Farlie Gumbel Morgenstern Distribution Research Articles

Farlie–Gumbel–Morgenstern Bivariate Moment Exponential Distribution and Its Inferences Based on Concomitants of Order Statistics

In this research, we design the Farlie–Gumbel–Morgenstern bivariate moment exponential distribution, a bivariate analogue of the moment exponential distribution, using the Farlie–Gumbel–Morgenstern approach. With the analysis of real-life data, the competitiveness of the Farlie–Gumbel–Morgenstern bivariate moment exponential distribution in comparison with the other Farlie–Gumbel–Morgenstern distributions is discussed. Based on the Farlie–Gumbel–Morgenstern bivariate moment exponential distribution, we develop the distribution theory of concomitants of order statistics and derive the best linear unbiased estimator of the parameter associated with the variable of primary interest (study variable). Evaluations are also conducted regarding the efficiency comparison of the best linear unbiased estimator relative to the respective unbiased estimator. Additionally, empirical illustrations of the best linear unbiased estimator with respect to the unbiased estimator are performed.

Ordered Variables and Their Concomitants under Extropy via COVID-19 Data Application

Extropy, as a complementary dual of entropy, has been discussed in many works of literature, where it is declared for other measures as an extension of extropy. In this article, we obtain the extropy of generalized order statistics via its dual and give some examples from well-known distributions. Furthermore, we study the residual and past extropy for such models. On the other hand, based on Farlie–Gumbel–Morgenstern distribution, we consider the residual extropy of concomitants of m-generalized order statistics and present this measure with some additional features. In addition, we provide the upper bound and stochastic orders of it. Finally, nonparametric estimation of the residual extropy of concomitants of m-generalized order statistics is included using simulated and real data connected with COVID-19 virus.

Parameter Estimation for Multi-state Coherent Series and Parallel Systems with Positively Quadrant Dependent Models

A multi-state coherent system consists of multiple components each of which passes through a sequence of states and it is of interest to estimate the distribution of time spent by the components in different states. Although not practical, for mathematical convenience, it is usually assumed that the times spent by the components in various states are independent of each other. This paper considers three-state series and parallel systems and is based on the assumption that times spent by the components in various states are positively quadrant dependent (PQD) and the corresponding dependence is modeled using a Farlie Gumbel Morgenstern (FGM) distribution. To begin with it is shown that even when marginal distributions are assumed exponential, the resulting likelihood function leads to a complicated expression making maximum likelihood (ML) based inference computationally challenging. A generalized method of moment (GMM) estimation is shown to be relatively simpler not only computationally but also the method works for arbitrary marginal distributions. The estimates obtained by GMM are shown to be uniformly consistent under some mild regularity conditions. Finite sample performances of the ML and GMM are illustrated using FGM distribution with various parametric marginal distributions. In case of exponential marginals, it is shown that GMM compares favorably to ML although the former method does not require parametric assumption for the marginals. The proposed methods are also illustrated using a real case study data of a rare type of head and neck cancer.

On bivariate ageing properties of exchangeable Farlie–Gumbel–Morgenstern distributions

ABSTRACTThis paper deals with bivariate Farlie–Gumbel–Morgenstern distributions. We build the TP2 (RR2) property of the residual lifetime and study the evolution of the dependence of the residual lifetime at large age. Also, we derive the aging property in the sense of the upper orthant order. Furthermore, in the context of IFR (DFR) marginals the residual lifetime at the age above the median is found to be increasing (decreasing) in the upper orthant order.

Uniform asymptotics for the ruin probabilities of a two-dimensional renewal risk model with dependent claims and risky investments

Consider a two-dimensional renewal risk model, in which the independent and identically distributed claim-size random vectors follow a common bivariate Farlie–Gumbel–Morgenstern distribution. Assuming that the surplus is invested in a portfolio whose return follows a Lévy process and that the claim-size distribution is heavy-tailed, uniformly asymptotic estimates for two kinds of finite-time ruin probabilities of the two-dimensional risk model are obtained.

The infinite-time ruin probability for a bidimensional renewal risk model with constant force of interest and dependent claims

ABSTRACTThis article studies a continuous-time bidimensional risk model, in which an insurer simultaneously confronts two kinds of claim sharing a common renewal claim-number process. Under the assumption that the claim size vectors form a sequence of independent and identically distributed random vectors following a common bivariate Farlie–Gumbel–Morgenstern distribution with extended regularly varying margins, we derive an explicit asymptotic formula for the corresponding infinite-time ruin probability.

Concomitants of [formula omitted]-generalized order statistics from generalized Farlie–Gumbel–Morgenstern distribution family

In this work, we study the concomitants of m-generalized order statistics (m-GOSs) from generalized Farlie–Gumbel–Morgenstern (GFGM) distribution family as a natural extension of the results by Beg and Ahsanullah (2008). We derive probability density functions, moments and recurrence relations between moments of concomitants of m-GOSs from the GFGM distribution family. Analogous results for order statistics and record values are presented as special cases. Moreover, using the features of the GFGM family and the moments of concomitants of order statistics, we propose three estimators of the dependence parameter of the GFGM family with Dagum distributed marginals.

Measures of information for concomitants of generalized order statistics from subfamilies of Farlie–Gumbel–Morgenstern distributions

In this paper, we study Shannon’s entropy and Fisher information number for concomitants of generalized order statistics from subfamilies of Farlie–Gumbel–Morgenstern when the marginal distributions are Weibull, exponential, Pareto and power function. Also, we provide some numerical results of Shannon entropy and Fisher information number for concomitants of order statistics.

Ruin with insurance and financial risks following the least risky FGM dependence structure

Recently, Chen (2011) studied the finite-time ruin probability in a discrete-time risk model in which the insurance and financial risks form a sequence of independent and identically distributed random pairs with common bivariate Farlie–Gumbel–Morgenstern (FGM) distribution. The parameter θ of the FGM distribution governs the strength of dependence, with a smaller value of θ corresponding to a less risky situation. For the subexponential case with −1<θ≤1, a general asymptotic formula for the finite-time ruin probability was derived. However, the derivation there is not valid for the least risky case θ=−1. In this paper, we complete the study by extending it to θ=−1. The new formulas for θ=−1 look very different from, but are intrinsically consistent with, the existing one for −1<θ≤1, and they offer a quantitative understanding on how significantly the asymptotic ruin probability decreases when θ switches from its normal range to its negative extremum.

Asymptotic finite-time ruin probability for a bidimensional renewal risk model with constant interest force and dependent subexponential claims

This paper considers a bidimensional renewal risk model with constant interest force and dependent subexponential claims. Under the assumption that the claim size vectors form a sequence of independent and identically distributed random vectors following a common bivariate Farlie–Gumbel–Morgenstern distribution, we derive for the finite-time ruin probability an explicit asymptotic formula.

New families of symmetric/asymmetric copulas

In 2004, Rodríguez-Lallena and Úbeda-Flores have introduced a class of bivariate copulas which generalizes some known families such as the Farlie–Gumbel–Morgenstern distributions. In 2006, Dolati and Úbeda-Flores presented multivariate generalizations of this class, also they investigated symmetry, dependence concepts and measuring the dependence among the components of each classes. In this paper, a new method of constructing binary copulas is introduced, extending the original study of Rodríguez-Lallena and Úbeda-Flores to new families of symmetric/asymmetric copulas. Several properties and parameters of newly introduced copulas are included. Among others, relationship of our construction method with several kinds of ordinal sums of copulas is clarified.

The tail probability of the product of dependent random variables from max-domains of attraction

In this article, we investigate the tail probability of the product of finitely many non-negative dependent random variables. They follow distributions from max-domains of attraction of extreme value distributions and their dependence is modeled via a multivariate Farlie–Gumbel–Morgenstern distribution. For each of the Fréchet, Gumbel and Weibull cases, we obtain an explicit asymptotic formula for the tail probability of the product. Our study extends a few known results in the literature.

From the Huang–Kotz FGM distribution to Baker’s bivariate distribution

Huang and Kotz (1999) [17] considered a modification of the Farlie–Gumbel–Morgenstern (FGM) distribution, introducing additional parameters, and paved the way for many research papers on modifications of FGM distributions allowing high correlation. The first part of the present paper is a review of recent developments on bivariate Huang–Kotz FGM distributions and their extensions. In the second part a class of new bivariate distributions based on Baker’s system of bivariate distributions is considered. It is shown that for a model of a given order, this class of distributions with fixed marginals which are based on pairs of order statistics constructed from the bivariate sample observations of dependent random variables allows higher correlation than Baker’s system. It also follows that under certain conditions determined by Lin and Huang (2010) [21], the correlation for these systems converges to the maximum Fréchet–Hoeffding upper bound as the sample size tends to infinity.

The product of two dependent random variables with regularly varying or rapidly varying tails

Let X and Y be two nonnegative and dependent random variables following a generalized Farlie–Gumbel–Morgenstern distribution. In this short note, we study the impact of a dependence structure of X and Y on the tail behavior of X Y . We quantify the impact as the limit, as x → ∞ , of the quotient of Pr ( X Y > x ) and Pr ( X ∗ Y ∗ > x ) , where X ∗ and Y ∗ are independent random variables identically distributed as X and Y , respectively. We obtain an explicit expression for this limit when X is regularly varying or rapidly varying tailed.

New constructions of diagonal patchwork copulas

We present a method for constructing symmetric copulas which generalizes the diagonal patchwork construction of copulas procedure. We also show how it is related to a new construction of a generalized Farlie–Gumbel–Morgenstern distribution and to the copula transforms.

Likelihood inference for exchangeable continuous data with covariates and varying cluster sizes; use of the Farlie–Gumbel–Morgenstern model

This article investigates the Farlie–Gumbel–Morgenstern class of models for exchangeable continuous data. We show how the model specification can account for both individual and cluster level covariates, we derive insights from comparisons with the multivariate normal distribution, and we discuss maximum likelihood inference when a sample of independent clusters of varying sizes is available. We propose a method for maximum likelihood estimation which is an alternative to direct numerical maximization of the likelihood that sometimes exhibits non-convergence problems. We describe an algorithm for generating samples from the exchangeable multivariate Farlie–Gumbel–Morgenstern distribution with any marginals, using the structural properties of the distribution. Finally, we present the results of a simulation study designed to assess the properties of the maximum likelihood estimators, and we illustrate the use of the FGM distributions with the analysis of a small data set from a developmental toxicity study.

New Approach of Directional Dependence in Exchange Markets Using Generalized FGM Copula Function

This article presents an application of copula methodology in exchange markets. In this article, we consider the concept of directional dependence given by Sungur (2005). We also consider and study directional dependence for generalized Farlie–Gumbel–Morgenstern (FGM) distributions, which are a member of the Rodríguez-Lallena and Úbeda-Flores (2004) family, C(u, v) = uv + f(u)g(v). Examples of the generalized FGM distributions are provided with exchange market data of the Euro, Canadian dollar, Korean Won, Japanese Yen, and Hong Kong dollar against the U.S. dollar.

An order-statistics-based method for constructing multivariate distributions with fixed marginals

A new system of multivariate distributions with fixed marginal distributions is introduced via the consideration of random variates that are randomly chosen pairs of order statistics of the marginal distributions. The distributions allow arbitrary positive or negative Pearson correlations between pairs of random variates and generalise the Farlie–Gumbel–Morgenstern distribution. It is shown that the copulas of these distributions are special cases of the Bernstein copula. Generation of random numbers from the distributions is described, and formulas for the Kendall and grade (Spearman) correlations are given. Procedures for data fitting are described and illustrated with examples.

Concomitants of generalized order statistics from Farlie–Gumbel–Morgenstern distributions

Generalized order statistics constitute a unified model for ordered random variables that includes order statistics and record values among others. Here, we consider concomitants of generalized order statistics for the Farlie–Gumbel–Morgenstern bivariate distributions and study recurrence relations between their moments. We derive the joint distribution of concomitants of two generalized order statistics and obtain their product moments. Application of these results is seen in establishing some well known results given separately for order statistics and record values and obtaining some new results.

Test of independence for generalized Farlie–Gumbel–Morgenstern distributions

Given a pair of absolutely continuous random variables ( X , Y ) distributed as the generalized Farlie–Gumbel–Morgenstern (GFGM) distribution, we develop a test for testing the hypothesis: X and Y are independent vs. the alternative; X and Y are positively (negatively) quadrant dependent above a preassigned degree of dependence. The proposed test maximizes the minimum power over the alternative hypothesis. Also it possesses a monotone increasing power with respect to the dependence parameter of the GFGM distribution. An asymptotic distribution of the test statistic and an approximate test power are also studied.