Farlie–Gumbel–Morgenstern Copula

• Farlie-Gumbel-Morgenstern (FGM) copula is a simple analytic model that introduces controlled, weak symmetric dependence among arbitrary marginal distributions.
• Its mathematical construction separates marginal behavior from joint dependence, with a maximum linear correlation of |ρ| ≤ 1/3, suitable for moderate dependencies.
• Generalizations via functional extensions and orthogonal expansions overcome limitations by capturing richer dependence structures and enabling flexible high-dimensional modeling.

The Farlie–Gumbel–Morgenstern (FGM) copula family consists of analytic constructions that introduce controlled, explicit dependence between random variables with arbitrary given marginals. The classical bivariate FGM copula is defined for $(u_1,u_2)\in[0,1]^2$ as

$$C^{\mathrm{FGM}}(u_1,u_2; \kappa) = u_1u_2 + \kappa u_1u_2(1-u_1)(1-u_2), \quad \kappa\in[-1,1],$$

parameterizing the dependence between $u_1$ and $u_2$. This copula is among the simplest bivariate copula forms, widely adopted in probability, statistics, reliability, risk theory, and applied fields (e.g., astrophysics, queueing, and wireless communication) when a moderate degree of stochastic dependence is to be encoded between non-Gaussian variables.

1. Mathematical Construction and Core Properties

Let $F_1(x_1)$, $F_2(x_2)$ be univariate cdfs of random variables $X_1$ and $X_2$. Sklar's theorem asserts that any bivariate cdf $G(x_1, x_2)$ may be written as $C[F_1(x_1), F_2(x_2)]$ for some copula $C$. The FGM construction is

$$G(x_1, x_2) = F_1(x_1) F_2(x_2) \bigl[1+\kappa (1-F_1(x_1))(1-F_2(x_2)) \bigr],$$

while the corresponding joint density is

$$g(x_1,x_2) = f_1(x_1) f_2(x_2) \left[1+\kappa (2F_1(x_1)-1)(2F_2(x_2)-1)\right].$$

Here $f_i$ is the marginal density of $F_i$. The copula separates the marginal structure from dependence, with the additional $(2F_1-1)(2F_2-1)$ factor introducing the joint dependence.

Crucially, the attainable range of linear correlation is $|\rho| \leq 1/3$; thus, the FGM copula is valid only for cases of weak dependence. For dependence exceeding this limit, alternative copula constructions (such as the Gaussian copula) are necessary (Takeuchi, 2010).

The multivariate ($d$-dimensional) extension is

$$C(u_1, \ldots, u_d) = \prod_{j=1}^d u_j \left[1+\sum_{k=2}^d \sum_{1\leq i_1 < \ldots < i_k \leq d} \theta_{i_1, \ldots, i_k} \prod_{l=1}^k (1-u_{i_l})\right],$$

with a low-dimensional illustration and constraints on $\theta$ parameters that ensure copula validity (Blier-Wong et al., 2022).

2. Limitations and Motivated Generalizations

The limited range $|\rho| \le 1/3$, the absence of tail dependence, and the fact that the FGM structure can only model symmetric, relatively weak dependence led to significant generalizations:

• Functional-parameter extensions: The bivariate extension introduces functions $\varphi$ and $\theta$:

$$C_{\theta, \varphi}(u,v) = uv + \theta(\max\{u,v\}) \cdot \varphi(\min\{u,v\})$$

where necessary and sufficient regularity/monotonicity conditions on $\varphi$ and $\theta$ are established to ensure copula validity (Amblard et al., 2011). This construction enables Spearman’s $\rho$ to range up to 1 and positive tail dependence up to any value in $[0,1]$.

• Legendre polynomial (orthogonal) expansions: Any symmetric, absolutely continuous copula with square-integrable density can be represented as

$$C(u,v) = uv + \sum_{k=1}^\infty \lambda_k \left[\int_0^u \phi_k(x)dx\right]\left[\int_0^v \phi_k(y)dy\right]$$

where $\{\phi_k\}$ is an orthonormal system (e.g., shifted Legendre polynomials) and $\lambda_k$ are parameters tied to dependence structure (Longla, 2023). The FGM copula is the special case where only one quadratic basis function is used (Longla et al., 2023).

These generalizations yield richer dependence properties, with higher-order parameters encoding more complex (even asymmetric) and tail behaviors.

3. Probabilistic and Geometric Structure in High Dimensions

A key theoretical advance is the probabilistic representation of high-dimensional FGM copulas based on exchangeable Bernoulli vectors (Blier-Wong et al., 2022, Blier-Wong et al., 2022):

• Convex hull of extreme points: The $d$-variate exchangeable FGM parameter set forms the convex hull of finitely many “extreme points”, ensuring tractable characterization of admissible dependence structures.
• Sampling and estimation: Stochastic sampling and maximum likelihood estimation exploit this probabilistic structure—sampling is performed by drawing an exchangeable Bernoulli vector $(I_1, \ldots, I_d)$, with each $u_j$ sampled conditionally, while parameter estimation is facilitated via expectation-maximization techniques over the convex hull.

In practical construction with mixtures involving Bernoulli and Coxian-2 distributions, the stochastic representation ensures tractable simulation and closed-form expressions for dependence measures, enhancing the ability to model asymmetric and negative dependence in high dimensions (Blier-Wong et al., 2022).

4. Applications in Risk, Queueing, and Engineering Sciences

Risk models with dependence: Within classical and collective risk models, an FGM copula models dependence between inter-claim times and claim severities. This modifies surplus ruin probability and moments of the aggregate loss, leading to closed-form expressions for the Laplace transform of the survival probability and explicit formulas for moments, variance, Value-at-Risk (VaR), and Tail Value-at-Risk (TVaR) (Qazvini, 2020, Blier-Wong et al., 2022). The cumulative effect of even moderate FGM dependence can be substantial for risk aggregation and capital allocation.

Queueing systems: In single-server queues, the FGM copula models dependence between service and inter-arrival times, altering the steady-state distribution of maximum and minimum customer overlap times. For positive dependence, these overlap times increase, with direct implications for the analysis of contagion and customer interaction in service systems (Dimitriou, 17 Sep 2025).

Wireless communications: For MAC channels where fading coefficients are correlated, the FGM copula is used to derive closed-form expressions for the outage probability (OP). Negative dependence (anti-correlation of channel coefficients) is found to decrease the OP and improve system efficiency (Entezami et al., 20 Dec 2024).

Bivariate models for reliability: The FGM copula is employed to couple novel marginals (e.g., inverse Topp-Leone) in bivariate lifetime models, with estimation by MLE or Bayesian (MCMC) methods, yielding improved fit for heterogeneous reliability data (Tyagi, 2022).

5. Inference and Estimation: Bernstein Smoothing and Orthogonal Expansions

Nonparametric and semiparametric inference exploits the analytic structure of FGM-type copulas:

• Moment estimators: For the extended/Legendre FGM copula, unbiased, consistent, and asymptotically normal estimators for the dependence parameters are formed by sample averages of orthogonal polynomial–based function products (Longla et al., 2023, Hamadou et al., 8 Sep 2025).
• Bernstein smoothing: For lower-tail measures (lower-tail Spearman’s $\rho$), Bernstein-smoothed estimators based on the FGM copula demonstrably achieve up to 70% lower mean squared error (MSE) in deep-tail regions relative to unsmoothed empirical estimators, especially for moderate or weak dependence and small samples (Ouimet et al., 10 Jun 2025).
• Central limit theorems: The Legendre expansion estimators and MLEs are both shown to be asymptotically normal (with explicit covariance matrices), facilitating inference and hypothesis testing for independence (Longla et al., 2023, Hamadou et al., 8 Sep 2025).

6. Local Dependence and Structural Properties

The relative local dependence of the FGM copula is characterized by

$$r^{\mathrm{FGM}}_\theta(u,v) = \frac{4\theta}{\bigl[1 + \theta(2u-1)(2v-1)\bigr]^3},$$

which is invariant under monotonic marginal transformations and peaks near the center of the unit square, decaying towards the edges. This localizes the “interaction” encoded by the copula, highlighting its context-specific suitability for moderate, symmetric, center-concentrated dependence structures (Sukeda et al., 24 Jul 2024).

Positive measure inducing property (PMI): For $\alpha \ge 0$, the FGM copula is PMI, meaning it concentrates more copula mass near the diagonal and induces ordering among concordance measures—e.g., $3\gamma\geq 2\rho$ (Gini’s gamma and Spearman’s rho), providing testable constraints for model selection (Fuchs et al., 2023).

7. Summary Table: FGM Copula Family and Application Domains

Feature/Extension	Mathematical Expression	Key Application Domains
Classical bivariate FGM	$uv + \kappa uv(1-u)(1-v)$	Astrophysics, basic risk/queueing, simple fits
Bivariate extension (Amblard et al., 2011)	$uv + \theta(\max(u,v))\varphi(\min(u,v))$	Finance, hydrology, tail-dependent systems
Multivariate FGM	$\prod_j u_j [1+ \sum \theta_{\cdots}\prod(1-u_j)]$	Insurance risk aggregation, dependence modeling
Orthogonal (Legendre) extension	$$uv + \lambda_1 \Phi_1(u)\Phi_1(v) + \lambda_2\Phi_2(u)\Phi_2(v)$$	Flexible high-dim modeling, engineering data
PMI property	$c(u,v) - c(1-u,v) - c(u,1-v) + c(1-u,1-v)\ge 0$	Concordance tests, copula model selection
Local dependence	$$r^\mathrm{FGM}_\theta(u,v) = 4\theta/[1 + \theta(2u-1)(2v-1)]^3$$	Dependence mapping, diagnostic tools

References and Impact

• The FGM copula remains a core tool for constructing joint distributions with tractable, interpretable, and explicit dependence for weak to moderate associations (Takeuchi, 2010, Amblard et al., 2011, Longla, 2023).
• Its generalizations (via parametric functions or orthogonal expansions) address modeling needs in high-/multi-dimensional, asymmetric, or tail-sensitive applications (Blier-Wong et al., 2022, Blier-Wong et al., 2022, Longla et al., 2023, Hamadou et al., 8 Sep 2025).
• In applied contexts, the FGM copula informs statistical risk analysis (Qazvini, 2020, Blier-Wong et al., 2022), queueing system performance (Dimitriou, 17 Sep 2025), and performance analysis of communication channels (Entezami et al., 20 Dec 2024).
• Advanced inference techniques, including nonparametric (smoothed) lower-tail concordance estimation (Ouimet et al., 10 Jun 2025) and tests derived from PMI orderings (Fuchs et al., 2023), bolster the arsenal for empirical model validation and copula selection in complex data environments.

The FGM copula family—despite intrinsic limitations in maximal dependence—serves as both a pedagogical prototype and a practical component for analytic and computational solutions in multidimensional probability modeling. Contemporary work on its extensions continues to expand its theoretical scope and applied impact across statistical and engineering sciences.