My apologies if this is a naive question but does anyone happen to know if the equation below (and/or the distribution it describes) is commonly know by a specific name?
$f(i) = ( a / i^k ) b^i$
(where $a$, $b$ and $k$ are constants; and, FWIW, for my purposes $i$ should be a positive integer denoting rank according to frequency)
This equation appears in Simon (1955), but he doesn't give it a specific name and always refers to it by its example number in his paper.
This is not my field so I'm not familiar with the relevant literature so I don't know how often this particular distribution crops up and if it is common enough to warrant a specific name (pointers to places where it does appear would also be appreciated though!).
The same formula is also mentioned in Tambovtsev & Martindale (2007), who call it a "Yule distribution" but, as far as I can glean, this seems wrong since it appears "Yule(-Simon)" refers to a different distribution; T&M seem to have mislabelled it based on a careless reading of Simon, who does indeed mention the above equation in the proximity of a discussion of Yule (1924).
Instead, Simon cites Champernowne (1953) for this equation; however, calling it a "Champernowne distribution" doesn't seem appropriate either since, isn't that already the name for a different distribution?
Help, comments and corrections would be gratefully appreciated!
References
Champernowne, D. G. 1953. A Model of Income Distribution. The Economic Journal 63(250): 318–51
Simon, H. A. 1955. On a class of skew distribution functions. Biometrika 42(3–4). 425–40.
Tambovtsev, Y. A. & C. Martindale. 2007. Phoneme frequencies follow a Yule distribution. SKASE Journal of Theoretical Linguistics 4(2). 1–11.
Yule, G. U. 1924. A mathematical theory of evolution, based on the conclusions of Dr J. C. Willis, F.R.S. Philosophical Transactions B 213(21).
As far as I can tell, it doesn't quite have a name, but perhaps the following makes sense? The logarithmic distribution has pmf \begin{align*} p(i) = -\frac{1}{\log(1-p)}\frac{p^i}{i} \end{align*} So this would be a special case when $k = 1$ in your parametrization, and $a$ is then determined to ensure the pmf sums to 1. The reason for this name is because the pmf is derived from the series expansion of the logarithmic function. In your formulation, the pmf would indeed equal \begin{align*} p(i) = \text{Li}_{k}(b) \frac{b^i}{i^k} \end{align*} Hence, if we would to continue the tradition of naming distributions based off series expansions from which they are derived, this could be called the "Polylogarithmic distribution". No results show up when I google this, though.