aicCOP: Akaike Information Criterion between a Fitted Coupla and an Empirical Copula

Description

Compute the Akaike information criterion (AIC) $\mathrm{AIC}_\mathbf{C}$ (Chen and Guo, 2019, p. 29), which is computed using mean square error $\mathrm{MSE}_\mathbf{C}$ as

$$\mathrm{MSE}_\mathbf{C} = \frac{1}{n}\sum_{i=1}^n \bigl(\mathbf{C}_n(u_i,v_i) - \mathbf{C}_{\Theta_m}(u_i, v_i)\bigr)^2\mbox{ and}$$ $$\mathrm{AIC}_\mathbf{C} = 2m + n\log(\mathrm{MSE}_\mathbf{C})\mbox{,}$$

where $\mathbf{C}_n(u_i,v_i)$ is the empirical copula (empirical joint probability) for the $i$th observation, $\mathbf{C}_{\Theta_m}(u_i, v_i)$ is the fitted copula having $m$ parameters in $\Theta$. The $\mathbf{C}_n(u_i,v_i)$ comes from EMPIRcop. The $\mathrm{AIC}_\mathbf{C}$ is in effect saying that the best copula will have its joint probabilities plotting on a 1:1 line with the empirical joint probabilities, which is an $\mathrm{AIC}_\mathbf{C} = -\infty$. From the $\mathrm{MSE}_\mathbf{C}$ shown above, the root mean square error rmseCOP and Bayesian information criterion (BIC) bicCOP can be computed. These goodness-of-fits can assist in deciding one copula favorability over another, and another goodness-of-fit using the absolute differences between $\mathbf{C}_n(u,v)$ and $\mathbf{C}_{\Theta_m}(u, v)$ is found under statTn.

Usage

aicCOP(u, v=NULL, cop=NULL, para=NULL, m=NA, ...)

Value

The value for $\mathrm{AIC}_\mathbf{C}$ is returned.

Arguments

u: Nonexceedance probability $u$ in the $X$ direction;
v: Nonexceedance probability $v$ in the $Y$ direction; If not given, then a second column from argument u is attempted;
cop: A copula function;
para: Vector of parameters or other data structure, if needed, to pass to the copula;
m: The number of parameters in the copula, which is usually determined by length of para if m=NA, but some complex compositions of copulas are difficult to authoritatively probe for total parameter lengths and mixing coefficients; and
...: Additional arguments to pass to either copula (likely most commonly to the empirical copula).

Author

W.H. Asquith

References