# How many observations to estimate a parameter of an Archimedean copula?

Let's consider for example the bivariate Gumbel copula. $$C(u_1, u_2)=\exp \left[-\left(\left(-\ln\left(u_1\right)\right)^{\theta}+\left(-\ln\left(u_2\right)\right)^{\theta}\right)^{\frac{1}{\theta}}\right].$$ In R there are some functions (such as fitCopula) which estimate the parameter $\theta$. Do you maybe have a reference which suggests how many observations $(u_1, u_2)$ I should consider to reliably estimate $\theta$? And is this number maybe influenced by the number of time series (I have a Gumbel copula with 9 variables)?

• What would constitute "reliable" for your purposes? – Glen_b May 29 '16 at 11:49
• Sorry if I was being vague, I meant in terms of both unbiasedness and consistency... – Kondo May 29 '16 at 12:16
• Consistency is a property in the limit, i.e. as $n\to\infty$ and "unbiasedness" is either true or it isn't -- unless you have a very unusual estimator it will hold for all finite $n$ or none of them - it's not going to help find $n$. I was imagining a specific numerical criterion, e.g. one relating to the s.e. or a C.I. width or maybe MSE/RMSE – Glen_b May 29 '16 at 21:57
• Oh ok, I now get what you mean. Well I wouldn't know whether one of these criteria is superior to the other, maybe RMSE/MSE? – Kondo May 29 '16 at 22:35
• I think you'll probably need to use simulation to relate RMSE to sample size (I'm not aware of algebraic results, but perhaps there are some); from that you should be able to get approximate sample size, but someone more knowledge might be able to offer something better than that suggestion. – Glen_b May 29 '16 at 22:52