241 research outputs found
Operator Regular Variation of Multivariate Liouville Distributions
Operator regular variation reveals general power-law distribution tail decay
phenomena using operator scaling, that includes multivariate regular variation
with scalar scaling as a special case. In this paper, we show that a
multivariate Liouville distribution is operator regularly varying if its
driving function is univariate regularly varying. Our method focuses on
operator regular variation of multivariate densities, which implies, as we also
show in this paper, operator regular variation of the multivariate
distributions. This general result extends the general closure property of
multivariate regular variation established by de Haan and Resnick in 1987
Neural Likelihoods via Cumulative Distribution Functions
We leverage neural networks as universal approximators of monotonic functions
to build a parameterization of conditional cumulative distribution functions
(CDFs). By the application of automatic differentiation with respect to
response variables and then to parameters of this CDF representation, we are
able to build black box CDF and density estimators. A suite of families is
introduced as alternative constructions for the multivariate case. At one
extreme, the simplest construction is a competitive density estimator against
state-of-the-art deep learning methods, although it does not provide an easily
computable representation of multivariate CDFs. At the other extreme, we have a
flexible construction from which multivariate CDF evaluations and
marginalizations can be obtained by a simple forward pass in a deep neural net,
but where the computation of the likelihood scales exponentially with
dimensionality. Alternatives in between the extremes are discussed. We evaluate
the different representations empirically on a variety of tasks involving tail
area probabilities, tail dependence and (partial) density estimation.Comment: 10 page
Modelling multivariate extremes through angular-radial decomposition of the density function
We present a new framework for modelling multivariate extremes, based on an
angular-radial representation of the probability density function. Under this
representation, the problem of modelling multivariate extremes is transformed
to that of modelling an angular density and the tail of the radial variable,
conditional on angle. Motivated by univariate theory, we assume that the tail
of the conditional radial distribution converges to a generalised Pareto (GP)
distribution. To simplify inference, we also assume that the angular density is
continuous and finite and the GP parameter functions are continuous with angle.
We refer to the resulting model as the semi-parametric angular-radial (SPAR)
model for multivariate extremes. We consider the effect of the choice of polar
coordinate system and introduce generalised concepts of angular-radial
coordinate systems and generalised scalar angles in two dimensions. We show
that under certain conditions, the choice of polar coordinate system does not
affect the validity of the SPAR assumptions. However, some choices of
coordinate system lead to simpler representations. In contrast, we show that
the choice of margin does affect whether the model assumptions are satisfied.
In particular, the use of Laplace margins results in a form of the density
function for which the SPAR assumptions are satisfied for many common families
of copula, with various dependence classes. We show that the SPAR model
provides a more versatile framework for characterising multivariate extremes
than provided by existing approaches, and that several commonly-used approaches
are special cases of the SPAR model. Moreover, the SPAR framework provides a
means of characterising all `extreme regions' of a joint distribution using a
single inference. Applications in which this is useful are discussed
Quantile Coherency: A General Measure for Dependence between Cyclical Economic Variables
In this paper, we introduce quantile coherency to measure general dependence
structures emerging in the joint distribution in the frequency domain and argue
that this type of dependence is natural for economic time series but remains
invisible when only the traditional analysis is employed. We define estimators
which capture the general dependence structure, provide a detailed analysis of
their asymptotic properties and discuss how to conduct inference for a general
class of possibly nonlinear processes. In an empirical illustration we examine
the dependence of bivariate stock market returns and shed new light on
measurement of tail risk in financial markets. We also provide a modelling
exercise to illustrate how applied researchers can benefit from using quantile
coherency when assessing time series models.Comment: paper (49 pages) and online supplement (31 pages), R codes to
replicate the figures in the paper are available at
https://github.com/tobiaskley/quantile_coherency_replicatio
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