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A discrete distribution with mass function

$$p(x;k) = \frac{k}{(x+k)(x+k-1)},\quad x = 1,2,\ldots$$

arises on page 9 of this paper.

For $k=1$ it is a Yule-Simon distribution with $\rho=1$, but I haven't found any other examples.

Does it have a name? Does it appear in any other contexts? Is there a simple stochastic process that might generate it?

Simon Byrne
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2 Answers2

13

It's a discrete power law.

(This is a description--whose meaning will be made precise below--rather than a technical term. The phrase "discrete power law" has a slightly different technical meaning, as indicated by @Cardinal in comments to this answer.)

To see this, observe that the partial fraction decomposition can be written

$$p(x;k) = \frac{k}{(x+k)(x+k-1)} = \frac{1}{1 + (x-1)/k} - \frac{1}{1 + x/k}.$$

The CDF telescopes into a closed form:

$$\eqalign{ &\text{CDF}(i) = \sum_{x=1}^i p(x;k) \\ = &[\frac{1}{1 + 0/k} - \frac{1}{1 + 1/k}] + [\frac{1}{1 + 1/k} - \frac{1}{1 + 2/k}] + \cdots + [\frac{1}{1 + (i-1)/k} - \frac{1}{1 + i/k}] \\ = &\frac{1}{1 + 0/k} + [- \frac{1}{1 + 1/k} + \frac{1}{1 + 1/k}] + [ - \frac{1}{1 + 2/k} + \cdots + \frac{1}{1 + (i-1)/k}] - \frac{1}{1 + i/k} \\ = &1 + 0 + \cdots + 0 - \frac{1}{1 + i/k} \\ = &\frac{i}{i+k}. }$$

(Incidentally, because this is easily inverted, it immediately provides an efficient way to generate random variables from this distribution: simply compute $\lceil \frac{k u}{1 - u} \rceil$ where $u$ is uniformly distributed on $(0,1)$.)

Differentiating this expression with respect to $i$ shows how the CDF can be written as an integral,

$$\text{CDF}(i) = \frac{i}{i+k} = \int_0^i \frac{dt/k}{(1 + t/k)^2} = \sum_{x=1}^i \int_{x-1}^x \frac{dt/k}{(1 + t/k)^2},$$

whence

$$p(x;k) = \int_{x-1}^x \frac{dt/k}{(1 + t/k)^2}.$$

This form of writing it exhibits $k$ as a scale parameter for the family of (continuous) distributions determined by the density

$$f(\xi)d\xi = (1 + \xi)^{-2}\, d\xi$$

and shows how $p(x;k)$ is the discretized version of $f$ (scaled by $k$) obtained by integrating the continuous probability over the interval from $x-1$ to $x$. That's obviously a power law with exponent $-2$. This observation gives you an entrance into extensive literature on power laws and how they arise in science, engineering, and statistics, which may suggest many answers to your last two questions.

whuber
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  • (+1) From the probability mass function, it's clear that $p(x; k) \sim k x^{-2}$ as $x \to \infty$, which seems to be enough to conclude that it's a power-law distribution. In fact, $p(x;k) x^2 / k \uparrow 1$ as $x \to \infty$. – cardinal Aug 26 '11 at 00:29
  • @cardinal You're right, but there's a limitation to this argument: it only shows that $p$ is asymptotically a power law. The calculations show that it's exactly a discretized version of a power law. – whuber Aug 26 '11 at 15:47
  • I'm not quite sure about the distinction you're trying to draw. Unfortunately, I haven't gotten the chance to think about it carefully, but it appears you are defining a discrete power law distribution as one that is a discretized version of a continuous power law distribution. Am I interpreting your comment correctly? At any rate, when I see reference to discrete power laws in the literature, the usual definition seems to be the weaker (i.e., asymptotic) one I've used. (cont.) – cardinal Aug 26 '11 at 19:01
  • (Cont.) On the other hand, a Zipf distribution would seem to be as pure of a discrete power law as possible, yet I do not believe it can be generated as a discretization of a continuous power law. Have I misinterpreted your intent? (By the way, your development above is quite nice. The recognition of the the telescoping sum for the cdf is great, as is the recognition of an easy sampling scheme.) – cardinal Aug 26 '11 at 19:02
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Okay, after a bit more investigation, I found some more details.

It's a special case of a continuous mixture of a geometric distribution with a Beta, so could be called a Beta-geometric distribution. Specifically, if: $$P \sim \mathrm{Beta}(1,k) $$ and: $$X|P \sim \mathrm{Geometric}(P)$$ then the marginal distribution of $Y = X+1$ has this distribution. As such, it's a special case of a Beta-Negative binomial distribution.

It has a couple of other interesting properties:

  • It has an infinite mean
  • It describes its own tail distribution: if $X$ has this distribution with parameter $k$, then $X-t | X>t$ has parameter $t+k$.
Simon Byrne
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