A new algorithm based on copulas for VaR valuation with empirical calculations

This paper concerns the application of copula functions in VaR valuation. The copula function is used to model the dependence structure of multivariate assets. After the introduction of the traditional Monte Carlo simulation method and the pure copula method we present a new algorithm based on mixture copula functions and the dependence measure, Spearman’s rho. This new method is used to simulate daily returns of two stock market indices in China, Shanghai Stock Composite Index and Shenzhen Stock Composite Index, and then empirically calculate six risk measures including VaR and conditional VaR. The results are compared with those derived from the traditional Monte Carlo method and the pure copula method. From the comparison we show that the dependence structure between asset returns plays a more important role in valuating risk measures comparing with the form of marginal distributions.     

Smooth semiparametric and nonparametric Bayesian estimation of bivariate densities from bivariate histogram data

Penalized B-splines combined with the composite link model are used to estimate a bivariate density from a histogram with wide bins. The goals are multiple: they include the visualization of the dependence between the two variates, but also the estimation of derived quantities like Kendall’s tau, conditional moments and quantiles. Two strategies are proposed: the first one is semiparametric with flexible margins modeled using B-splines and a parametric copula for the dependence structure; the second one is nonparametric and is based on Kronecker products of the marginal B-spline bases. Frequentist and Bayesian estimations are described. A large simulation study quantifies the performances of the two methods under different dependence structures and for varying strengths of dependence, sample sizes and amounts of grouping. It suggests that Schwarz’s BIC is a good tool for classifying the competing models. The density estimates are used to evaluate conditional quantiles in two applications in social and in medical sciences.     

Estimation of Kendall’s tau from censored data

This paper considers the nonparametric estimation of Kendall’s tau for bivariate censored data. Under censoring, there have been some papers discussing the nonparametric estimation of Kendall’s tau, such as Wang and Wells (2000), Oakes (2008) and Lakhal et al. (2009). In this article, we consider an alternative approach to estimate Kendall’s tau. The main idea is to replace a censored event-time by a proper imputation. Thus, it induces three estimators, say , , and . We also apply the bootstrap method to estimate the variance of , and and to construct the corresponding confidence interval. Furthermore, we analyze two data sets by the suggested approach, and compare these practical estimators of Kendall’s tau in simulation studies.     

On the performance of some non-parametric estimators of the conditional survival function with interval-censored data

Simple nonparametric estimators of the conditional distribution of a response variable given a continuous covariate are often useful in survival analysis. Since a few nonparametric estimation options are available, a comparison of the performance of these options may be of value to determine which approach to use in a given application. In this note, we compare various nonparametric estimators of the conditional survival function when the response is subject to interval- and right-censoring. The estimators considered are a generalization of Turnbull’s estimator proposed by Dehghan and Duchesne (2011) and two nonparametric estimators for complete or right-censored data used in conjunction with imputation methods, namely the Nadaraya-Watson and generalized Kaplan-Meier estimators. We study the finite sample integrated mean squared error properties of all these estimators by simulation and compare them to a semi-parametric estimator. We propose a rule-of-thumb based on simple sample summary statistics to choose the most appropriate among these estimators in practice.     

Dependence tree structure estimation via copula

We propose an approach for dependence tree structure learning via copula. A nonparametric algorithm for copula estimation is presented. Then a Chow-Liu like method based on dependence measure via copula is proposed to estimate maximum spanning bivariate copula associated with bivariate dependence relations. The main advantage of the approach is that learning with empirical copula focuses on dependence relations among random variables, without the need to know the properties of individual variables as well as without the requirement to specify parametric family of entire underlying distribution for individual variables. Experiments on two real-application data sets show the effectiveness of the proposed method.     

Efficiently sampling nested Archimedean copulas

Efficient sampling algorithms for both Archimedean and nested Archimedean copulas are presented. First, efficient sampling algorithms for the nested Archimedean families of Ali-Mikhail-Haq, Frank, and Joe are introduced. Second, a general strategy how to build a nested Archimedean copula from a given Archimedean generator is presented. Sampling this copula involves sampling an exponentially tilted stable distribution. A fast rejection algorithm is developed for the more general class of tilted Archimedean generators. It is proven that this algorithm reduces the complexity of the standard rejection algorithm to logarithmic complexity. As an application it is shown that the fast rejection algorithm outperforms existing algorithms for sampling exponentially tilted stable distributions involved, e.g., in nested Clayton copulas. Third, with the additional help of randomization of generator parameters, explicit sampling algorithms for several nested Archimedean copulas based on different Archimedean families are found. Additional results include approximations and some dependence properties, such as Kendall’s tau and tail dependence parameters. The presented ideas may also apply in the more general context of sampling distributions given by their Laplace-Stieltjes transforms.     

Eliminating bias due to censoring in Kendall’s tau estimators for quasi-independence of truncation and failure

While the currently available estimators for the conditional Kendall’s tau measure of association between truncation and failure are valid for testing the null hypothesis of quasi-independence, they are biased when the null does not hold. This is because they converge to quantities that depend on the censoring distribution. The magnitude of the bias relative to the theoretical Kendall’s tau measure of association between truncation and failure due to censoring has not been studied, and so its importance in real problems is not known. We quantify this bias in order to assess the practical usefulness of the estimators. Furthermore, we propose inverse probability weighted versions of the conditional Kendall’s tau estimators to remove the effects of censoring and provide asymptotic results for the estimators. In simulations, we demonstrate the decrease in bias achieved by these inverse probability weighted estimators. We apply the estimators to the Channing House data set and an AIDS incubation data set.     

Orthogonality-projection-based estimation for semi-varying coefficient models with heteroscedastic errors

This paper is concerned with the estimation in semi-varying coefficient models with heteroscedastic errors. An iterated two-stage orthogonality-projection-based estimation is proposed. This method can easily be used to estimate the model parametric and nonparametric parts, as well as the variance function, and in the estimators the parametric part and nonparametric part do not affect each other. Under some mild conditions, the consistency, conditional biases, conditional variances and asymptotic normality of the resulting estimators are studied explicitly. Moreover, some simulation studies are carried out to examine the finite sample performance of the proposed methods. Finally, the methodologies are illustrated by a real data set.     

Quantile curves and dependence structure for bivariate distributions

Within the context of a general bivariate distribution an intuitive method is presented in order to study the dependence structure of the two distributions. A set of points—level curve—which accumulate the same probability for a fixed quadrant is considered. This procedure provides four level curves which can be considered as the boundary of a generalization of the real interquantile interval. It is shown that the accumulated probability among the level curves depends on the dependence structure of the distribution function where the dependence structure is given by the notion of copula. Furthermore, the case when the marginal distributions are independent is investigated. This result is used to find out positive or negative dependence properties for the variables. Finally, a nonparametric test for independence with a local dependence meaning is performed and applied to different data sets.     

Asymptotically efficient estimation of the conditional expected shortfall

A procedure for efficient estimation of the trimmed mean of a random variable conditional on a set of covariates is proposed. For concreteness, the focus is on a financial application where the trimmed mean of interest corresponds to the conditional expected shortfall, which is known to be a coherent risk measure. The proposed class of estimators is based on representing the estimator as an integral of the conditional quantile function. Relative to the simple analog estimator that weights all conditional quantiles equally, asymptotic efficiency gains may be attained by giving different weights to the different conditional quantiles while penalizing excessive departures from uniform weighting. The approach presented here allows for either parametric or nonparametric modeling of the conditional quantiles and the weights, but is essentially nonparametric in spirit. The asymptotic properties of the proposed class of estimators are established. Their finite sample properties are illustrated through a set of Monte Carlo experiments and an empirical application¹.     

Conditional quantile estimation based on optimal quantization: From theory to practice

Small-sample properties of a nonparametric estimator of conditional quantiles based on optimal quantization, that was recently introduced (Charlier et al., 2015), are investigated. More precisely, (i) the practical implementation of this estimator is discussed (by proposing in particular a method to properly select the corresponding smoothing parameter, namely the number of quantizers) and (ii) its finite-sample performances are compared to those of classical competitors. Monte Carlo studies reveal that the quantization-based estimator competes well in all cases and sometimes dominates its competitors, particularly when the regression function is quite complex. A real data set is also treated. While the main focus is on the case of a univariate covariate, simulations are also conducted in the bivariate case.     

Nonparametric estimation of pair-copula constructions with the empirical pair-copula

A pair-copula construction is a decomposition of a multivariate copula into a structured system, called regular vine, of bivariate copulae or pair-copulae. The standard practice is to model these pair-copulae parametrically, inducing a model risk, with errors potentially propagating throughout the vine structure. The empirical pair-copula provides a nonparametric alternative, which is conjectured to still achieve the parametric convergence rate. Its main advantage for the user is that it does not require the choice of parametric models for each of the pair-copulae constituting the construction. It can be used as a basis for inference on dependence measures, for selecting an appropriate vine structure, and for testing for conditional independence.     

A nonparametric approach to weighted estimating equations for regression analysis with missing covariates

Missing data often occur in regression analysis. Imputation, weighting, direct likelihood, and Bayesian inference are typical approaches for missing data analysis. The focus is on missing covariate data, a common complication in the analysis of sample surveys and clinical trials. A key quantity when applying weighted estimators is the mean score contribution of observations with missing covariate(s), conditional on the observed covariates. This mean score can be estimated parametrically or nonparametrically by its empirical average using the complete case data in case of repeated values of the observed covariates, typically assuming categorical or categorized covariates. A nonparametric kernel based estimator is proposed for this mean score, allowing the full exploitation of the continuous nature of the covariates. The performance of the kernel based method is compared to that of a complete case analysis, inverse probability weighting, doubly robust estimators and multiple imputation, through simulations.     

Comparison of nonparametric methods in nonlinear mixed effects models

During the drug development, nonlinear mixed effects models are routinely used to study the drug’s pharmacokinetics and pharmacodynamics. The distribution of random effects is of special interest because it allows to describe the heterogeneity of the drug’s kinetics or dynamics in the population of individuals studied. Parametric models are widely used, but they rely on a normality assumption which may be too restrictive. In practice, this assumption is often checked using the empirical distribution of random effects’ empirical Bayes estimates. Unfortunately, when data are sparse (like in patients phase III clinical trials), this method is unreliable. In this context, nonparametric estimators of the random effects distribution are attractive. Several nonparametric methods (estimators and their associated computation algorithms) have been proposed but their use is limited. Indeed, their practical and theoretical properties are unclear and they have a reputation for being computationally expensive. Four nonparametric methods in comparison with the usual parametric method are evaluated. Statistical and computational features are reviewed and practical performances are compared in simulation studies mimicking real pharmacokinetic analyses. The nonparametric methods seemed very useful when data are sparse. On a simple pharmacokinetic model, all the nonparametric methods performed roughly equivalently. On a more challenging pharmacokinetic model, differences between the methods were clearer.     

On comparing two sequences of numbers and its applications to clustering analysis

A conceptual problem that appears in different contexts of clustering analysis is that of measuring the degree of compatibility between two sequences of numbers. This problem is usually addressed by means of numerical indexes referred to as sequence correlation indexes. This paper elaborates on why some specific sequence correlation indexes may not be good choices depending on the application scenario in hand. A variant of the Product-Moment correlation coefficient and a weighted formulation for the Goodman-Kruskal and Kendall’s indexes are derived that may be more appropriate for some particular application scenarios. The proposed and existing indexes are analyzed from different perspectives, such as their sensitivity to the ranks and magnitudes of the sequences under evaluation, among other relevant aspects of the problem. The results help suggesting scenarios within the context of clustering analysis that are possibly more appropriate for the application of each index.     

Efficient estimation of a semiparametric dynamic copula model

A new semiparametric dynamic copula model is proposed where the marginals are specified as parametric GARCH-type processes, and the dependence parameter of the copula is allowed to change over time in a nonparametric way. A straightforward two-stage estimation method is given by local maximum likelihood for the dependence parameter, conditional on consistent first stage estimates of the marginals. First, the properties of the estimator are characterized in terms of bias and variance and the bandwidth selection problem is discussed. The proposed estimator attains the semiparametric efficiency bound and its superiority is demonstrated through simulations. Finally, the wide applicability of the model in financial time series is illustrated, and it is compared with traditional models based on conditional correlations.     

Concordance and consensus

This paper deals with the measurement of concordance and the construction of consensus in preference data, either in the form of preference rankings or in the form of response distributions with Likert-items. We propose a set of axioms of concordance in preference orderings and a new class of concordance measures. The measures outperform classic measures like Kendall’s τ and W and Spearman’s ρ in sensitivity and apply to large sets of orderings instead of just to pairs of orderings. For sets of N orderings of n items, we present very efficient and flexible algorithms that have a time complexity of only O(Nn²). Remarkably, the algorithms also allow for fast calculation of all longest common subsequences of the full set of orderings. We experimentally demonstrate the performance of the algorithms. A new and simple measure for assessing concordance on Likert-items is proposed.     

Nonparametric estimation of the conditional tail index and extreme quantiles under random censoring

The estimation of the tail index and extreme quantiles of a heavy-tailed distribution is addressed when some covariate information is available and the data are randomly right-censored. Several estimators are constructed by combining a moving-window technique (for tackling the covariate information) and the inverse probability-of-censoring weighting method. The asymptotic normality of these estimators is established and their finite-sample properties are investigated via simulations. A comparison with alternative estimators is provided. Finally, the proposed methodology is illustrated on a medical dataset.     

Nonparametric conditional hazard rate estimation: A local linear approach

A new nonparametric estimator for the conditional hazard rate is proposed, which is defined as the ratio of local linear estimators for the conditional density and survivor function. The resulting hazard rate estimator is shown to be pointwise consistent and asymptotically normally distributed under appropriate conditions. Furthermore, plug-in bandwidths based on normal and uniform reference distributions and minimizing the asymptotic mean squared error are derived. In terms of the mean squared error the new estimator is highly competitive in comparison to existing estimators for the conditional hazard rate. Moreover, its smoothing parameters are relatively robust to misspecification of the reference distributions, which facilitates bandwidth selection. Additionally, the new hazard rate estimator is conveniently calculated using standard software for local linear regression. The use of the local linear hazard rate is illustrated in an application to kidney transplant data.     

Modeling the yearly Value-at-Risk for operational risk in Chinese commercial banks

In this paper, we explore the loss data collection exercise for operational risk in Chinese commercial banks from 1999 to first half of 2006. Firstly, the above data are bootstrapped to analyze the capital allocation for a medium-scaled commercial bank in China. Secondly, for every selected cell, we calibrate two truncated distributions to fit the loss severity, one for ‘normal’ losses and the other for the ‘extreme’ losses. Moreover, a more realistic dependence structure - multivariate t copula function is used to measure the relation among the selected cells. In the final, the simulation results suggest that substantial savings can be achieved through measuring the dependence by means of multivariate t copula function than by means of perfect positive dependence.