Full Likelihood and Inference Functions for Marginal Parameters

Why the Estimation Route Matters

A copula model separates the marginal distributions of individual risks from their dependence structure. Estimation must therefore address both parts: the marginal parameters describe each risk separately, while the copula parameter describes how the risks move together. The estimation route matters because the copula likelihood depends on the fitted marginal parameters through the transformed observations.

Two standard approaches are joint maximum likelihood, which estimates all parameters simultaneously, and inference functions for margins (IFM), which first estimates each margin and then estimates the copula parameters conditional on those marginal estimates. They can produce different estimates from the same data. IFM is computationally convenient, but its second stage does not generally maximize the full likelihood.

Definition: Suppose $(X_i,Y_i)$, $i=1,\ldots,n$, are independent observations from a continuous bivariate model with marginal densities $f_X(x;\alpha)$, $f_Y(y;\beta)$, distribution functions $F_X(x;\alpha)$, $F_Y(y;\beta)$, and copula density $c(u,v;\gamma)$. The joint density is $$ f_{X,Y}(x,y;\alpha,\beta,\gamma) =f_X(x;\alpha)f_Y(y;\beta)c\!\left(F_X(x;\alpha),F_Y(y;\beta);\gamma\right). $$ Here $\alpha,\beta$ are marginal parameters and $\gamma$ is a copula parameter. This density factorization assumes continuous margins and an absolutely continuous copula. It does not directly apply to discrete margins, for which probability masses and copula nonuniqueness require separate treatment.

Joint Maximum Likelihood and IFM

For paired observations, the full log-likelihood is the sum of three contributions: one from each marginal density and one from the copula density evaluated at the marginal probability transforms. Write $u_i=F_X(x_i;\alpha)$ and $v_i=F_Y(y_i;\beta)$. Then

\ell(\alpha,\beta,\gamma) =\sum_{i=1}^n\log f_X(x_i;\alpha) +\sum_{i=1}^n\log f_Y(y_i;\beta) +\sum_{i=1}^n\log c(u_i,v_i;\gamma).

Joint maximum likelihood maximizes this entire expression over $(\alpha,\beta,\gamma)$ at once. IFM instead uses two stages: estimate each set of marginal parameters by maximizing its own marginal likelihood, then maximize the copula log-likelihood with those marginal estimates held fixed. “Inference functions” refers to solving the parameter estimation problem for each component in sequence.

Fit the margins.
Estimate $\alpha$ from the $X$ observations and $\beta$ from the $Y$ observations, usually by separate maximum-likelihood fits.

Transform the observations.
Compute $u_i=F_X(x_i;\hat{\alpha})$ and $v_i=F_Y(y_i;\hat{\beta})$ using the fitted marginal distribution functions.

Fit the copula.
Maximize $\sum_i\log c(u_i,v_i;\gamma)$ over the valid parameter space for $\gamma$.

Assess uncertainty appropriately.
The estimated margins are themselves uncertain. Standard errors for the copula estimate should account for that uncertainty, for example by a suitable sandwich method or a bootstrap that repeats both stages.

The distinction in stage two is important: IFM treats the fitted transforms as fixed while estimating the copula parameter. Full maximum likelihood allows a change in a marginal parameter to affect both the marginal-density terms and the copula-density term. Therefore a marginal estimate that is optimal for the marginal likelihood alone need not be optimal for the joint likelihood.

Worked Example: Factoring a Full Likelihood

A hypothetical portfolio records paired continuous losses $X$ and $Y$. Specify exponential marginal models with rates $\lambda$ and $\mu$, so $f_X(x;\lambda)=\lambda e^{-\lambda x}$ and $F_X(x;\lambda)=1-e^{-\lambda x}$ for $x\geq0$, with analogous expressions for $Y$. The copula is an FGM copula with parameter $\theta$, whose density is $c(u,v;\theta)=1+\theta(1-2u)(1-2v)$, $-1\leq\theta\leq1$. Write the contribution of one observed pair $(x,y)$ to the full likelihood and explain the estimation distinction.

Method. Substitute the specified marginal densities and transforms into the joint density factorization. For one pair,

f_{X,Y}(x,y;\lambda,\mu,\theta) =\lambda e^{-\lambda x}\mu e^{-\mu y} \left[1+\theta(1-2(1-e^{-\lambda x}))(1-2(1-e^{-\mu y}))\right].

Since $1-2(1-e^{-\lambda x})=2e^{-\lambda x}-1$, this can also be written

f_{X,Y}(x,y;\lambda,\mu,\theta) =\lambda\mu e^{-\lambda x-\mu y} \left[1+\theta(2e^{-\lambda x}-1)(2e^{-\mu y}-1)\right].

The log-likelihood contribution is therefore $\log\lambda-\lambda x+\log\mu-\mu y+ \log[1+\theta(2e^{-\lambda x}-1)(2e^{-\mu y}-1)]$. For $n$ pairs, sum this expression over the observations. Joint maximum likelihood optimizes the sum over $\lambda,\mu,\theta$ simultaneously. IFM first estimates the two rates from their separate exponential marginal likelihoods, then substitutes those rates into the transformed observations and maximizes only the copula terms over $\theta$.

Check. Setting $\theta=0$ makes the bracket equal to one, leaving the product of the two exponential densities. This is the independence model. A common mistake is to retain the copula factor but omit one marginal density; that expression is not the joint density.

Why Full and Marginal Scores Differ

The reason can be seen by differentiating the full log-likelihood with respect to a marginal parameter. For example, let $a_i=1-2F_X(x_i;\alpha)$ and $b_i=1-2F_Y(y_i;\beta)$ under the FGM copula. Its density is $1+\theta a_i b_i$. The copula contribution to the score for $\alpha$ is

\frac{\partial}{\partial\alpha}\log(1+\theta a_i b_i) =\frac{\theta b_i\,(\partial a_i/\partial\alpha)} {1+\theta a_i b_i}.

The full score for $\alpha$ includes this term as well as the derivative of the $X$-marginal log-likelihood. At the marginal-only estimate, the marginal score may be zero, but the copula contribution need not be zero. Thus the full score may not vanish there.

This calculation also explains why merely fitting the margins and then reporting their estimates as full maximum-likelihood estimates is incorrect. The copula can alter the best-fitting marginal parameters because the transformed values $u_i,v_i$ depend on those parameters. The same principle applies to other differentiable copula families, though the copula score takes a family-specific form.

Worked Example: Checking the Full Score at a Marginal Estimate

At a candidate marginal parameter value $\lambda$, suppose the marginal log-likelihood score is zero. For three paired observations, consider the FGM copula with $\theta=0.5$. The relevant values $(a_i,b_i,\partial a_i/\partial\lambda)$ are $(-0.05527,0.8,-0.94473)$, $(0.2,0.3,0)$, and $(-0.55374,-0.8,-0.89252)$. Determine whether the full score for $\lambda$ is zero.

Method. Add the three copula contributions using $\theta b_i(\partial a_i/\partial\lambda)/(1+\theta a_i b_i)$. The marginal contribution is zero by assumption.

For the first observation, the denominator is $1+0.5(-0.05527)(0.8)=0.977892$; the numerator is $0.5(0.8)(-0.94473)=-0.377892$. The contribution is $-0.377892/0.977892\approx-0.3864$.

For the second observation, $\partial a_i/\partial\lambda=0$, so its contribution is zero.

For the third observation, the denominator is $1+0.5(-0.55374)(-0.8)=1.221496$; the numerator is $0.5(-0.8)(-0.89252)=0.357008$. The contribution is $0.357008/1.221496\approx+0.2923$.

The total copula contribution is $-0.3864+0+0.2923\approx-0.0942$. Adding the zero marginal score leaves a nonzero full score, so this marginal-only estimate is not stationary for the full likelihood at the stated candidate. In particular, the third term is positive because both factors $-0.8$ and $-0.89252$ are negative. Treating it as negative would give the wrong score.

Estimating an FGM Copula Parameter After Marginal Fitting

A small calculation illustrates the copula stage of IFM. The FGM density is nonnegative on the closed unit square when $-1\leq\theta\leq1$, and strictly positive on its interior for that parameter range. It can equal zero at some boundary points when $\theta=1$ or $\theta=-1$. For fixed transformed observations, the copula log-likelihood is $\sum_i\log[1+\theta(1-2u_i)(1-2v_i)]$.

To derive its score, put $w_i=(1-2u_i)(1-2v_i)$. Differentiating each log term gives $w_i/(1+\theta w_i)$, so an interior maximum must satisfy $\sum_i w_i/(1+\theta w_i)=0$. The second derivative is $-\sum_i w_i^2/(1+\theta w_i)^2$, which is nonpositive wherever the likelihood is defined. Thus the log-likelihood is concave in $\theta$; if the score root is outside the allowed interval, compare the likelihood at the relevant boundary instead.

Worked Example: IFM Estimate for an FGM Copula

Suppose a hypothetical calibration set has produced fitted continuous marginal models. For two paired validation observations, their fitted transforms are $(u_1,v_1)=(0.1,0.25)$ and $(u_2,v_2)=(0.8,0.25)$. Estimate the FGM parameter by the copula stage of IFM.

Method. The marginal fitting stage is complete; use its fitted transforms without refitting them in this copula step. Calculate $w_i=(1-2u_i)(1-2v_i)$. For the first pair, $w_1=(0.8)(0.5)=0.4$. For the second, $w_2=(-0.6)(0.5)=-0.3$.

The copula log-likelihood is $\ell_C(\theta)=\log(1+0.4\theta)+\log(1-0.3\theta)$. Its score is

\ell_C'(\theta)=\frac{0.4}{1+0.4\theta} -\frac{0.3}{1-0.3\theta}.

Set this equal to zero and cross-multiply, noting that the denominators are positive on $[-1,1]$: $0.4(1-0.3\theta)-0.3(1+0.4\theta)=0$. Hence $0.1-0.24\theta=0$, and $\hat{\theta}=0.1/0.24=5/12\approx0.4167$, to four decimal places. This is inside the allowed interval. The second derivative is negative at this value, and indeed throughout the parameter interval, so it is the maximum. As an additional check, $\ell_C(0)=0$, while $\ell_C(5/12)=\log(7/6)+\log(7/8)=\log(49/48)>0$.

The estimate describes positive association for this fitted FGM model. A common mistake is to solve the score equation and accept any root without checking the parameter range. In this example the root is valid; in another data set the maximizing value could lie at a boundary.

Interpreting the Two Approaches

Under suitable regularity conditions, both joint maximum likelihood and IFM can provide consistent parameter estimates when the specified model is correct and the sample grows. They are not generally numerically identical, and IFM is not generally as statistically efficient as joint maximum likelihood because its stages do not optimize the same full objective. In practice, IFM can be attractive when the marginal models are straightforward to fit and joint optimization is difficult or unstable.

The distinction also affects uncertainty estimates. Treating fitted transforms as if they were known can understate uncertainty in the copula estimate. A standard error calculation should reflect estimation of both the marginal and copula parameters. A bootstrap intended to do this should resample paired observations and repeat the marginal fits and copula fit in every replicate; resampling only the final transformed pairs misses the first-stage uncertainty.

For an FGM copula, the density is $c(u,v;\theta)=1+\theta(1-2u)(1-2v)$. A useful check is that the fitted parameter lies in $[-1,1]$ and each density factor used in the log-likelihood is positive for the observed interior transforms. A zero factor would make the log-likelihood negative infinity. Other copula families have their own parameter restrictions and likelihood checks.

Key takeaway: Joint maximum likelihood optimizes marginal and copula contributions together. IFM fits the margins first and then estimates dependence conditional on those fitted margins. Because marginal parameters also affect the copula terms, a marginal-only estimate need not be a full-likelihood estimate.

Check Your Understanding

Use the stated method and check parameter restrictions before accepting a copula estimate.

For a continuous bivariate model, identify the three terms in the joint log-likelihood factorization and state which parameters are optimized jointly under full maximum likelihood. (Answer: The two marginal log-density terms and the copula log-density term; full maximum likelihood jointly optimizes both marginal parameter sets and the copula parameter.)
An IFM fit gives $\hat{\alpha}=2$ and $\hat{\beta}=3$. Explain what values are used to estimate the copula parameter and whether the second stage re-estimates $\alpha,\beta$. (Answer: Use $u_i=F_X(x_i;2)$ and $v_i=F_Y(y_i;3)$; the second stage holds $\alpha=2,\beta=3$ fixed.)
For an FGM fit, the transformed pairs produce $w_1=0.3$ and $w_2=-0.2$. Solve the interior score equation and check the parameter range. (Answer: $0.3/(1+0.3\theta)-0.2/(1-0.2\theta)=0$ gives $0.1-0.12\theta=0$, so $\hat{\theta}=0.8333$, which is admissible.)
At a marginal estimate, the marginal score for a parameter is zero but the sum of the copula contributions to its full score is $0.07$. Is this point stationary for the full likelihood in that parameter? (Answer: No; the full score is $0+0.07=0.07$, not zero.)
For FGM observations with $w_1=0.25$ and $w_2=-0.20$, compute the score at $\theta=1$ and say whether this endpoint is a candidate maximum based on the one-sided direction into the parameter space. (Answer: The score is $0.25/1.25-0.20/0.80=0.20-0.25=-0.05$; moving left from $1$ increases the log-likelihood locally, so $\theta=1$ is not a maximum.)

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