statTn: The Tn Statistic of a Fitted Copula to an Empirical Copula

Description

Compute the $T_n(p)$ statistic of Genest et al. (2011) that is defined as $$T_n(p) = \sum_{i=1}^n \big|\mathbf{C}_n(u_i, v_i) - \mathbf{C}_{\Theta_n}(u_i, v_i)\big|^p\mbox{,}$$ where $\mathbf{C}_n(u,v)$ is the empirical copula, $\mathbf{C}_{\Theta_n}(u,v)$ is the fitted copula with estimated parameters $\Theta_n$ from the sample of size $n$. The $T_n$ for $p = 2$ is reported by those authors to be of general purpose and overall performance in large scale simulation studies. The extension here for arbitary exponent $p$ is made for flexibility. Alternatively the definition could be associated with the statistic $T_n(p)^{1/p}$ in terms of a root $1/p$ of the summation as shown above.

The $T_n$ statistic is obviously a form of deviation between the empirical (nonparametric) and parametric fitted copula. The distribution of this statistic through Monte Carlo simulation could be used for inference. The inference is based on that a chosen parametric model is suitably close to the empirical copula. The $T_n(p)$ statistic has an advantage of being relatively straightforward to understand and explain to stakeholders and decision makers, is attractive for being suitable in a wide variety of circumstances, but intuitively might have limited statistical power in some situations for it looks at whole copula structure and not say at tail dependency. Finally, other goodness-of-fits using the squared differences between $\mathbf{C}_n(u,v)$ and $\mathbf{C}_{\Theta_m}(u, v)$ are aicCOP, bicCOP, and rmseCOP.

Usage

statTn(u, v=NULL, cop=NULL, para=NULL, p=2, proot=FALSE, ...)

Value

The value for $T_n$ is returned dependent on the specification of $p$ and whether rooting of the result is desired.

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;
p: The value for $p$, and the default follows that of Genest et al. (2011);
proot: A logical controling whether the $T_n$ returned be rooted by $1/p$, and the default follows that of Genest et al. (2011); and
...: Additional arguments to pass to the copula function and (or) the empirical copula.

Author

W.H. Asquith

References