Considering that θ in Frank copula is a function of the Kendall's tau as follows:
I would like to know the range of Kendall's tau that can be used in the Frank copula.
Thank you in advance for any helps.
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Could any close voter please indicate what's unclear about this? – Glen_b Nov 12 '19 at 22:28
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$\tau = 1+\frac{4}{\theta}[D_1(\theta)-1]$ with $D_1(\theta)= \frac{1}{\theta}\int_0^\theta \frac{t}{e^t-1} dt$,
where is the $D_n(x)$ is the Debye function. This is set as an exercise in Nelsen's book.
As far as I can see, the possible range of population $\tau$ values appears to be $(-1,1)\,\text{\\}\,\{0\}$. That is, between $-1$ and $1$, but not including the endpoints; you get as close to $1$ or $-1$ as you like by choosing sufficiently large parameter values with sign the same as $\tau$, and (strictly speaking) omitting $0$ (though again, you can get as close as you like by choosing the parameter close to $0$; however the independence copula is normally regarded as included).
Glen_b
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Thank you for the response. I would like to know the range of τ as θ ∈ R/0. Can we determine a range for τ explicitly? – Saba Ghotbi Oct 30 '19 at 01:32
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The table at the bottom of this copula page gives Kendall's tau as a function of $\theta$. In general,
... Joe, H. (2014). Dependence Modeling with Copulas. Chapman & Hall / CRC.
contains many formulas and other results on specific copula families.
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Thank you for the response. I would like to know the range of τ as θ ∈ R/0. Can we determine a range for τ explicitly? – Saba Ghotbi Oct 30 '19 at 01:32