Extremal memory of stochastic volatility with an application to tail shape inference

We characterize joint tails and tail dependence for a class of stochastic volatility processes. We derive the exact joint tail shape of multivariate stochastic volatility with innovations that have a regularly varying distribution tail. This is used to give four new characterizations of tail dependence. In three cases tail dependence is a non-trivial function of linear volatility memory parametrically represented by tail scales, while tail power indices do not provide any relevant dependence information. Although tail dependence is associated with linear volatility memory, tail dependence itself is nonlinear. In the fourth case a linear function of tail events and exceedances is linearly independent. Tail dependence falls in a class that implies the celebrated Hill (1975) tail index estimator is asymptotically normal, while linear independence of nonlinear tail arrays ensures the asymptotic variance is the same as the iid case. We illustrate the latter finding by simulation.     

A new class of models for bivariate joint tails

Summary.  A fundamental issue in applied multivariate extreme value analysis is modelling dependence within joint tail regions. The primary focus of this work is to extend the classical pseudopolar treatment of multivariate extremes to develop an asymptotically motivated representation of extremal dependence that also encompasses asymptotic independence. Starting with the usual mild bivariate regular variation assumptions that underpin the coefficient of tail dependence as a measure of extremal dependence, our main result is a characterization of the limiting structure of the joint survivor function in terms of an essentially arbitrary non-negative measure that must satisfy some mild constraints. We then construct parametric models from this new class and study in detail one example that accommodates asymptotic dependence, asymptotic independence and asymmetry within a straightforward parsimonious parameterization. We provide a fast simulation algorithm for this example and detail likelihood-based inference including tests for asymptotic dependence and symmetry which are useful for submodel selection. We illustrate this model by application to both simulated and real data. In contrast with the classical multivariate extreme value approach, which concentrates on the limiting distribution of normalized componentwise maxima, our framework focuses directly on the structure of the limiting joint survivor function and provides significant extensions of both the theoretical and the practical tools that are available for joint tail modelling.     

A semi-parametric model for multivariate extreme values

Threshold methods for multivariate extreme values are based on the use of asymptotically justified approximations of both the marginal distributions and the dependence structure in the joint tail. Models derived from these approximations are fitted to a region of the observed joint tail which is determined by suitably chosen high thresholds. A drawback of the existing methods is the necessity for the same thresholds to be taken for the convergence of both marginal and dependence aspects, which can result in inefficient estimation. In this paper an extension of the existing models, which removes this constraint, is proposed. The resulting model is semi-parametric and requires computationally intensive techniques for likelihood evaluation. The methods are illustrated using a coastal engineering application.     

Tail dependence for skew Laplace distribution and skew Cauchy distribution

ABSTRACT

Coefficient of tail dependence measures the strength of dependence in the tail of a bivariate distribution and it has been found useful in the risk management. In this paper, we derive the upper tail dependence coefficient for a random vector following the skew Laplace distribution and the skew Cauchy distribution, respectively. The result shows that skew Laplace distribution is asymptotically independent in upper tail, however, skew Cauchy distribution has asymptotic upper tail dependence.     

Likelihood Inference for Multivariate Extreme Value Distributions Whose Spectral Vectors have known Conditional Distributions

Multivariate extreme value statistical analysis is concerned with observations on several variables which are thought to possess some degree of tail dependence. The main approaches to inference for multivariate extremes consist in approximating either the distribution of block component‐wise maxima or the distribution of the exceedances over a high threshold. Although the expressions of the asymptotic density functions of these distributions may be characterized, they cannot be computed in general. In this paper, we study the case where the spectral random vector of the multivariate max‐stable distribution has known conditional distributions. The asymptotic density functions of the multivariate extreme value distributions may then be written through univariate integrals that are easily computed or simulated. The asymptotic properties of two likelihood estimators are presented, and the utility of the method is examined via simulation.     

Estimation and inference of the joint conditional distribution for multivariate longitudinal data using nonparametric copulas

In this paper we study estimating the joint conditional distributions of multivariate longitudinal outcomes using regression models and copulas. For the estimation of marginal models, we consider a class of time-varying transformation models and combine the two marginal models using nonparametric empirical copulas. Our models and estimation method can be applied in many situations where the conditional mean-based models are not good enough. Empirical copulas combined with time-varying transformation models may allow quite flexible modelling for the joint conditional distributions for multivariate longitudinal data. We derive the asymptotic properties for the copula-based estimators of the joint conditional distribution functions. For illustration we apply our estimation method to an epidemiological study of childhood growth and blood pressure.     

CONDITIONAL EXPECTATION FORMULAE FOR COPULAS

Not only are copula functions joint distribution functions in their own right, they also provide a link between multivariate distributions and their lower‐dimensional marginal distributions. Copulas have a structure that allows us to characterize all possible multivariate distributions, and therefore they have the potential to be a very useful statistical tool. Although copulas can be traced back to 1959, there is still much scope for new results, as most of the early work was theoretical rather than practical. We focus on simple practical tools based on conditional expectation, because such tools are not widely available. When dealing with data sets in which the dependence throughout the sample is variable, we suggest that copula‐based regression curves may be more accurate predictors of specific outcomes than linear models. We derive simple conditional expectation formulae in terms of copulas and apply them to a combination of simulated and real data.     

Inference for Multi‐dimensional High‐frequency Data with an Application to Conditional Independence Testing

We find the asymptotic distribution of the multi‐dimensional multi‐scale and kernel estimators for high‐frequency financial data with microstructure. Sampling times are allowed to be asynchronous and endogenous. In the process, we show that the classes of multi‐scale and kernel estimators for smoothing noise perturbation are asymptotically equivalent in the sense of having the same asymptotic distribution for corresponding kernel and weight functions. The theory leads to multi‐dimensional stable central limit theorems and feasible versions. Hence, they allow to draw statistical inference for a broad class of multivariate models, which paves the way to tests and confidence intervals in risk measurement for arbitrary portfolios composed of high‐frequently observed assets. As an application, we enhance the approach to construct a test for investigating hypotheses that correlated assets are independent conditional on a common factor.     

Partial derivatives and confidence intervals of bivariate tail dependence functions

Bivariate extreme value theory was used to estimate a rare event (see de Haan and de Ronde [1998. Sea and wind: multivariate extremes at work. Extremes 1, 7–45]). This procedure involves estimating a tail dependence function. There are several estimators for the tail dependence function in the literature, but their limiting distributions depend on partial derivatives of the tail dependence function. In this paper smooth estimators are proposed for estimating partial derivatives of bivariate tail dependence functions and their asymptotic distributions are derived as well. A simulation study is conducted to compare different estimators of partial derivatives in terms of both mean squared errors and coverage accuracy of confidence intervals of the bivariate tail dependence function based on these different estimators of partial derivatives.     

SEMIPARAMETRIC ESTIMATION OF THE ERROR DISTRIBUTION IN MULTIVARIATE REGRESSION USING COPULAS

A semiparametric method is developed to estimate the dependence parameter and the joint distribution of the error term in the multivariate linear regression model. The nonparametric part of the method treats the marginal distributions of the error term as unknown, and estimates them using suitable empirical distribution functions. Then the dependence parameter is estimated by either maximizing a pseudolikelihood or solving an estimating equation. It is shown that this estimator is asymptotically normal, and a consistent estimator of its large sample variance is given. A simulation study shows that the proposed semiparametric method is better than the parametric ones available when the error distribution is unknown, which is almost always the case in practice. It turns out that there is no loss of asymptotic efficiency as a result of the estimation of regression parameters. An empirical example on portfolio management is used to illustrate the method.     

Asymptotic theory for maximum likelihood estimates in reduced-rank multivariate generalized linear models

Reduced-rank regression is a dimensionality reduction method with many applications. The asymptotic theory for reduced rank estimators of parameter matrices in multivariate linear models has been studied extensively. In contrast, few theoretical results are available for reduced-rank multivariate generalized linear models. We develop M-estimation theory for concave criterion functions that are maximized over parameter spaces that are neither convex nor closed. These results are used to derive the consistency and asymptotic distribution of maximum likelihood estimators in reduced-rank multivariate generalized linear models, when the response and predictor vectors have a joint distribution. We illustrate our results in a real data classification problem with binary covariates.     

Modelling Dependence within Joint Tail Regions

Standard approaches for modelling dependence within joint tail regions are based on extreme value methods which assume max-stability, a particular form of joint tail dependence. We develop joint tail models based on a broader class of dependence structure which provides a natural link between max-stable models and weaker forms of dependence including independence and negative association. This approach overcomes many of the problems that are encountered with standard methods and is the basis for a Poisson process representation that generalizes existing bivariate results. We apply the new techniques to simulated and environmental data, and demonstrate the marked advantage that the new approach offers for joint tail extrapolation.     

Tail-weighted measures of dependence

Multivariate copula models are commonly used in place of Gaussian dependence models when plots of the data suggest tail dependence and tail asymmetry. In these cases, it is useful to have simple statistics to summarize the strength of dependence in different joint tails. Measures of monotone association such as Kendall's tau and Spearman's rho are insufficient to distinguish commonly used parametric bivariate families with different tail properties. We propose lower and upper tail-weighted bivariate measures of dependence as additional scalar measures to distinguish bivariate copulas with roughly the same overall monotone dependence. These measures allow the efficient estimation of strength of dependence in the joint tails and can be used as a guide for selection of bivariate linking copulas in vine and factor models as well as for assessing the adequacy of fit of multivariate copula models. We apply the tail-weighted measures of dependence to a financial data set and show that the measures better discriminate models with different tail properties compared to commonly used risk measures – the portfolio value-at-risk and conditional tail expectation.     

Multivariate τ-estimators for location and scatter

We discuss the robustness and asymptotic behaviour of τ-estimators for multivariate location and scatter. We show that τ-estimators correspond to multivariate M-estimators defined by a weighted average of redescending ψ-functions, where the weights are adaptive. We prove consistency and asymptotic normality under weak assumptions on the underlying distribution, show that τ-estimators have a high breakdown point, and obtain the influence function at general distributions. In the special case of a location-scatter family, τ-estimators are asymptotically equivalent to multivariate S-estimators defined by means of a weighted ψ-function. This enables us to combine a high breakdown point and bounded influence with good asymptotic efficiency for the location and covariance estimator.     

A family of models for uniform and serial dependence in repeated measurements studies

Data arising from a randomized double-masked clinical trial for multiple sclerosis have provided particularly variable longitudinal repeated measurements responses. Specific models for such data, other than those based on the multivariate normal distribution, would be a valuable addition to the applied statistician's toolbox. A useful family of multivariate distributions can be generated by substituting the integrated intensity of one distribution into a second (outer) distribution. The parameters in the second distribution are then used to create a dependence structure among observations on a unit. These may either be a form of serial dependence for longitudinal data or of uniform dependence within clusters. These are respectively analogous to the Kalman filter of state space models and to copulas, but they have the major advantage that they do not require any explicit integration. One useful outer distribution for constructing such multivariate distributions is the Pareto distribution. Certain special models based on it have previously been used in event history analysis, but those considered here have much wider application.     

Bivariate Tail Dependence and the Generation of Multivariate Extreme Value Distributions

We define, in a probabilistic way, a parametric family of multivariate extreme value distributions. We derive its copula, which is a mixture of several complete dependent copulas and total independent copulas, and the bivariate tail dependence and extremal coefficients. Based on the obtained results for these coefficients, we propose a method to build multivariate extreme value distributions with prescribed tail/extremal coefficients. We illustrate the results with examples.     

Bayesian joint modeling of correlated counts data with application to adverse birth outcomes

In disease mapping, health outcomes measured at the same spatial locations may be correlated, so one can consider joint modeling the multivariate health outcomes accounting for their dependence. The general approaches often used for joint modeling include shared component models and multivariate models. An alternative way to model the association between two health outcomes, when one outcome can naturally serve as a covariate of the other, is to use ecological regression model. For example, in our application, preterm birth (PTB) can be treated as a predictor for low birth weight (LBW) and vice versa. Therefore, we proposed to blend the ideas from joint modeling and ecological regression methods to jointly model the relative risks for LBW and PTBs over the health districts in Saskatchewan, Canada, in 2000–2010. This approach is helpful when proxy of areal-level contextual factors can be derived based on the outcomes themselves when direct information on risk factors are not readily available. Our results indicate that the proposed approach improves the model fit when compared with the conventional joint modeling methods. Further, we showed that when no strong spatial autocorrelation is present, joint outcome modeling using only independent error terms can still provide a better model fit when compared with the separate modeling.     

A simple nonparametric test for bivariate symmetry about a line

Over the years many researchers have dealt with testing the hypotheses of symmetry in univariate and multivariate distributions in the parametric and nonparametric setup. In a multivariate setup, there are several formulations of symmetry, for example, symmetry about an axis, joint symmetry, marginal symmetry, radial symmetry, symmetry about a known point, spherical symmetry, and elliptical symmetry among others. In this paper, for the bivariate case, we formulate a concept of symmetry about a straight line passing through the origin in a plane and accordingly develop a simple nonparametric test for testing the hypothesis of symmetry about a straight line. The proposed test is based on a measure of deviance between observed counts of bivariate samples in suitably defined pairs of sets. The exact null distribution and non-null distribution, for specified classes of alternatives, of the test statistics are obtained. The null distribution is tabulated for sample size from n=5 up to n=30. The null mean, null variance and the asymptotic null distributions of the proposed test statistics are also obtained. The empirical power of the proposed test is evaluated by simulating samples from the suitable class of bivariate distributions. The empirical findings suggest that the test performs reasonably well against various classes of asymmetric bivariate distributions. Further, it is advocated that the basic idea developed in this work can be easily adopted to test the hypotheses of exchangeability of bivariate random variables and also bivariate symmetry about a given axis which have been considered by several authors in the past.     

Statistical inference for restricted partially linear varying coefficient errors-in-variables models

As a useful extension of partially linear models and varying coefficient models, the partially linear varying coefficient model is useful in statistical modelling. This paper considers statistical inference for the semiparametric model when the covariates in the linear part are measured with additive error and some additional linear restrictions on the parametric component are available. We propose a restricted modified profile least-squares estimator for the parametric component, and prove the asymptotic normality of the proposed estimator. To test hypotheses on the parametric component, we propose a test statistic based on the difference between the corrected residual sums of squares under the null and alterative hypotheses, and show that its limiting distribution is a weighted sum of independent chi-square distributions. We also develop an adjusted test statistic, which has an asymptotically standard chi-squared distribution. Some simulation studies are conducted to illustrate our approaches.     

Max‐infinitely divisible models and inference for spatial extremes

For many environmental processes, recent studies have shown that the dependence strength is decreasing when quantile levels increase. This implies that the popular max‐stable models are inadequate to capture the rate of joint tail decay, and to estimate joint extremal probabilities beyond observed levels. We here develop a more flexible modeling framework based on the class of max‐infinitely divisible processes, which extend max‐stable processes while retaining dependence properties that are natural for maxima. We propose two parametric constructions for max‐infinitely divisible models, which relax the max‐stability property but remain close to some popular max‐stable models obtained as special cases. The first model considers maxima over a finite, random number of independent observations, while the second model generalizes the spectral representation of max‐stable processes. Inference is performed using a pairwise likelihood. We illustrate the benefits of our new modeling framework on Dutch wind gust maxima calculated over different time units. Results strongly suggest that our proposed models outperform other natural models, such as the Student‐t copula process and its max‐stable limit, even for large block sizes.