Kullback-Leibler divergence, while the second and third approaches use
estimates of asymptotic variances of Doss's (2010) IS estimator and Geyer's
(1994) reverse logistic estimator, respectively. Thus, we provide consistent
spectral variance estimators for these asymptotic variances. The proposed
methods for selecting proposal densities are illustrated using various detailed
examples
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