Farlie Gumbel Morgenstern Copula Research Articles

Exchangeable FGM copulas

AbstractCopulas provide a powerful and flexible tool for modeling the dependence structure of random vectors, and they have many applications in finance, insurance, engineering, hydrology, and other fields. One well-known class of copulas in two dimensions is the Farlie–Gumbel–Morgenstern (FGM) copula, since its simple analytic shape enables closed-form solutions to many problems in applied probability. However, the classical definition of the high-dimensional FGM copula does not enable a straightforward understanding of the effect of the copula parameters on the dependence, nor a geometric understanding of their admissible range. We circumvent this issue by analyzing the FGM copula from a probabilistic approach based on multivariate Bernoulli distributions. This paper examines high-dimensional exchangeable FGM copulas, a subclass of FGM copulas. We show that the dependence parameters of exchangeable FGM copulas can be expressed as a convex hull of a finite number of extreme points. We also leverage the probabilistic interpretation to develop efficient sampling and estimating procedures and provide a simulation study. Throughout, we discover geometric interpretations of the copula parameters that assist one in decoding the dependence of high-dimensional exchangeable FGM copulas.

New bivariate family of distributions based on any copula function: Statistical properties

In this paper, new bivariate family of distributions based on any copula is established. Of this, we introduce new bivariate Topp-Leone family based on Farlie-Gumbel-Morgenstern (FGM) copula. As a special case, we concentrate our study in the new bivariate Topp-Leone-Exponential-Exponential (BFGMTLEE) distribution based on FGM copula. Some of its properties are developed such as product moments, moment generating function and entropy.

A Collection of New Trigonometric- and Hyperbolic-FGM-Type Copulas

Copula analysis was created to explain the dependence of two or more quantitative variables. Due to the need for in-depth data analysis involving complex variable relationships, there is always a need for new copula models with original features. As a modern example, for the analysis of circular or periodic data types, trigonometric copulas are particularly attractive and recommended. This is, however, an underexploited topic. In this article, we propose a new collection of eight trigonometric and hyperbolic copulas, four based on the sine function and the others on the tangent function, all derived from the construction of the famous Farlie–Gumbel–Morgenstern copula. In addition to their original trigonometric and hyperbolic functionalities, the proposed copulas have the feature of depending on three parameters with complementary roles: one is a dependence parameter; one is a shape parameter; and the last can be viewed as an angle parameter. In our main findings, for each of the eight copulas, we determine a wide range of admissible values for these parameters. Subsequently, the capabilities, features, and functions of the new copulas are thoroughly examined. The shapes of the main functions of some copulas are illustrated graphically. Theoretically, symmetry in general, stochastic dominance, quadrant dependence, tail dependence, Archimedean nature, correlation measures, and inference on the parameters are investigated. Some copula shapes are illustrated with the help of figures. On the other hand, some two-dimensional inequalities are established and may be of separate interest.

Stochastic representation of FGM copulas using multivariate Bernoulli random variables

A one-to-one correspondence between Fréchet's class of multivariate Bernoulli distribution with symmetric marginals and the well-known family of Farlie-Gumbel-Morgenstern (FGM) copulas is established. A new stochastic representation of the family of d-variate FGM copulas is introduced. The representation is bijective: from any d-variate Bernoulli distribution, one may define a corresponding d-variate FGM copula; and for any d-variate FGM copula, one finds the corresponding d-variate Bernoulli distribution. The proposed stochastic representation has many advantages, notably establishing stochastic orders, constructing subclasses of FGM copulas and sampling. In particular, one may use the stochastic representation to develop computational methods to perform sampling from subclasses of FGM copulas, which scale well to large dimensions.

Risk Analysis of the Copula Dependent Aggregate Discounted Claims with Weibull Inter-Arrival Time

We model the recursive moments of aggregate discounted claims, assuming the inter-claim arrival time follows a Weibull distribution to accommodate overdispersed and underdispersed data set. We use a copula to represent the dependence structure between the inter-claim arrival time and its subsequent claim amount. We then use the Laplace inversion via the Gaver-Stehfest algorithm to solve numerically the first and second moments, which takes the form of a Volterra integral equation (VIE). We compute the average and variance of the aggregate discounted claims under the Farlie-Gumbel-Morgenstern (FGM) copula and conduct a sensitivity analysis under various Weibull inter-claim parameters and claim-size parameters. The comparison between the equidispersed, overdispersed and underdispersed counting processes shows that when claims arrive at times that vary more than is expected, insured lives can expect to pay higher premium, and vice versa for the case of claims arriving at times that vary less than expected. Upon comparing the Weibull risk process with an equivalent Poisson process, we also found that copulas with a wider range of dependency parameter such as the Frank and Heavy Right Tail (HRT), have a greater impact on the value of moments as opposed to modeling under FGM copula with weak dependence structure.

Beta Linear Failure Rate Geometric Distribution with Applications

In this paper we introduce bivariate Weibull distributions derived from copula functions in presence of cure fraction, censored data and covariates. Two copla functions are explored: the FGM (Farlie - Gumbel Morgenstern) copula and the Gumbel copula. Inferences for the proposed models are obtained under the Bayesian approach, using standard MCMC (Markov Chain Monte Carlo) methods. An illustration of the proposed methodology is given considering a medical data set.

Expanded Fréchet Model: Mathematical Properties, Copula, Different Estimation Methods, Applications and Validation Testing

The extreme value theory is expanded by proposing and studying a new version of the Fréchet model. Some new bivariate type extensions using Farlie–Gumbel–Morgenstern copula, modified Farlie–Gumbel–Morgenstern copula, Clayton copula, and Renyi’s entropy copula are derived. After a quick study for its properties, different non-Bayesian estimation methods under uncensored schemes are considered, such as the maximum likelihood estimation method, Anderson–Darling estimation method, ordinary least square estimation method, Cramér–von-Mises estimation method, weighted least square estimation method, left-tail Anderson–Darling estimation method, and right-tail Anderson–Darling estimation method. Numerical simulations were performed for comparing the estimation methods using different sample sizes for three different combinations of parameters. The Barzilai–Borwein algorithm was employed via a simulation study. Three applications were presented for measuring the flexibility and the importance of the new model for comparing the competitive distributions under the uncensored scheme. Using the approach of the Bagdonavicius–Nikulin goodness-of-fit test for validation under the right censored data, we propose a modified chi-square goodness-of-fit test for the new model. The modified goodness-of-fit statistic test was applied for the right censored real data set, called leukemia free-survival times for autologous transplants. Based on the maximum likelihood estimators on initial data, the modified goodness-of-fit test recovered the loss in information while the grouping data and followed chi-square distributions. All elements of the modified goodness-of-fit criteria tests are explicitly derived and given.

A New Parametric Life Family of Distributions: Properties, Copula and Modeling Failure and Service Times

A new family of probability distributions is defined and applied for modeling symmetric real-life datasets. Some new bivariate type G families using Farlie–Gumbel–Morgenstern copula, modified Farlie–Gumbel–Morgenstern copula, Clayton copula and Renyi’s entropy copula are derived. Moreover, some of its statistical properties are presented and studied. Next, the maximum likelihood estimation method is used. A graphical assessment based on biases and mean squared errors is introduced. Based on this assessment, the maximum likelihood method performs well and can be used for estimating the model parameters. Finally, two symmetric real-life applications to illustrate the importance and flexibility of the new family are proposed. The symmetricity of the real data is proved nonparametrically using the kernel density estimation method.

Transmuted Topp-Leone Weibull Lifetime Distribution: Statistical Properties and Different Method of Estimation

In this work we focus on proposing a new lifetime Weibull type model called the transmuted Topp-Leone Weibull and studying its properties. We derive some new bivariate and multivariate transmuted Topp-Leone Weibull versions using “Farlie Gumbel Morgenstern (FGM) Copula”, “modified FGM Copula”, “Clayton Copula” and “Renyi's entropy Copula”. The estimation of its unknown parameters is carried out by considering different method of estimation. The statistical performances of all methods are studied by two real data sets and a numerical Monte Carlo simulation. The Cramer-Von Mises method is the best method for modeling the carbon fibers data. The maximum likelihood method is the best method for modeling the Leukemia data, however all other methods performed well.

A new bivariate Poisson distribution via conditional specification: properties and applications

ABSTRACT In this article, we discuss a bivariate Poisson distribution whose conditionals are univariate Poisson distributions and the marginals are not Poisson which exhibits negative correlation. Some useful structural properties of this distribution namely marginals, moments, generating functions, stochastic ordering are investigated. Simple proofs of negative correlation, marginal over-dispersion, distribution of sum and conditional given the sum are also derived. The distribution is shown to be a member of the multi-parameter exponential family and some natural but useful consequences are also outlined. Parameter estimation with maximum likelihood is implemented. Copula-based simulation experiments are carried out using Bivariate Normal and the Farlie–Gumbel–Morgenstern copulas to assess how the model behaves in dealing with the situation. Finally, the distribution is fitted to seven bivariate count data sets with an inherent negative correlation to illustrate suitability.

Copula function‐based analysis of outage probability and coverage region for wireless multiple access communications with correlated fading channels

From practical and theoretical viewpoints, performance analysis of communications with correlated fading channels is important. In this study, the authors exploit a novel approach based on Copula function concept to investigate the impact of correlation of Rayleigh fading channels on communication performances. One of the most convenient ways to describe the dependence between several variables is using Copula functions. For this purpose, by applying the Farlie–Gumbel–Morgenstern Copula function, they derive closed-form expressions for outage probability as well as coverage region in a wireless multiple access channel. It is shown that the fading correlation improves the outage probability and coverage region for negative dependence structure. Specifically, whenever the dependence structure tends to negative values, the outage and coverage region performance improvement is increased. Finally, the efficiency of the analytical results is illustrated numerically.

A new extension of the FGM copula with an application in reliability

We propose a new symmetric extension of the bivariate Farlie-Gumbel-Morgenstern (FGM) copula with given marginals. The main feature of this new extension is the higher dependence level it permits between random variables. It is shown that the range of Spearman’s Rho is We use this new extension to estimate reliability in a dependent stress-strength model with an application to the Egyptian finance system.

Choice probabilities of random utility maximization models when the errors distribution is a polynomial copula with Gumbel marginals

ABSTRACT The expression of the choice probabilities of RUM models in which the utility errors distribution is a polynomial copula with Gumbel marginals is derived. In particular, it is shown that the expression of the Choice Probability Generating Function can be easily obtained from the polynomial copula of the distribution. This result is particularized for two types of polynomial copulas: the multivariate Farlie–Gumbel–Morgenstern copulas and the Baker's copulas. The correlation coefficient of the errors with these copulas is also obtained. The corresponding RUM models may account for relatively large negative and positive correlation. The RUM models introduced in this work are applied to two samples of interurban trips with three alternatives: airplane, rail and car. The results of the fit show that the models presented in this work may account for the correct correlation signbetween the alternative errors and yield a better fit than a nested logit model.

Dependent Competing Risks Modeling for Mechanical Systems Under Component and Performance Failure

This paper establishes a competing failure analysis model for complex mechanical system under component failure and performance failure considering degradation. Traditionally, mechanical system is composed by a number of components. Meanwhile, mechanical system has the ability to accomplish its specific performance. Therefore, mechanism may fail because of two kinds of failure modes, the component failure due to degradation (such as component wear) and the performance failure (system couldn't complete performance). The two failure modes compete with each other because as soon as one mode occurs the system just fails. The component will degrade with time as system operates as well as the system performance. In this paper, Brownian motion (BM) with nonlinear drift is used to model the degradation of components based on which component failure is analyzed. The function of performance measurement is built by surrogate and performance failure is analyzed by it at different working circulation. Farlie–Gumbel–Morgenstern (FGM) copula is introduced to describe the dependence. The system reliability is analyzed by FGM copula as well as competing failure probability for each failure mode. Finally, a numerical example and an engineering case study are used to illustrate the proposed model.

Estimation of a common mean vector in bivariate meta-analysis under the FGM copula

ABSTRACTWe propose a bivariate Farlie–Gumbel–Morgenstern (FGM) copula model for bivariate meta-analysis, and develop a maximum likelihood estimator for the common mean vector. With the aid of novel mathematical identities for the FGM copula, we derive the expression of the Fisher information matrix. We also derive an approximation formula for the Fisher information matrix, which is accurate and easy to compute. Based on the theory of independent but not identically distributed (i.n.i.d.) samples, we examine the asymptotic properties of the estimator. Simulation studies are given to demonstrate the performance of the proposed method, and a real data analysis is provided to illustrate the method.

Reliability modeling for a discrete time multi-state system with random and dependent transition probabilities

In a random environment, state transition probabilities of a multi-state system can change as the environment changes. Thus, a dynamic reliability model with random and dependent transition probabilities is developed for non-repairable discrete-time multi-state system in this article. The dependence among the random state transition probabilities of the system is modeled by a copula function. By probability argument and random process theory, we obtain explicit expressions of some reliability characteristics and joint survival function of random time spent by the system in all working states (partially and completely working states). A special case is considered when the state transition probabilities are dependent random variables with power distribution, and the dependence structure is modeled by Farlie–Gumbel–Morgenstern copula. Numerical examples are also presented to demonstrate the developed model and perform a comparison for the models with random and fixed transition probabilities.

Dependence in a background risk model

Many copula families, including the classes of Archimedean, elliptical and Liouville copulas, may be written as the survival copula of a random vector R×(Y1,Y2), where R is a strictly positive random variable independent of the random vector (Y1,Y2) . A unified framework is presented for studying the dependence structure underlying this stochastic representation, which is called the background risk model. Formulas for the copula, Kendall’s tau and tail dependence coefficients are obtained and special cases are detailed. The usefulness of the construction for model building is illustrated with an extension of Archimedean copulas with completely monotone generators, based on the Farlie–Gumbel–Morgenstern copula. In particular, explicit expressions for the distribution and the Tail-Value-at-Risk of the aggregated risk RY1+RY2 are available in a generalization of the widely used multivariate Pareto-II model.

A bivariate count model with discrete Weibull margins

Multivariate discrete data arise in many fields (statistical quality control, epidemiology, failure and reliability analysis, etc.) and modelling such data is a relevant task. Here we consider the construction of a bivariate model with discrete Weibull margins, based on Farlie–Gumbel–Morgenstern copula, analyse its properties especially in terms of attainable correlation, and propose several methods for the point estimation of its parameters. Two of them are the standard one-step and two-step maximum likelihood procedures; the other two are based on an approximate method of moments and on the method of proportion, which represent intuitive alternatives for estimating the dependence parameter. A Monte Carlo simulation study is presented, comprising more than one hundred artificial settings, which empirically assesses the performance of the different estimation techniques in terms of statistical properties and computational cost. For illustrative purposes, the model and related inferential procedures are fitted and applied to two datasets taken from the literature, concerning failure data, presenting either positive or negative correlation between the two observed variables. The applications show that the proposed bivariate discrete Weibull distribution can model correlated counts even better than existing and well-established joint distributions.

Low Flow Regimes of the Tarim River Basin, China: Probabilistic Behavior, Causes and Implications

Droughts are a frequent occurrence in Xinjiang, China, and therefore fundamental to determining their hydrologic characteristics is low flow analysis. To that end, 11 probability distribution functions and 26 copulas functions were employed to analyze the changing characteristics of low flow regime (defined as seven-day low flow) of the Tarim River Basin. Results indicated that: (1) The Wakeby distribution satisfactorily described the probabilistic behavior of the low flow regime. According to Akaike Information Criterion (AIC), Bayesian Information Criterions (BIC), maximum likelihood, and other residual-based metrics, Tawn copula, Farlie–Gumbel–Morgenstern copula and Frank copula were the best choice and used in this current study. (2) After 1987, hydrological droughts of longer return periods were prone to higher occurrence frequency. (3) The low flow volume has been increasing in recent years due to the temperature-induced increase of snowmelt and increasing precipitation. However, hydrological droughts can be expected to occur due to the massive increase in water demand from the development of irrigated agriculture, increasing arable land and livestock farming. As a result, the water shortage in the lower Tarim River Basin will be increasingly severe under the influence of climate change and human activities. To alleviate the shortage would call for the development of water-saving agricultural irrigation, water-saving technology, conservation of eco-environment and sustainable development of local socio-economy.

A new extension of the FGM copula for negative association

ABSTRACTIn this paper we introduced a single parameter, absolutely continuous and radially symmetric bivariate extension of the Farlie-Gumbel-Morgenstern (FGM) family of copulas. Specifically, this extension measures the higher negative dependencies than most FGM extensions available in literature. Closed-form formulas for distribution, quantile, density, conditional distribution, regression, Spearman's rho, Kendall's tau, and Gini's gamma are obtained. In addition, a formula for random variate generations is presented in closed-form to facilitate simulation studies. We conduct both paired and multiple comparisons with Frank, Gaussian, and Plackett copulas to investigate the performance based on Vuong's test. Furthermore, the new copula is compared with Frank, Gaussian, and Plackett copulas using both Kolmogorov-Smirnov and Cramér-von Mises type test statistics. Finally, a bivariate dataset is analyzed to compare and illustrate the flexibility of the new copula for negative dependence.