Two-parameter Poisson-Dirichlet distribution of innovation arrival processes
Following the literature on endogenous growth, assume that the economy grows by experiencing innovations over time. Suppose that there are two kinds of innovation. An innovation, when it occurs, either raises productivity and hence output of one of the existing clusters (sectors) or creates an entirely new cluster (of initial size 1).
After the arrival of the nth innovation, there will be a total of Kn clusters (sectors) of sizes ni , i = 1, ... ,k, say, n = щ + n2 + ••• + nk , that is, the total number of clusters is Kn = k.
In this two-parameter Poisson-Dirichlet process, we have either
because an innovation occurs in one of these existing clusters with probability Pi, or
These give a probability rate of the ( n +1) th innovation attaching to one of the existing clusters or creating a new cluster.
There are two parameters, a and 0. Parameter в is the same as that used by Ewens (1972) and used Hoppe (1984) to cast the model as an urn model. The parameter a was introduced by Pitman (2002) when he extended the Ewens model. It controls the rate at which new types of so far unobserved innovation arrive.
Yamato and Sibuya (2000) derived the expression for E(Kn )
Solving this recursion equation, they have also obtained the closed form expression for E(Kn ).
Using Stirling's formula we obtain an approximate expression
Yamato and Sibuya (2000) have also calculated the coefficient of variation of Kn normalized by na to be
It is also known that
where L is called the Mittag-Leffler random variable, to which we return later.
Carlton (1999) has proved that an estimate of a based on n pieces of data converges almost surely to it
when в is known but a is not. Also see Hoppe (1984), Pitman (2002), and Yamato and Sibuya (2000).
Note that Kn/n a is not self-averaging, for 0< a < 1 , but is selfaveraging with a = 0.