A new family of trivariate proper quasi-copulas

summary:In this paper, we provide a new family of trivariate proper quasi-copulas. As an application, we show that $W^{3}$ – the best-possible lower bound for the set of trivariate quasi-copulas (and copulas) – is the limit member of this family, showing how the mass of $W^3$ is distributed on the plane $x+y+z=2$ of $[0,1]^3$ in an easy manner, and providing the generalization of this result to $n$ dimensions

Efficient estimation of parameters in marginals in semiparametric multivariate models

We consider a general multivariate model where univariate marginal distributions are known up to a parameter vector and we are interested in estimating that parameter vector without specifying the joint distribution, except for the marginals. If we assume independence between the marginals and maximize the resulting quasi-likelihood, we obtain a consistent but inefficient QMLE estimator. If we assume a parametric copula (other than independence) we obtain a full MLE, which is efficient but only under a correct copula specification and may be biased if the copula is misspecified. Instead we propose a sieve MLE estimator (SMLE) which improves over QMLE but does not have the drawbacks of full MLE. We model the unknown part of the joint distribution using the Bernstein-Kantorovich polynomial copula and assess the resulting improvement over QMLE and over misspecified FMLE in terms of relative efficiency and robustness. We derive the asymptotic distribution of the new estimator and show that it reaches the relevant semiparametric efficiency bound. Simulations suggest that the sieve MLE can be almost as efficient as FMLE relative to QMLE provided there is enough dependence between the marginals. We demonstrate practical value of the new estimator with several applications. First, we apply SMLE in an insurance context where we build a flexible semi-parametric claim loss model for a scenario where one of the variables is censored. As in simulations, the use of SMLE leads to tighter parameter estimates. Next, we consider financial risk management examples and show how the use of SMLE leads to superior Value-at-Risk predictions. The paper comes with an online archive which contains all codes and datasets

Multivariate hydrological frequency analysis and risk mapping

In hydrological frequency analysis, it is difficult to apply standard statistical methods to derive multivariate probability distributions of the characteristics of hydrologic or hydraulic variables except under the following restrictive assumptions: (1) variables are assumed independent, (2) variables are assumed to have the same marginal distributions, and (3) variables are assumed to follow or are transformed to normal distribution. Relaxing these assumptions when deriving multivariate distributions of the characteristics of correlated hydrologic and hydraulic variables. The copula methodology is applied to perform multivariate frequency analysis of rainfall, flood, low-flow, water quality, and channel flow, using data from the Amite river basin in Louisiana. And finally, the risk methodology is applied to analyze flood risks. Through the study, it was found that (1) copula method was found reasonably well to be applied to derive the multivariate hydrological frequency model compared with other conventional methods, i.e., multivariate normal approach, N-K model approach, independence transformation approach etc.; (2) nonstationarity was found more or less existed in the rainfall and streamflow time series, but according to the nonstationary test, in most cases, the stationarity assumption may be approximately valid; (3) the multivariate frequency analysis coupling nonstationarity indicated that the stationary assumption was valid for both bivariate and trivariate analysis; and (4) risk, defined by both flooding event and the damage caused by the scenario, showed the difference from that defined by T-year return period design event and the probability of total damage with the comparison indicating that only one character, i.e., T-year event or probability of total damage was not adequate to define the risk

Modeling and Analysis of Repeated Ordinal Data Using Copula Based Likelihoods and Estimating Equation Methods

Repeated or longitudinal ordinal data occur in many fields such as biology, epidemiology, and finance. These data normally are analyzed using both likelihood and non-likelihood methods. The first part of this dissertation discusses the multivariate ordered probit model which is a likelihood method based on latent variables. We show that this latent variable model belong to a very general class of Copula models. We use the copula representation for the multivariate ordered probit model to obtain maximum likelihood estimates of the parameters. We apply the methodology in the analysis of real life data examples. Though likelihood methods are preferable, there are computational challenges implementing them. Alternatives are the non-likelihood models. These are partially specified models, that is, in these models only the functional forms of the marginals are known but joint distributions are unknown. In addition, the dependence among the observations is modeled using an appropriate correlation structure. The second part of the dissertation outlines the estimating equations approach for the analysis of longitudinal ordinal data for these non-likelihood models. We study the asymptotic properties of the estimates for both likelihood and non-likelihood methods. Comparisons based on simulations show that the maximum likelihood estimates arising from copula models are more efficient than the estimates obtained from estimating equations. The third part of this dissertation describes how ordinal data can be viewed as multinomial random vectors and points out the theoretical challenges in finding restrictions on the correlation parameters for dependent multinomial random vectors

Factor copula models for item response data

Factor or conditional independence models based on copulas are proposed for multivariate discrete data such as item responses. The factor copula models have interpretations of latent maxima/minima (in comparison with latent means) and can lead to more probability in the joint upper or lower tail compared with factor models based on the discretized multivariate normal distribution (or multidimensional normal ogive model). Details on maximum likelihood estimation of parameters for the factor copula model are given, as well as analysis of the behavior of the log-likelihood. Our general methodology is illustrated with several item response data sets, and it is shown that there is a substantial improvement on existing models both conceptually and in fit to data