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(uv, cop=NULL, para=NULL, p=2, proot=FALSE, ...)

Arguments

An R two column matrix or data.frame of a sample of nonexceedance probabilities $u$ and $v$;

cop

A copula function;

para

Vector of parameters or other data structure, if needed, to pass to the copula;

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.

Value

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

References

Genest, C., Kojadinovic, I., Ne<U+0161>lehov<U+00E1>, J., and Yan, J., 2011, A goodness-of-fit test for bivariate extreme-value copulas: Bernoulli, v. 17, no. 1, pp. 253--275.

Examples

Run this code

# NOT RUN {
# Example here is just for Tn. For the example below, the PSP copula is quite different
# from the Gumbel-Hougaard copula and thus, the hatTn would be expected to be different
# from those of the Gumbel-Hougaard and certainly not too near to zero.
sampleUV  <- simCOP(n=60, cop=PSP, graphics=FALSE)   # random sample
hatTau <- cor(samUV$U, samUV$V, method="kendall") # Kendall Tau
hatTn  <- statTn(sampleUV, cop=GHcop, para=GHcop(tau=hatTau)$para,
                 ctype="bernstein", bernprogress=TRUE)
# approximate range 0.0141 -- 0.030
# hatTn in this case is by itself is somewhat uninformative and requires
# Monte Carlo to put an individual value into context.
# }

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