add yule_simon_lcdf - #3276
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SteveBronder
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lgtm!
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Summary
With this PR the lcdf of Yule-Simin distribution are added.
See issue #3237
Expressions involved:
CDF:
$$ \begin{align} & P(Y \leq y) = F(y,\alpha) = 1-\frac{ \Gamma(\alpha+1) \Gamma(y+1) }{\Gamma(\alpha+y+1)}, \quad y=1,2,..., \alpha>0 \end{align} $$
The partial derivatives w.r.t log cdf:
$$ \begin{align} \frac{\partial \log F(y,\alpha)}{\partial \alpha} &= - \frac{C}{1-C} \cdot \frac{\mathrm{d} \log C}{\mathrm{d} \alpha} = - \frac{C}{1-C} \cdot [\psi(\alpha+1)-\psi(y+\alpha+1)] \end{align} $$
where $C$ is the ccdf function, $\psi$ is the digamma function.
Tests
Test is written follow the guide.
Side Effects
No.
Release notes
yule_simon_lcdf is available if merged.
Checklist
Copyright holder: Zhi Ling
The copyright holder is typically you or your assignee, such as a university or company. By submitting this pull request, the copyright holder is agreeing to the license the submitted work under the following licenses:
- Code: BSD 3-clause (https://opensource.org/licenses/BSD-3-Clause)
- Documentation: CC-BY 4.0 (https://creativecommons.org/licenses/by/4.0/)
the basic tests are passing
the code is written in idiomatic C++ and changes are documented in the doxygen
the new changes are tested