160 research outputs found
Nonparametric estimation of the tree structure of a nested Archimedean copula
One of the features inherent in nested Archimedean copulas, also called
hierarchical Archimedean copulas, is their rooted tree structure. A
nonparametric, rank-based method to estimate this structure is presented. The
idea is to represent the target structure as a set of trivariate structures,
each of which can be estimated individually with ease. Indeed, for any three
variables there are only four possible rooted tree structures and, based on a
sample, a choice can be made by performing comparisons between the three
bivariate margins of the empirical distribution of the three variables. The set
of estimated trivariate structures can then be used to build an estimate of the
target structure. The advantage of this estimation method is that it does not
require any parametric assumptions concerning the generator functions at the
nodes of the tree.Comment: 25 pages, 9 figure
Nested Archimedean copulas: a new class of nonparametric tree structure estimators
Any nested Archimedean copula is defined starting from a rooted phylogenetic
tree, for which a new class of nonparametric estimators is presented. An
estimator from this new class relies on a two-step procedure where first a
binary tree is built and second is collapsed if necessary to give an estimate
of the target tree structure. Several examples of estimators from this class
are given and the performance of each of these estimators, as well as of the
only known comparable estimator, is assessed by means of a simulation study
involving target structures in various dimensions, showing that the new
estimators, besides being faster, usually offer better performance as well.
Further, among the given examples of estimators from the new class, one of the
best performing one is applied on three datasets: 482 students and their
results to various examens, 26 European countries in 1979 and the percentage of
workers employed in different economic activities, and 104 countries in 2002
for which various health-related variables are available. The resulting
estimated trees offer valuable insights on the analyzed data. The future of
nested Archimedean copulas in general is also discussed
Modeling Dependencies in Finance using Copulae
In this paper we provide a review of copula theory with applications to finance. We illustrate the idea on the bivariate framework and discuss the simple, elliptical and Archimedean classes of copulae. Since the cop- ulae model the dependency structure between random variables, next we explain the link between the copulae and common dependency measures, such as Kendall's tau and Spearman's rho. In the next section the copulae are generalized to the multivariate case. In this general setup we discuss and provide an intensive literature review of estimation and simulation techniques. Separate section is devoted to the goodness-of-fit tests. The importance of copulae in finance we illustrate on the example of asset allocation problems, Value-at-Risk and time series models. The paper is complemented with an extensive simulation study and an application to financial data.Distribution functions, Dimension Reduction, Risk management, Statistical models
HMM in dynamic HAC models
Understanding the dynamics of high dimensional non-normal dependency structure is a challenging task. This research aims at attacking this problem by building up a hidden Markov model (HMM) for Hierarchical Archimedean Copulae (HAC), where the HAC represent a wide class of models for high dimensional dependency, and HMM is a statistical technique to describe time varying dynamics. HMM applied to HAC provide flexible modeling for high dimensional non Gaussian time series. Consistency results for both parameters and HAC structures are established in an HMM framework. The model is calibrated to exchange rate data with a VaR application, where the model’s performance is compared with other dynamic models, and in the second application we simulate rainfall process.Hidden Markov model, Hierarchical Archimedean Copulae, Multivariate Distribution
copulaedas: An R Package for Estimation of Distribution Algorithms Based on Copulas
The use of copula-based models in EDAs (estimation of distribution
algorithms) is currently an active area of research. In this context, the
copulaedas package for R provides a platform where EDAs based on copulas can be
implemented and studied. The package offers complete implementations of various
EDAs based on copulas and vines, a group of well-known optimization problems,
and utility functions to study the performance of the algorithms. Newly
developed EDAs can be easily integrated into the package by extending an S4
class with generic functions for their main components. This paper presents
copulaedas by providing an overview of EDAs based on copulas, a description of
the implementation of the package, and an illustration of its use through
examples. The examples include running the EDAs defined in the package,
implementing new algorithms, and performing an empirical study to compare the
behavior of different algorithms on benchmark functions and a real-world
problem
Recommended from our members
Copulae: On the Crossroads of Mathematics and Economics
The central focus of the workshop was on copula theory as well as applications to multivariate stochastic modelling. The programme was intrinsically interdisciplinary and represented areas with much recent progress. The workshop included talks and dynamic discussions on construction, estimation and various applications of copulas to finance, insurance, hydrology, medicine, risk management and related fields
Bayesian model averaging over tree-based dependence structures for multivariate extremes
Describing the complex dependence structure of extreme phenomena is
particularly challenging. To tackle this issue we develop a novel statistical
algorithm that describes extremal dependence taking advantage of the inherent
hierarchical dependence structure of the max-stable nested logistic
distribution and that identifies possible clusters of extreme variables using
reversible jump Markov chain Monte Carlo techniques. Parsimonious
representations are achieved when clusters of extreme variables are found to be
completely independent. Moreover, we significantly decrease the computational
complexity of full likelihood inference by deriving a recursive formula for the
nested logistic model likelihood. The algorithm performance is verified through
extensive simulation experiments which also compare different likelihood
procedures. The new methodology is used to investigate the dependence
relationships between extreme concentration of multiple pollutants in
California and how these pollutants are related to extreme weather conditions.
Overall, we show that our approach allows for the representation of complex
extremal dependence structures and has valid applications in multivariate data
analysis, such as air pollution monitoring, where it can guide policymaking
On structure, family and parameter estimation of hierarchical Archimedean copulas
Research on structure determination and parameter estimation of hierarchical
Archimedean copulas (HACs) has so far mostly focused on the case in which all
appearing Archimedean copulas belong to the same Archimedean family. The
present work addresses this issue and proposes a new approach for estimating
HACs that involve different Archimedean families. It is based on employing
goodness-of-fit test statistics directly into HAC estimation. The approach is
summarized in a simple algorithm, its theoretical justification is given and
its applicability is illustrated by several experiments, which include
estimation of HACs involving up to five different Archimedean families.Comment: 63 pages, one attachment in attachment.pd
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