The inference of functions for margins (IFM) method is the most common method used to estimate the copula parameters. In the context of copula modeling, “inference of functions for margins” refers to the process of estimating the probability distribution functions (PDFs) or cumulative distribution functions (CDFs) of individual variables (margins) within a multivariate model. This step is important because copula models typically involve two main components: the joint distribution (captured by the copula) and the marginal distributions (captured by the margins). The IFM is specifically concerned with estimating the functions that describe the individual marginal distributions.
The IFM is a two-step method. The first step is to estimate the parameters of the marginal distribution of each variable. The marginal distributions are estimated by maximizing the log of the likelihood function, which is given by: \(\begin{align} \hat{\alpha}=\arg\max\sum_{t=1}^{N}\ln f_i(x_i^t;\alpha_i) \end{align}\) where $\hat{\alpha}$ is the estimated parameter of $\alpha$, and $f_i$ is the probability density function (PDF) of $X_i$ for $i = 1, 2$. The marginal distribution functions used are
The fittest marginal univariate distribution is selected on the basis of
The p-value is estimated using Anderson–Darling statistic to perform the Anderson–Darling hypothesis test with the 5% significance level. If the p-value is larger than 5%, then the test fails to reject the null hypothesis, that is, the data are from a population with the selected distribution, $F(\cdot)$.
The second step of the IFM method is to estimate the copula parameter by maximizing the log-likelihood function of copula density, which is given by: \(\begin{align} \hat{\theta}=\arg\max\sum_{t=1}^{N}\ln c(F_1(x_1^t|\hat{\alpha}_1),F_2(x_2^t|\hat{\alpha}_2);\theta) \end{align}\) where $\hat{\theta}$ is the estimated parameter of $\theta$, and $c$ is the copula density function. The copula functions used are
The fittest copula function is selected on the basis of several statistical tests, such as
The theoretical frequency is obtained from the model of copula function, while the empirical frequency is estimated using Gringorten’s formula [2].
Using the Cramer–von Mises statistic, the p-value is estimated to perform the Cramer–von Mises hypothesis test with the 5% significance level. If the p-value is greater than 5%, then the test fails to reject the null hypothesis, i.e., the data are from the selected theoretical copula, $C(\cdot)$.
For more details, read article [3].
References
[1] Berg, D. (2009). Copula goodness-of-fit testing: an overview and power comparison. The European Journal of Finance, 15(7-8), 675–701.
[2] Gringorten, I. I. (1963). A plotting rule for extreme probability paper. Journal of Geophysical Research, 68(3), 813-814.
[3] Najib, M. K., Nurdiati, S., & Sopaheluwakan, A. (2022). Multivariate fire risk models using copula regression in Kalimantan, Indonesia. Natural Hazards, 113(2), 1263-1283.
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