Types Of Marginal Distributions Research Articles

Multidimensional portfolio risk measurement: a mixed copula approach

It is an increasingly challenging task to explore the risk measurement for multidimensional portfolios with nonlinear correlative assets. A risk measurement scheme based on the mixed copula theory is proposed in this paper, where the mixed copula is constructed by the linear combination of three single Archimedean copulas, embodying greater flexibility than single copula in connecting different types of marginal distributions. In the scenario, ARMA-EGARCH model with t innovation is employed to fit marginal distributions, and the parameter values of the mixed copulas are inferred by maximum likelihood estimation (MLE) method, and interior point algorithm is used to calculate the extreme values of the MLE, VaR and CVaR, corresponding to the optimal portfolio with the minimum risk. Finally, an empirical study on five international stock market indexes in Europe is performed to verify the feasibility and effectiveness of the scheme.

Multidimensional portfolio risk measurement: a mixed copula approach

It is an increasingly challenging task to explore the risk measurement for multidimensional portfolios with nonlinear correlative assets. A risk measurement scheme based on the mixed copula theory is proposed in this paper, where the mixed copula is constructed by the linear combination of three single Archimedean copulas, embodying greater flexibility than single copula in connecting different types of marginal distributions. In the scenario, ARMA-EGARCH model with t innovation is employed to fit marginal distributions, and the parameter values of the mixed copulas are inferred by maximum likelihood estimation (MLE) method, and interior point algorithm is used to calculate the extreme values of the MLE, VaR and CVaR, corresponding to the optimal portfolio with the minimum risk. Finally, an empirical study on five international stock market indexes in Europe is performed to verify the feasibility and effectiveness of the scheme.

Assessing the copula selection for bivariate frequency analysis based on the tail dependence test

The flood characteristics, namely, peak, duration and volume provide important information for the design of hydraulic structures, water resources planning, reservoir management and flood hazard mapping. Flood is a complex phenomenon defined by strongly correlated characteristics such as peak, duration and volume. Therefore, it is necessary to study the simultaneous, multivariate, probabilistic behaviour of flood characteristics. Traditional multivariate parametric distributions have widely been applied for hydrological applications. However, this approach has some drawbacks such as the dependence structure between the variables, which depends on the marginal distributions or the flood variables that have the same type of marginal distributions. Copulas are applied to overcome the restriction of traditional bivariate frequency analysis by choosing the marginals from different families of the probability distribution for flood variables. The most important step in the modelling process using copula is the selection of copula function which is the best fit for the data sample. The choice of copula may significantly impact the bivariate quantiles. Indeed, this study indicates that there is a huge difference in the joint return period estimation using the families of extreme value copulas and no upper tail copulas (Frank, Clayton and Gaussian) if there exists asymptotic dependence in the flood characteristics. This study suggests that the copula function should be selected based on the dependence structure of the variables. From the results, it is observed that the result from tail dependence test is very useful in selecting the appropriate copula for modelling the joint dependence structure of flood variables. The extreme value copulas with upper tail dependence have proved that they are appropriate models for the dependence structure of the flood characteristics and Frank, Clayton and Gaussian copulas are the appropriate copula models in case of variables which are diagnosed as asymptotic independence.

EM algorithm in Gaussian copula with missing data

Rank-based correlation is widely used to measure dependence between variables when their marginal distributions are skewed. Estimation of such correlation is challenged by both the presence of missing data and the need for adjusting for confounding factors. In this paper, we consider a unified framework of Gaussian copula regression that enables us to estimate either Pearson correlation or rank-based correlation (e.g. Kendall’s tau or Spearman’s rho), depending on the types of marginal distributions. To adjust for confounding covariates, we utilize marginal regression models with univariate location-scale family distributions. We establish the EM algorithm for estimation of both correlation and regression parameters with missing values. For implementation, we propose an effective peeling procedure to carry out iterations required by the EM algorithm. We compare the performance of the EM algorithm method to the traditional multiple imputation approach through simulation studies. For structured types of correlations, such as exchangeable or first-order auto-regressive (AR-1) correlation, the EM algorithm outperforms the multiple imputation approach in terms of both estimation bias and efficiency.

Probability Distribution of Rainfall-Runoff Using Entropy Theory

Using the entropy theory, the Shannon entropy-based joint probability distribution of rainfall amount and runoff depth (runoff volume per unit area) is derived. In order to evaluate the accuracy of the entropy-based joint distribution, the study applies the following approaches: (1) comparison of the entropy determined from the derived joint distribution with that of the empirical joint entropy calculated based on the kernel density with the optimized kernel bandwidth, (2) the root mean square error (RMSE) of the joint probability density derived from the joint distribution and the empirical joint density, and (3) testing the derived joint distribution using the chi-square goodness-of-fit test for multivariate distributions. In addition, the marginal distributions of rainfall and runoff derived from the joint distribution are compared with the empirical marginal and fitted gamma distributions and are evaluated using the Kolmogorov-Smirnov (K-S) goodness-of-fit test and RMSE. With the joint distribution appropriately determined, the conditional distribution of runoff depth is derived for a given rainfall depth. The derived distributions are evaluated using rainfall-runoff data from small agricultural experimental watersheds near Riesel (Waco), Texas, maintained by the USDA-ARS. Results of the study indicate that: (1) the Shannon entropy theory can be satisfactorily applied to model the bivariate rainfall and runoff distribution; (2) it can be applied to model the bivariate random variables when convenient bivariate distributions are not able to capture the dependence between them; and (3) because the Shannon entropy uses constraints to formulate the joint distribution, it may be applied to the bivariate random variables incorporating different types of marginal distributions.

Copula‐based flood frequency (COFF) analysis at the confluences of river systems

AbstractMany civil infrastructures are located near the confluence of two streams, where they may be subject to inundation by high flows from either stream or both. These infrastructures, such as highway bridges, are designed to meet specified performance objectives for floods of a specified return period (e.g. the 100 year flood). Because the flooding of structures on one stream can be affected by high flows on the other stream, it is important to know the relationship between the coincident exceedence probabilities on the confluent stream pair in many hydrological engineering practices. Currently, the National Flood Frequency Program (NFF), which was developed by the US Geological Survey (USGS) and based on regional analysis, is probably the most popular model for ungauged site flood estimation and could be employed to estimate flood probabilities at the confluence points. The need for improved infrastructure design at such sites has motivated a renewed interest in the development of more rigorous joint probability distributions of the coincident flows. To accomplish this, a practical procedure is needed to determine the crucial bivariate distributions of design flows at stream confluences. In the past, the copula method provided a way to construct multivariate distribution functions. This paper aims to develop the Copula‐based Flood Frequency (COFF) method at the confluence points with any type of marginal distributions via the use of Archimedean copulas and dependent parameters. The practical implementation was assessed and tested against the standard NFF approach by a case study in Iowa's Des Moines River. Monte Carlo simulations proved the success of the generalized copula‐based joint distribution algorithm. Copyright © 2009 John Wiley & Sons, Ltd.

IDF Curves Using the Frank Archimedean Copula

This study aims at deriving intensity-duration-frequency (IDF) curves from bivariate rainfall frequency analysis using the copula method, which does not assume the rainfall variables to be independent or normal or have the same type of marginal distributions. The bivariate distributions are then employed to determine conditional return periods and IDF curves needed for urban drainage design. The IDF curves are tested using rainfall data from catchments in Louisiana and are also compared with those of Technical Paper 40 (TP-40) of the U.S. National Weather Service as well as those published in the literature.

Gumbel–Hougaard Copula for Trivariate Rainfall Frequency Analysis

Joint distributions of rainfall intensity, duration, and depth or those of rainfall intensity and duration, rainfall depth and duration, and rainfall intensity and depth are important in hydrologic design and floodplain management. Considering the dependence among rainfall intensity, depth, and duration, multivariate rainfall frequency distributions have been derived using one of three fundamental assumptions. Either the rainfall intensity, duration, and depth have been assumed independent, or they each have the same type of marginal probability distribution or they have been assumed to have the normal distribution or have been transformed to have the normal distribution. In reality, however, rainfall intensity, duration, and depth are dependent, do not follow, in general, the normal distribution, and do not have the same type of marginal distributions. This study aims at deriving trivariate rainfall frequency distributions using the Gumbel–Hougaard copula which does not assume the rainfall variables to be independent or normal or have the same type of marginal distributions. The trivariate distribution is then employed to determine joint conditional return periods, and is tested using rainfall data from the Amite River Basin in Louisiana.

Using Gaussian copula to simulate repetitive projects

An important requirement for simulating repetitive projects is to treat correlations inherent in the repetition of same crews working at various locations. To attain the requirement, this study develops a new Monte Carlo simulation model implementing a Gaussian copula in conjunction with the inverse‐transform method to generate correlated duration samples in repetitive projects that have pre‐specified marginal distributions and pairwise rank correlations. The proposed model is equipped with an automatic approximation procedure to adjust an infeasible correlation matrix, if necessary. The proposed model is statistically verified through a real‐life residential apartment project. The simulation results are compared to two conventional analyses (PERT and simulation without correlation) to show the aggregated impact of correlations. The proposed model contributes to the state‐of‐the‐art in handling non‐linear dependencies among activity durations that may have non‐normal distributions. Moreover, it is flexible in the ways of correlation assessments (qualitative or quantitative), the magnitudes of correlations (weak to strong), and the types of marginal distributions (symmetrical or skewed).

Bivariate rainfall frequency distributions using Archimedean copulas

Joint distributions of rainfall intensity and depth, rainfall intensity and duration, or rainfall depth and duration are important in hydrologic design and floodplain management. Multivariate rainfall frequency distributions have usually been derived using one of three fundamental assumptions: (1) Either rainfall variables (e.g., intensity, depth, and duration) have each the same type of the marginal probability distribution, (2) the variables have been assumed to have joint normal distribution or have been transformed and assumed to have joint normal distribution, or (3) they have been assumed independent-a trivial case. In reality, however, rainfall variables are dependent, do not follow, in general, the normal distribution, and do not have the same type of marginal distributions. This study aims at deriving bivariate rainfall frequency distributions using the copula method in which four Archimedean copulas were examined and compared. The advantage of the copula method is that no assumption is needed for the rainfall variables to be independent or normal or have the same type of marginal distributions. The bivariate distributions are then employed to determine joint and conditional return periods, and are tested using rainfall data from the Amite River basin in Louisiana, United States.

A heuristic approach for the generation of multivariate random samples with specified marginal distributions and correlation matrix

This paper presents a heuristic approach for multivariate random number generation. Our aim is to generate multivariate samples with specified marginal distributions and correlation matrix, which can be incorporated into risk analysis models to conduct simulation studies. The proposed sampling approach involves two distinct steps: first a univariate random sample from each specified probability distribution is generated; then a heuristic combinatorial optimization procedure is used to rearrange the generated univariate samples, in order to obtain the desired correlations between them. The combinatorial optimization step is performed with a simulated annealing algorithm, which changes only the positions and not the values of the numbers generated in the first step. The proposed multivariate sampling approach can be used with any type of marginal distributions: continuous or discrete, parametric or non-parametric, etc.

Teaching Singular Distributions to Undergraduates

Abstract Singular distributions are seldom covered in undergraduate probability courses, although they are of interest in statistics and, as is shown by example, can easily arise through extending mixed discrete and continuous distributions to two or more dimensions. A representation is given that makes the construction of a class of singular distributions in two dimensions simple to carry out. This representation is also used to characterize the types of marginal distributions that members of this class can have.