Farlie-Gumbel-Morgenstern Research Articles

Bivariate iterated Farlie–Gumbel–Morgenstern stress–strength reliability model for Rayleigh margins: Properties and estimation

In this paper, we propose bivariate iterated Farlie–Gumbel–Morgenstern (FGM) due to [Huang and Kotz (1984). Correlation structure in iterated Farlie-Gumbel-Morgenstern distributions. Biometrika 71(3), 633–636. https://doi.org/10.2307/2336577] with Rayleigh marginals. The dependence stress–strength reliability function is derived with its important reliability characteristics. Estimates of dependence reliability parameters are obtained. We analyse the effects of dependence parameters on the reliability function. We found that the upper bound of the positive correlation coefficient is attaining to 0.41 under a single iteration with Rayleigh marginals. A comprehensive comparison between classical FGM with iterated FGM copulas is graphically examined to assess the over or under estimation of reliability with respect to α and β. We propose a two-phase estimation procedure for estimating the reliability parameters. A Monte-Carlo simulation study is conducted to assess the finite sample behaviour of the proposed reliability estimators. Finally, the proposed estimators are examined and validated with real data sets.

Collective risk models with FGM dependence

ABSTRACT We study copula-based collective risk models when the dependence structure is defined by a Farlie-Gumbel-Morgenstern (FGM) copula. By leveraging a one-to-one correspondence between the class of FGM copulas and multivariate symmetric Bernoulli distributions, we find convenient representations for the moments and Laplace-Stieltjes transform for the aggregate random variable defined from collective risk models with FGM dependence. We examine different components of this collective risk model, aiming to better understand the impact of the assumed dependence between a claim's frequency and severity. Relying on stochastic ordering, we analyze the impact of dependence on the aggregate claim amount random variable. Even if the FGM copula may only induce moderate dependence, we illustrate through numerical examples that the cumulative effect of FGM dependence can lead to substantial variations in key risk measures on aggregate random variables defined from collective risk models.

Analyzing symmetric distributions by utilizing extropy measures based on order statistics

Quantification of the uncertainty of distribution functions, by using entropy and extropy, is important in many statistical analyses. Inspired by this, our study uses extropy and several related measures (including the cumulative residual extropy, cumulative past extropy, and extropy-inaccuracy measure) of order statistics (OSs) to offer multiple characterizations of symmetric continuous distributions. We demonstrate that a defining feature of symmetric distributions is the equality of these measures of upper and lower OSs. Using concomitants of OSs based on the bivariate distributions belonging to the Farlie-Gumbel-Morgenstern (FGM) family, the same characteristic is demonstrated for these measures. Finally, a real data set is used to illustrate the applicability of the suggested test.

On Bivariate Distributions with Singular Part

There are many families of bivariate distributions with given marginals. Most families, such as the Farlie–Gumbel–Morgenstern (FGM) and the Ali–Mikhail–Haq (AMH), are absolutely continuous, with an ordinary probability density. In contrast, there are few families with a singular part or a positive mass on a curve. We define a general condition useful to detect the singular part of a distribution. By continuous extension of the bivariate diagonal expansion, we define and study a wide family containing these singular distributions, obtain the probability density, and find the canonical correlations and functions. The set of canonical correlations is described by a continuous function rather than a countable sequence. An application to statistical inference is given.

Construction of New Bivariate Semilinear Copulas Based on the Ali–Mikhail–Haq, Farlie–Gumbel–Morgenstern and Plackett Families

Copulas are quite a popular, flexible, and useful tool for multivariate modeling in many fields of applications, including finance and insurance, actuarial sciences, biomedical studies, geostatistics, and hydrology. In this paper, we use the concept of diagonal function to build new bivariate copulas based on the well known and important Ali–Mikhail–Haq (AMH), Farlie–Gumbel–Morgenstern (FGM) and Plackett families. We also study some dependence properties of the newly constructed semilinear copulas, namely, the Spearman’s [Formula: see text] coefficient. In particular, we find that the range of this rank correlation coefficient varies between weak and moderate values for the transformed AMH and FGM families, although it varies between moderate and strong values for the transformed Plackett families. Moreover, in all cases, the Spearman’s [Formula: see text] coefficient reaches negative and positive values. Finally, we present a synthetic data generation algorithm, as well as a classical method of estimation of the copula association parameter [Formula: see text] for each of the six transformed bivariate families.

Bivariate Epanechnikov-exponential distribution: statistical properties, reliability measures, and applications to computer science data

<p>One important area of statistical theory and its applications to bivariate data modeling is the construction of families of bivariate distributions with specified marginals. This motivates the proposal of a bivariate distribution employing the Farlie-Gumbel-Morgenstern (FGM) copula and Epanechnikov exponential (EP-EX) marginal distribution, denoted by EP-EX-FGM. The EP-EX distribution is a complementing distribution, not a rival, to the exponential (EX) distribution. Its simple function shape and dependence on a single scale parameter make it an ideal choice for marginals in the suggested new bivariate distribution. The statistical properties of the EP-EX-FGM model are examined, including product moments, coefficient of correlation between the internal variables, moment generating function, conditional distribution, concomitants of order statistics (OSs), mean residual life function, and vitality function. In addition, we calculated reliability and information measures including the hazard function, reversed hazard function, positive quadrant dependence feature, bivariate extropy, bivariate weighted extropy, and bivariate cumulative residual extropy. Estimating model parameters is accomplished by utilizing maximum likelihood, asymptotic confidence intervals, and Bayesian approaches. Finally, the advantage of EP-EX-FGM over the bivariate Weibull FGM distribution, bivariate EX-FGM distribution, and bivariate generalized EX-FGM distribution is illustrated using actual data sets.</p>

Inference and other aspects for $ q- $Weibull distribution via generalized order statistics with applications to medical datasets

<abstract><p>This work utilizes generalized order statistics (GOSs) to study the $ q $-Weibull distribution from several statistical perspectives. First, we explain how to obtain the maximum likelihood estimates (MLEs) and utilize Bayesian techniques to estimate the parameters of the model. The Fisher information matrix (FIM) required for asymptotic confidence intervals (CIs) is generated by obtaining explicit expressions. A Monte Carlo simulation study is conducted to compare the performances of these estimates based on type Ⅱ censored samples. Two well-established measures of information are presented, namely extropy and weighted extropy. In this context, the order statistics (OSs) and sequential OSs (SOSs) for these two measures are studied based on this distribution. A bivariate $ q $-Weibull distribution based on the Farlie-Gumbel-Morgenstern (FGM) family and its relevant concomitants are studied. Finally, two concrete instances of medical real data are ultimately provided.</p></abstract>

Bivariate Unit-Weibull Distribution: Properties and Inference

In this article, we introduce a novel bivariate probability distribution that is absolutely continuous. Considering the Farlie–Gumbel–Morgenstern (FGM) copula and the unit-Weibull distribution, we can obtain a bivariate unit-Weibull distribution. We evaluate the main properties of the new proposal and use two estimation methods to estimate the parameter for the bivariate probability distribution. A brief Monte Carlo simulation study is conducted to assess the behavior of the employed estimation method and the characteristics of the estimators. Ultimately, as an illustration, a real-life application is presented, demonstrating the utility of the proposal.

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.

Scrutiny of a More Flexible Counterpart of Huang–Kotz FGM’s Distributions in the Perspective of Some Information Measures

In this work, we reveal some distributional traits of concomitants of order statistics (COSs) arising from the extended Farlie–Gumbel–Morgenstern (FGM) bivariate distribution, which was developed and studied in recent work. The joint distribution and product moments of COSs for this family are discussed. Moreover, some useful recurrence relations between single and product moments of concomitants are obtained. In addition, the asymptotic behavior of the concomitant’s rank for order statistics (OSs) is studied. The information measures, differential entropy, Kullback–Leibler (KL) distance, Fisher information number (FIN), and cumulative past inaccuracy (CPI) are theoretically and numerically studied.

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.

Modified Weighted Uniform Distribution And Its Bivariate Extension

In this paper a new version of weighted uniform distribution is constructed and studied. The statistical properties of the new distribution including the behavior of hazard and reversed hazard functions, moments, the central moments, moment generating function, mean, variance, coefficient of skewness, coefficient of kurtosis, median, mode, quantile , stochastic ordering and order statistics are also obtained, a simulation study and real data applications are performed. Moreover, a bivariate extension of the new distribution named the bivariate modified uniform (BMWU) distribution is introduced. The proposed bivariate distribution is of type Farlie–Gumbel–Morgenstern (FGM) copula. The BMWU distribution has modified weighted uniform marginal distributions. The joint cumulative distribution function, the joint survival function, the joint probability density function, the joint hazard rate function and the statistical properties of the BMWU distribution are also derived.

Modified Weighted Rayleigh Distribution and Its Bivariate Extension

In this paper, a new version of weighted Rayleigh distribution is constructed and studied. The statistical properties of the new distribution including the behavior of hazard and reversed hazard functions, moments, the central moments, moment generating function, mean, variance, coefficient of skewness, coefficient of kurtosis, median, mode, quantiles, stochastic ordering, exact information matrix and order statistics are also obtained, a simulation study and real data applications are performed. Furthermore, a bivariate extension of the new distribution called the bivariate modified Rayleigh (BMWR) distribution is introduced. The proposed bivariate distribution is of type Farlie–Gumbel–Morgenstern (FGM) copula. The BMWR distribution has modified weighted Rayleigh marginal distributions. The joint cumulative distribution function, the joint survival function, the joint probability density function, the joint hazard rate function and the statistical properties of the BMWR distribution are also derived.

Bivariate power Lomax distribution with medical applications.

In this paper, a bivariate power Lomax distribution based on Farlie-Gumbel-Morgenstern (FGM) copulas and univariate power Lomax distribution is proposed, which is referred to as BFGMPLx. It is a significant lifetime distribution for modeling bivariate lifetime data. The statistical properties of the proposed distribution, such as conditional distributions, conditional expectations, marginal distributions, moment-generating functions, product moments, positive quadrant dependence property, and Pearson's correlation, have been studied. The reliability measures, such as the survival function, hazard rate function, mean residual life function, and vitality function, have also been discussed. The parameters of the model can be estimated through maximum likelihood and Bayesian estimation. Additionally, asymptotic confidence intervals and credible intervals of Bayesian's highest posterior density are computed for the parameter model. Monte Carlo simulation analysis is used to estimate both the maximum likelihood and Bayesian estimators.

Modeling the Optimal Combination of Proportional and Stop-Loss Reinsurance with Dependent Claim and Stochastic Insurance Premium

This paper investigates an optimal reinsurance policy using a risk model with dependent claim and insurance premium by assuming that the insurance premium is random. Their dependence structure is modeled using Sarmanov’s bivariate exponential distribution and the Farlie–Gumbel–Morgenstern (FGM) copula-based bivariate exponential distribution. The reinsurance premium paid by the insurer to the reinsurer is fixed and is charged by the expected value premium principle (EVPP) and standard deviation premium principle (SDPP). The main objective of this paper is to determine the proportion and retention limit of the optimal combination of proportional and stop-loss reinsurance for the insurer. Specifically, with a constrained reinsurance premium, we use the minimization of the Value-at-Risk (VaR) of the insurer’s net cost. When determining the optimal proportion and retention limit, we provide some numerical examples to illustrate the theoretical results. We show that the dependence parameter, the probability of claim occurrence, and the confidence level have effects on the optimal VaR of the insurer’s net cost.

Survival and Duration Analysis of MSMEs in Chiang Mai, Thailand: Evidence from the Post-COVID-19 Recovery

This study attempts to reveal the consequences of coronavirus disease 2019 (COVID-19) on micro, small, and medium enterprises (MSMEs) in Chiang Mai, Thailand. A total of 786 MSMEs were surveyed during May and August 2022, corresponding to the period when the recovery of businesses and livelihoods from the ongoing COVID-19 crisis became more perceptible. The perceptions of COVID-19’s impact on MSMEs and their survivability are explored and investigated. To achieve this goal, a copula-based sample selection survival model is introduced. This idea of the model is extended from the concept of the Cox proportional hazards model and copula-based sample selection model, enabling us to construct simultaneous equations—namely, the probability-of-failure equation (selection equation) and the duration-of-survival equation (time-to-event or outcome equation). Several copula functions with different dependence patterns are considered to join the failure equation and the duration-of-survival equation. By comparing the Akaike and Bayesian information criteria values of the candidate copulas, we find that Farlie–Gumbel–Morgenstern (FGM) copula performs the best-fit joint function in our analysis. Empirically, the results from this best-fit model reveal that the survival probability of MSMEs in the next year is around 80%. However, some MSMEs may not survive more than three months after the interview. Finally, our results also reveal that the tourism MSMEs have a lower chance of survival than the commercial and manufacturing MSMEs. Notably, the business size and the support schemes from the government—such as the debt restructuring process, the tax payment deadline extension, and the reduced social security contributions—exhibited a role in lengthening the survival duration of the non-surviving MSMEs.

Valuation of Equity-Linked Death Benefits on Two Lives with Dependence

The purpose of this paper is to investigate equity-linked death benefits for joint alive and last survivor individuals. Utilizing Farlie–Gumbel–Morgenstern (FGM) type dependency modeling framework, we first analyze the joint distribution of the couple (joint alive and last survival density) when marginal distributions follow mixed exponentials and weighted exponentials distributions. Then, we derive the price of the guaranteed minimum death benefit (GMDB) product. In addition, we provide closed analytical expressions of the price of some financial contingent claim contracts (classical and exotic options). Furthermore, we present some numerical results to support our theoretical results. We show in our numerical example that it is important to model the dependency between two lives (couple) since the price changes as the copula parameter changes.

A novel bivariate Lomax-G family of distributions: Properties, inference, and applications to environmental, medical, and computer science data

<abstract><p>This paper presents a novel family of bivariate continuous Lomax generators known as the BFGMLG family, which is constructed using univariate Lomax generator (LG) families and the Farlie Gumbel Morgenstern (FGM) copula. We have derived several structural statistical properties of our proposed bivariate family, such as marginals, conditional distribution, conditional expectation, product moments, moment generating function, correlation, reliability function, and hazard rate function. The paper also introduces four special submodels of the new family based on the Weibull, exponential, Pareto, and log-logistic baseline distributions. The study establishes metrics for local dependency and examines the significant characteristics of the proposed bivariate model. To provide greater flexibility, a multivariate version of the continuous FGMLG family are suggested. Bayesian and maximum likelihood methods are employed to estimate the model parameters, and a Monte Carlo simulation evaluates the performance of the proposed bivariate family. Finally, the practical application of the proposed bivariate family is demonstrated through the analysis of four data sets.</p></abstract>

Bivariate q-extended Weibull morgenstern family and correlation coefficient formulas for some of its sub-models

<abstract><p>A bivariate extension of a flexible univariate family is proposed. The new family is called bivariate q-extended Weibull Morgenstern family of distributions which can be constructed based on the Farlie-Gumbel-Morgenstern (FGM) copula technique. After introducing the new family, four sub-models are discussed in detail from the theoretical and numerical coefficient of correlation point of view with pointing to the effect of the $ q $ parameter.</p></abstract>

Some Information Measures Properties of the GOS-Concomitants from the FGM Family

In this paper we recall, extend and compute some information measures for the concomitants of the generalized order statistics (GOS) from the Farlie–Gumbel–Morgenstern (FGM) family. We focus on two types of information measures: some related to Shannon entropy, and some related to Tsallis entropy. Among the information measures considered are residual and past entropies which are important in a reliability context.