From Pareto II to q
New derivations corrected a type in Tsallis (2004) page 8 (see below). These gave consistence between Tsallis's derivations and Cosma Shalizi's MLE derivation. It also simplifies the whole exposition, moreso than represented here.
q exponential and q logarithm
The solution to , y(0)=1, is the q-exponential function (Tsallis 2004:5-6). A constant such as may be inserted before x and will carry over before x in equations 1 and 2)
if (1 + (1 - q)x) > 0); otherwise , where ,
The inverse of the q-exponential is the q-logarithmic function (Tsallis 2004:6):
where (ln1 x = ln x)
With an optional constant
Probability distribution and Cumulative probability distribution (CDF)
The probability distribution for (Tsallis 2004:6-8) normalized with is
The cumulative probability distribution for the q-exponential function in (1) (Tsallis 2004:8 -- for the typo in Tsallis see Ernesto Borges thesis 8.21!!!) is also a q-exponential function
- , with
- , with
- , where for is not normalizable. Thus, q cannot be fitted for a range at or above 2. (Note that does not enter into this equation).
The Shalizi MLE of q-exponential by Pareto II
See Cosma Shalizi, 2007 Maximum Likelihood Estimation for q-Exponential (Tsallis) Distributions. http://www.cscs.umich.edu/~crshalizi/research/tsallis-MLE These are classically known as Pareto II distributions. (Examples). Rather than using Shalizi's method, however, we procede to estimate q and directly, as follows (the two MLE approaches will be compared).
As related to Tsallis's exposition, above, Shalizi took as his cumulative distribution
Substituting to obtain Pareto II parameters, , we may take his equations (8) and (10) for the MLE estimation of in terms of
is the q for Tsallis equation 4, while Shalizi's is the for Tsallis equation 5.
It should be the case that the cumulative q-exponential asymptotes in the tail to a power-law slope of .
A direct two-parameter solution
Thurner, Kyriakopoulos, and Tsallis (2007:5) note that “A convenient procedure to perform a two-parameter fit is to take the q-logarithm of the distribution P, defined as”
if q > 1. Note that kappa does not enter this equation!
"This is done for a series of values of q. The function which can best be fit with a staight line determines the value of q, the slope being .
Deriving a direct MLE
Hence with slope if q > 1.
We solve this by numerical methods (starting with Excel solver, working to Matlab)
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http://prola.aps.org/pdf/PRE/v67/i1/e016106 computing the escort distribution -Itineration of the Internet over nonequilibrium stationary states in Tsallis statistics. Sumiyoshi Abe and Norikazu Suzuki
S. Abe and A. K. Rajagopal, J. Phys. A 33, 8733 ~2000. escort distribution