Empirical Comparison of Skewed t-copula Models for Insurance and Financial Data

Skewed t-copulas recently became popular as a modeling tool of non-linear dependence in statistics. In this paper we consider three different versions of skewed t-copulas introduced by Demarta and McNeill; Smith, Gan and Kohn; and Azzalini and Capitanio. Each of these versions represents a generalization of the symmetric t-copula model, allowing for a different treatment of lower and upper tails. Each of them has certain advantages in mathematical construction, inferential tools and interpretability. Our objective is to apply models based on different types of skewed t-copulas to the same financial and insurance applications. We consider comovements of stock index returns and times-to-failure of related vehicle parts under the warranty period. In both cases the treatment of both lower and upper tails of the joint distributions is of a special importance. Skewed t-copula model performance is compared to the benchmark cases of Gaussian and symmetric Student t-copulas. Instruments of comparison include information criteria, goodness-of-fit and tail dependence. A special attention is paid to methods of estimation of copula parameters. Some technical problems with the implementation of maximum likelihood method and the method of moments suggest the use of Bayesian estimation. We discuss the accuracy and computational efficiency of Bayesian estimation versus MLE. Metropolis-Hastings algorithm with block updates was suggested to deal with the problem of intractability of conditionals

A new approach to construction of objective priors: Hellinger information

Non-informative priors play crucial role in objective Bayesian analysis. Most popular ways of construction of non-informative priors are provided by the Jeffreys rule, matching probability principle, and reference prior approach. An alternative construction of non-informative priors is suggested based on the concept of Hellinger information related to Hellinger distance. Under certain regularity conditions, limit behavior of the Hellinger distance as the difference in the parameter values goes down to zero is closely related to Fisher information. In this case our approach generalizes the Jeffreys rule. However, what is more interesting, Hellinger information can be also used to describe information properties of the parametric set in non-regular situations, when Fisher information does not exist. Non-informative priors based on Hellinger information are studied for the non-regular class of distributions defined by Ghosal and Samanta and for some interesting examples outside of this class

Hellinger distance and non-informative priors

This paper introduces an extension of the Jeffreys’ rule to the construction of objective priors for non-regular parametric families. A new class of priors based on Hellinger information is introduced as Hellinger priors. The main results establish the relationship of Hellinger priors to the Jeffreys’ rule priors in the regular case, and to the reference and probability matching priors for the non-regular class introduced by Ghosal and Samanta. These priors are also studied for some non-regular examples outside of this class. Their behavior proves to be similar to that of the reference priors considered by Berger, Bernardo, and Sun, however some differences are observed. For the multi-parameter case, a combination of Hellinger priors and reference priors is suggested and some examples are considered

Hellinger distance and non-informative priors

This paper introduces an extension of the Jeffreys’ rule to the construction of objective priors for non-regular parametric families. A new class of priors based on Hellinger information is introduced as Hellinger priors. The main results establish the relationship of Hellinger priors to the Jeffreys’ rule priors in the regular case, and to the reference and probability matching priors for the non-regular class introduced by Ghosal and Samanta. These priors are also studied for some non-regular examples outside of this class. Their behavior proves to be similar to that of the reference priors considered by Berger, Bernardo, and Sun, however some differences are observed. For the multi-parameter case, a combination of Hellinger priors and reference priors is suggested and some examples are considered

Joint distribution of stock Indices: Methodological aspects of construction and selection of copula models

The paper discusses the practical aspects of modeling joint distribution of pairs of national stock indices via copula functions. Parameters of marginal distributions and the association parameter describing the dependence structure are estimated using empirical Bayes method numerically implemented with the help of random walk Metropolis algorithm. A comparison of parametric and semiparametric approaches to copula model construction is performed. The problem of selection of a class of pair copula functions approximating such empirical characteristics of stock indices dependence as Kendall’s concordance, joint empirical cumulative distribution function, and tail behavior

Bayesian model selection for hierarchical copulas and vines

Copula models provide an effective tool for modeling joint distributions. Model selection allowing to choose an appropriate subclass of copulas remains a critical issue for many applications. The paper suggests an implementation of Bayesian model selection procedure based on ideas of Bretthorst, Huard et al. It allows us to compare several classes of Archimedean copulas (Frank’s, Clayton’s, and and survival Gumbel-Hougaard families) and elliptical copulas (Gaussian and Student t-copulas). For dimensions higher than 2 we consider several types of hierarchical structures including nested Archimedean copulas, hierarchical Kendall copulas and vines. We consider a portfolio based on four national indices. Extreme market co-movements are modeled by the tail behavior of the joint distribution or index returns and currency exchange rates. Estimation of parameters within suggested copula families and hierarchical structures is carried out via empirical Bayes approach using random walk Metropolis algorithm and other Markov chain Monte Carlo techniques

An adaptive backward coupling Metropolis algorithm for truncated distributions

In the paper, a new version of the independent Metropolis algorithm is considered, combining adaptive proposals with perfect sampling. construction of a non-regenerative adaptive Metropolis algorithm follows. A backwards coupling procedure is applied after each adaptation in order to guarantee the St.ationarity of the target. The result is a perfect sample from the target distribution with undesirable positive autocorrelations suppressed by adaptations. Performance of the suggested algorithm is examined using several examples of uniform and Gaussian proposals for truncated non-Gaussian targets. These examples are related to a problem of detection of the point source of a scattered signal

Random rules and the ancient history of simulation

Modern approaches to simulation, involving Monte Carlo methods and randomized procedures of decision-making, are usually dated back to the mid-20-th century and the arrival of the computer era. Deeper history goes back to the 19-th and even 18-th centuries and involves such devices as Galton’s board and Buffon’s needle. However, one can argue that long before the invention of computers, older devices such as dice and their predecessors have been effectively used for games and divination. The idea of this paper is to review the use of ancient randomizing devices to trace the history of simulation and random rules of decision-making. Special attention will be paid to some contemporary cultures, which have preserved some unique elements of their ancient history: native cultures of the Americas, the Celtic civilizations of Ireland and Scotland, and the indigenous peoples of Northern and Central Asia (Altai and Siberia)