The range of Kendall tau for the Frank copula

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.

$$\tau = 1+\frac{4}{\theta}[D_1(\theta)-1]$$ 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).

• 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 at 1:32
• Sorry, I made an edit. – Glen_b -Reinstate Monica Nov 1 at 9:42

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.

• 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 at 1:32