Local dependence estimation using semiparametric archimedean copulas

The authors define a new semiparametric Archimedean copula family which has a flexible dependence structure. The generator of the family is a local interpolation of existing generators. It has locally‐defined dependence parameters. The authors present a penalized constrained least‐squares method to estimate and smooth these parameters. They illustrate the flexibility of their dependence model in a bi‐variate survival example.     

Mixture of extreme-value distributions: identifiability and estimation

In this paper, the mixture model of k extreme value distributions is investigated. Using the Laplace transform of extreme value distributions given in terms of the Krätzel function, we first prove the identifiability of the class of arbitrary mixtures of extreme-value distributions of type 1 and type 2. We then find the estimates for the parameters of the mixture of two extreme-value distributions, including the three different types, via the EM algorithm. The performance of the estimates is tested by Monte Carlo simulation.     

Nonparametric estimation of the lower tail dependence λL in bivariate copulas

The lower tail dependence λ_L is a measure that characterizes the tendency of extreme co-movements in the lower tails of a bivariate distribution. It is invariant with respect to strictly increasing transformations of the marginal distribution and is therefore a function of the copula of the bivariate distribution. λ_L plays an important role in modelling aggregate financial risk with copulas. This paper introduces three non-parametric estimators for λ_L. They are weakly consistent under mild regularity conditions on the copula and under the assumption that the number k = k(n) of observations in the lower tail, used for estimation, is asymptotically k ≈ √n. The finite sample properties of the estimators are investigated using a Monte Carlo simulation in special cases. It turns out that these estimators are biased, where amount and sign of the bias depend on the underlying copula, on the sample size n, on k, and on the true value of λ_L.     

Nonparametric estimation of copula-based measures of multivariate association from contingency tables

Nonparametric estimation of copula-based measures of multivariate association in a continuous random vector X=(X₁, …, X_d) is usually based on complete continuous data. In many practical applications, however, these types of data are not readily available; instead aggregated ordinal observations are given, for example, ordinal ratings based on a latent continuous scale. This article introduces a purely nonparametric and data-driven estimator of the unknown copula density and the corresponding copula based on multivariate contingency tables. Estimators for multivariate Spearman's rho and Kendall's tau are based thereon. The properties of these estimators in samples of medium and large size are evaluated in a simulation study. An increasing bias can be observed along with an increasing degree of association between the components. As it is to be expected, the bias is severely influenced by the amount of information available. Additionally, the influence of sample size is only marginal. We further give an empirical illustration based on daily returns of five German stocks.     

Nonparametric estimation of copula functions for dependence modelling

Copulas characterize the dependence among components of random vectors. Unlike marginal and joint distributions, which are directly observable, the copula of a random vector is a hidden dependence structure that links the joint distribution with its margins. Choosing a parametric copula model is thus a nontrivial task but it can be facilitated by relying on a nonparametric estimator. Here the authors propose a kernel estimator of the copula that is mean square consistent everywhere on the support. They determine the bias and variance of this estimator. They also study the effects of kernel smoothing on copula estimation. They then propose a smoothing bandwidth selection rule based on the derived bias and variance. After confirming their theoretical findings through simulations, they use their kernel estimator to formulate a goodness-of-fit test for parametric copula models.     

On the effect of long-range dependence on extreme value copula estimation with fixed marginals

ABSTRACT

We establish the existence of multivariate stationary processes with arbitrary marginal copula distributions and long-range dependence. The effect of long-range dependence on extreme value copula estimation is illustrated in the case of known marginals, by deriving functional limit theorems for a standard non parametric estimator of the Pickands dependence function and related parametric projection estimators. The asymptotic properties turn out to be very different from the case of iid or short-range dependent observations. Simulated and real data examples illustrate the results.     

Maximum likelihood estimation of skew-t copulas with its applications to stock returns

The multivariate Student-t copula family is used in statistical finance and other areas when there is tail dependence in the data. It often is a good-fitting copula but can be improved on when there is tail asymmetry. Multivariate skew-t copula families can be considered when there is tail dependence and tail asymmetry, and we show how a fast numerical implementation for maximum likelihood estimation is possible. For the copula implicit in a multivariate skew-t distribution, the fast implementation makes use of (i) monotone interpolation of the univariate marginal quantile function and (ii) a re-parametrization of the correlation matrix. Our numerical approach is tested with simulated data with data-driven parameters. A real data example involves the daily returns of three stock indices: the Nikkei225, S&P500 and DAX. With both unfiltered returns and GARCH/EGARCH filtered returns, we compare the fits of the Azzalini–Capitanio skew-t, generalized hyperbolic skew-t, Student-t, skew-Normal and Normal copulas.     

Projection estimators of Pickands dependence functions

The authors consider the construction of intrinsic estimators for the Pickands dependence function of an extreme‐value copula. They show how an arbitrary initial estimator can be modified to satisfy the required shape constraints. Their solution consists in projecting this estimator in the space of Pickands functions, which forms a closed and convex subset of a Hilbert space. As the solution is not explicit, they replace this functional parameter space by a sieve of finite‐dimensional subsets. They establish the asymptotic distribution of the projection estimator and its finite‐dimensional approximations, from which they conclude that the projected estimator is at least as efficient as the initial one.     

Semiparametric estimation of spatial long-range dependence

Estimation of the long-range dependence parameter in spatial processes using a semiparametric approach is studied. An extended formulation of the averaged periodogram method proposed in Robinson [1994. Semiparametric analysis of long memory time series. Ann. Statist. 22, 515–539] is derived, considering a certain homogeneous and isotropic behaviour of the spectral distribution in the low frequencies. The weak consistency of the estimator proposed is proved.     

Nonparametric estimation of some important indicators in reliability for semi-Markov processes

In this article we present nonparametric estimators of the mean time to failure (MTTF), the mean up time (MUT), and the mean down time (MDT) of an arbitrary finite semi-Markov process. Strong consistency and weak convergence is proved for the estimators of MTTF, MUT, and MDT, as the time interval of observation becomes large.     

Positive quadrant dependence tests for copulas

In this paper the interest is in testing the null hypothesis of positive quadrant dependence (PQD) between two random variables. Such a testing problem is important since prior knowledge of PQD is a qualitative restriction that should be taken into account in further statistical analysis, for example, when choosing an appropriate copula function to model the dependence structure. The key methodology of the proposed testing procedures consists of evaluating a “distance” between a nonparametric estimator of a copula and the independence copula, which serves as a reference case in the whole set of copulas having the PQD property. Choices of appropriate distances and nonparametric estimators of copula are discussed, and the proposed methods are compared with testing procedures based on bootstrap and multiplier techniques. The consistency of the testing procedures is established. In a simulation study the authors investigate the finite sample size and power performances of three types of test statistics, Kolmogorov–Smirnov, Cramér–von‐Mises, and Anderson–Darling statistics, together with several nonparametric estimators of a copula, including recently developed kernel type estimators. Finally, they apply the testing procedures on some real data. The Canadian Journal of Statistics 38: 555–581; 2010 © 2010 Statistical Society of Canada