# From Pareto II to q

## Note

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 $\frac{dy}{dx} = y^q \ \mathrm{(equation\ 1)}$, y(0)=1, is the q-exponential function (Tsallis 2004:5-6). A constant such as $\lambda$ may be inserted before x and will carry over before x in equations 1 and 2)

$e_q^x \equiv [1+(1-q)x]^{1/(1-q)} \ \mathrm{(equation\ 2)}$ if (1 + (1 - q)x) > 0); otherwise $e_q^x=0$, where $(e_1^x = e^x)$,

The inverse of the q-exponential is the q-logarithmic function (Tsallis 2004:6):

$ln_q\ x \equiv \frac{ x^{1-q} -1}{1-q} \ \mathrm{(Tsallis.6\ equation\ 3)}$ where (ln1 x = ln x)

$ln_q\ (e_q^x) \equiv x$

With an optional constant $\kappa$

$ln_q\ x/\kappa \equiv \frac{ (x/\kappa)^{1-q} -1}{1-q} \ \mathrm{(equation\ 3')}$ where $ln_1 \ x/\kappa = ln \ x/\kappa = ln \ x - ln \ \kappa$

## Probability distribution and Cumulative probability distribution (CDF)

The probability distribution for $e_q^x$ (Tsallis 2004:6-8) normalized with $\int_{0}^{\infin} dx\ p_{e_q^x}(x) = 1$ is

$p_{e_q^x}(x) \equiv (2-q)e_q^{-x} (x \ge 0) = (2-q)[1-(1-q)x]^{1/(1-q)} \ \mathrm{(Tsallis\ equation\ 4)}$

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

$P_{q,\kappa}(X) \equiv \int_{X}^{\infin} dx\ p_{e_q^x}(x) = e_{q_M}^{-X/\kappa}\ \mathrm{(Tsallis.8\ typo\ X/\kappa \ not\ X/q{_M})} = [1-(1-q_M)x/\kappa]^\frac{1}{1-q_M} = [1-(\frac{1-q}{2-q})x/\kappa]^\frac{2-q}{1-q} \ \mathrm{(equation\ 5)}$, with
$P_{q,\kappa}(X) = e_{q_M}^{-X/\kappa} = [1-(1-q_M)x/\kappa]^\frac{1}{1-q_M} = [1-(\frac{1-q}{2-q})x/\kappa]^\frac{2-q}{1-q} \ \mathrm{(equation\ 5)}$, with
$q_M \equiv 1/(2-q)$, where for $q \ge 2\ \ p(x)$ is not normalizable. Thus, q cannot be fitted for a range at or above 2. (Note that $\kappa$ 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 $\kappa$ directly, as follows (the two MLE approaches will be compared).

As related to Tsallis's exposition, above, Shalizi took as his cumulative distribution

$P_{q,\kappa}(X) \equiv e_{q_M}^{-X/\kappa} = [1 - (1-q)x/\kappa]^\frac{1}{(1-q)}, \ \mathrm{(Tsallis.8\ equation\ A\ =\ 1\ in\ Shalizi)}$,

Substituting to obtain Pareto II parameters, $, \theta, \sigma$, we may take his equations (8) and (10) for the MLE estimation of $\hat{\theta}, \hat{\sigma}$ in terms of $\hat{q}, \hat{\kappa}$

$q_M \equiv \frac{1}{(2-q)} = \frac{\theta+1}{\theta}\ \mathrm{hence} \ q_M = \frac{\theta+2}{\theta+1} \ne \frac{\hat{\theta}+1}{\hat{\theta}}\ \mathrm{as\ in\ Shalizi}\ \mathrm{and \ thus}$

$\hat{q} =\hat{q_M} = \frac{\hat{\theta}+2}{\hat{\theta}+1}$ is the q for Tsallis equation 4, while Shalizi's is the $q_M$ for Tsallis equation 5.

$\hat{\kappa} = \hat{\sigma}/\hat{\theta}$

It should be the case that the cumulative q-exponential asymptotes in the tail to a power-law slope of $1/(1-q_M)$.

## 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”

$Z_q(x) \equiv ln_q \ P_{q}(x) \approx \frac{[P_{q}(x)]^{1-q}-1}{1-q} = \frac{[(1-\frac{1-q}{2-q}x)^\frac{2-q}{1-q}]^{1-q}-1}{1-q} = \frac{q_M[[1-(1-q_M)x]^\frac{1}{1-q_M} -1]}{1-q_M}< 0$ if q > 1. Note that kappa does not enter this equation!

"This is done for a series of values of q. The function $Z_q(x)$ which can best be fit with a staight line determines the value of q, the slope being $\kappa$.

Figure from Thurner and Tsallis 2005:200 click for reference

## Deriving a direct MLE

Hence with slope $\kappa, x\kappa \approx \frac{[P_{q}(x)]^{1-q}-1}{(1-q)} = \frac{q_M[[1-(1-q_M)x]^\frac{1}{1-q_M} -1]}{1-q_M} > 0$ if q > 1.

We solve this by numerical methods (starting with Excel solver, working to Matlab)

## References

Chakravarti, I., R. Laha, and J. Roy, (1967). Kolmogorov-Smirnov (KS) test. Handbook of Methods of Applied Statistics, Volume I, John Wiley and Sons, pp. 392-394.

Clauset, Aaron, Cosma Rohilla Shalizi, and M. E. J. Newman. 2007. Power-law distributions in empirical data. http://arxiv.org/abs/0706.1062. R and Matlab software and documentation for the 2007 review article (29 June 2007). http://www.santafe.edu/~aaronc/powerlaws/

Harris, C. M. 1968. The Pareto Distribution As A Queue Service Discipline. Operations Research 16:307-313.

McGuire, B. A., E. S. Pearson and A. H. A. Wynn. 1952. The time intervals between industrial accidents. Biometrika 39(168):

Meehl, Paul. 1967 Theory testing in Psychology and Physics: A Methodological Paradox. Philosophy of Science 34(2)103-115. Reprinted 1970 (2006) in The Significance Test Controversy. Ed., Denton E. Morrison and Ramon E. Henkel. Chicago: Aldine.

Shalizi, Cosma. 2007 Maximum Likelihood Estimation for q-Exponential (Tsallis) Distributions. http://www.cscs.umich.edu/~crshalizi/research/tsallis-MLE

Silcock, H. 1954. The phenomenon of labour turnover. Journal of the Royal Statistical Society A 117: 429-440.]

Thurner, Stefan, Fragiskos Kyriakopoulos, Constantino Tsallis. 2007. Unified model for network dynamics exhibiting nonextensive statistics. Phys. Rev. E 76, 036111 (8 pages) published as [http://scitation.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=PLEEE8000076000003036111000001&idtype=cvips&gifs=yes

Thurner, Stefan, and Constantino Tsallis. 2005. Nonextensive aspects of self-organized scale-free gas-like networks. Europhys. Lett. 72 197-203 doi: 10.1209/epl/i2005-10221-1 http://www.iop.org/EJ/abstract/0295-5075/72/2/197/

Tsallis, Constantino. 1988. Possible generalization of Boltzmann-Gibbs statistics. J. Stat. Phys. 52:479-487.

Tsallis, Constantino. 2004. Nonextensive Statistical Mechanics: Construction and Physical Interpretation. Chapter 1, in Murray Gell-Mann and Constantino Tsallis, eds., Nonextensive Entropy: Interdisciplinary Applications. Santa Fe Institute Studies in the Sciences of Complexity. Oxford: Oxford University Press. pp. 1-53.

## Auxiliary references

http://arxiv.org/PS_cache/cond-mat/pdf/0401/0401140v2.pdfFinancial markets (cumulative same as Shalizi) http://tinyurl.com/yvsk92

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