Structured Coupling of Probability Loss Distributions: Assessing Joint Flood Risk in Multiple River Basins

Losses due to natural hazard events can be extraordinarily high and difficult to cope with. Therefore, there is considerable interest to estimate the potential impact of current and future extreme events at all scales in as much detail as possible. As hazards typically spread over wider areas, risk assessment must take into account interrelations between regions. Neglecting such interdependencies can lead to a severe underestimation of potential losses, especially for extreme events. This underestimation of extreme risk can lead to the failure of riskmanagement strategies when they are most needed, namely, in times of unprecedented events. In this article, we suggest a methodology to incorporate such interdependencies in risk via the use of copulas. We demonstrate that by coupling losses, dependencies can be incorporated in risk analysis, avoiding the underestimation of risk. Based on maximum discharge data of river basins and stream networks, we present and discuss different ways to couple loss distributions of basins while explicitly incorporating tail dependencies. We distinguish between coupling methods that require river structure data for the analysis and those that do not. For the later approach we propose a minimax algorithm to choose coupled basin pairs so that the underestimation of risk is avoided and the use of river structure data is not needed. The proposed methodology is especially useful for large‐scale analysis and we motivate and apply our method using the case of Romania. The approach can be easily extended to other countries and natural hazards.     

On Modeling Correlated Random Variables in Risk Assessment

Monte Carlo methods in risk assessment are finding increasingly widespread application. With the recognition that inputs may be correlated, the incorporation of such correlations into the simulation has become important. Most implementations rely upon the method of Iman and Conover for generating correlated random variables. In this work, alternative methods using copulas are presented for deriving correlated random variables. It is further shown that the particular algorithm or assumption used may have a substantial effect on the output results, due to differences in higher order bivariate moments.     

Mixed Marginal Copula Modeling

ABSTRACT

This article extends the literature on copulas with discrete or continuous marginals to the case where some of the marginals are a mixture of discrete and continuous components. We do so by carefully defining the likelihood as the density of the observations with respect to a mixed measure. The treatment is quite general, although we focus on mixtures of Gaussian and Archimedean copulas. The inference is Bayesian with the estimation carried out by Markov chain Monte Carlo. We illustrate the methodology and algorithms by applying them to estimate a multivariate income dynamics model. Supplementary materials for this article are available online.     

Assessing component reliability using lifetime data from systems

ABSTRACT

In this paper, we introduce a competing risks model for the lifetimes of components that differs from the classical competing risks models by the fact that it is not directly observable which component has failed. We propose two statistical methods for estimating the reliability of components from failure data on a system. Our methods are applied to simulated failure data, in order to illustrate the performance of the methods.     

On the identifiability of copulas in bivariate competing risks models

In competing risks models, the joint distribution of the event times is not identifiable even when the margins are fully known, which has been referred to as the “identifiability crisis in competing risks analysis” (Crowder, 1991). We model the dependence between the event times by an unknown copula and show that identification is actually possible within many frequently used families of copulas. The result is then extended to the case where one margin is unknown. The Canadian Journal of Statistics 41: 291–303; 2013 © 2013 Statistical Society of Canada     

Limiting Tail Dependence Copulas

The limiting lower-tail dependence copula (LLTDC) is defined as the copula of random variables which are right-truncated at thresholds tending to their left endpoints. This article shows LLTDCs are truncation-invariant and belong to the Ahmadi-Clayton family. Accordingly, it follows that limiting upper-tail dependence copulas are members of the survival Ahmadi-Clayton family.     

Calculating Bivariate Orthonormal Polynomials By Recurrence

Emerson gave recurrence formulae for the calculation of orthonormal polynomials for univariate discrete random variables. He claimed that as these were based on the Christoffel–Darboux recurrence relation they were more efficient than those based on the Gram–Schmidt method. This approach was generalised by Rayner and colleagues to arbitrary univariate random variables. The only constraint was that the expectations needed are well‐defined. Here the approach is extended to arbitrary bivariate random variables for which the expectations needed are well‐defined. The extension to multivariate random variables is clear.     

Copulas checker-type approximations: Application to quantiles estimation of sums of dependent random variables

Abstract

Several approximations of copulas have been proposed in the literature. By using empirical versions of checker-type copulas approximations, we propose non parametric estimators of the copula. Under some conditions, the proposed estimators are copulas and their main advantage is that they can be sampled from easily. One possible application is the estimation of quantiles of sums of dependent random variables from a small sample of the multivariate law and a full knowledge of the marginal laws. We show that estimations may be improved by including in an easy way in the approximated copula some additional information on the law of a sub-vector for example. Our approach is illustrated by numerical examples.     

A stochastic recurrence equations approach for score driven correlation models

We describe stationarity and ergodicity (SE) regions for a recently proposed class of score driven dynamic correlation models. These models have important applications in empirical work. The regions are derived from sufficiency conditions in Bougerol (1993 Bougerol, P. (1993). Kalman filtering with random coefficients and contractions. SIAM Journal on Control and Optimization 31(4):942–959.[Crossref], [Web of Science ®] , [Google Scholar]) and take a nonstandard form. We show that the nonstandard shape of the sufficiency regions cannot be avoided by reparameterizing the model or by rescaling the score steps in the transition equation for the correlation parameter. This makes the result markedly different from the volatility case. Observationally equivalent decompositions of the stochastic recurrence equation yield regions with different shapes and sizes. We use these results to establish the consistency and asymptotic normality of the maximum likelihood estimator. We illustrate our results with an analysis of time-varying correlations between U.K. and Greek equity indices. We find that also in empirical applications different decompositions can give rise to different conclusions regarding the stability of the estimated model.     

Predictive assessment of copula models

Copulas are powerful explanatory tools for studying dependence patterns in multivariate data. While the primary use of copula models is in multivariate dependence modelling, they also offer predictive value for regression analysis. This article investigates the utility of copula models for model‐based predictions from two angles. We assess whether, where, and by how much various copula models differ in their predictions of a conditional mean and conditional quantiles. From a model selection perspective, we then evaluate the predictive discrepancy between copula models using in‐sample and out‐of‐sample predictions both in bivariate and higher‐dimensional settings. Our findings suggest that some copula models are more difficult to distinguish in terms of their overall predictive power than others, and depending on the quantity of interest, the differences in predictions can be detected only in some targeted regions. The situations where copula‐based regression approaches would be advantageous over traditional ones are discussed using simulated and real data. The Canadian Journal of Statistics 47: 8–26; 2019 © 2018 Statistical Society of Canada