By little research , reasonable fit was obtained by Gumbel max distribution for ln(mod(1/x), Gumbel min for ln(mod(x) , also Frechet, but the value of parameters in Frechet is around 2*10^8.
Extreme Value type I distribution-Gumbel
Gumbel max and min , however , behave rather reasonably. They seem to be in the region indincated by Skewness/curtosis analysis, while Frechet is just close. See picture.
Of course, with current accuracy these numbers may be false, but interestingly, involving Gumbel distributions as first approximation leads to establishing connection between Euler-Macheroni Constant 0,5772156649 and \( \Omega \) constant=0,567143 since in Gumbel distribution:
\( Mean = \mu+\beta*0,5772156649 = \Omega \)
According to my limited accuracy possibilites, for Gumbel max (corresponding to ln(mod(1/x)) iterations))
\( \mu = 0,023562377 \)
\( \beta= 0,941738056 \)
This distribution has interesting variance which only depends on \( \beta \):
\( Variance= {(\pi^2/6)}*\beta= 1,64493*0,9417380= 1.458.... \)
and median
\( Median = \mu-\beta*\ln(\ln(2)) = 0.368721542 \)
Points of inflection:
\( X = \mu+-\beta*\ln(0.5*(3+5^{1/2}))= \mu+-0,96242*\beta=+0.877.. - 0.83.. \)
And fixed Skewness= 1.1395.. and Curtosis =3+ 12/5 = 5.4
The last 2 I can not match exactly from my data , I am few percent off, but I guess since it is continuous distribution more accurate calculations with much more points are needed to decide.
The other statistical parameters of this interesting distribution involve Gamma and digamma functions, while \( {\pi^2/6} \) obviosuly implies Zeta(2).
I find this exercise interesting, definitely worth iterating. I just wonder what continuous /negative iteration of it might mean.
Ivars
Extreme Value type I distribution-Gumbel
Gumbel max and min , however , behave rather reasonably. They seem to be in the region indincated by Skewness/curtosis analysis, while Frechet is just close. See picture.
Of course, with current accuracy these numbers may be false, but interestingly, involving Gumbel distributions as first approximation leads to establishing connection between Euler-Macheroni Constant 0,5772156649 and \( \Omega \) constant=0,567143 since in Gumbel distribution:
\( Mean = \mu+\beta*0,5772156649 = \Omega \)
According to my limited accuracy possibilites, for Gumbel max (corresponding to ln(mod(1/x)) iterations))
\( \mu = 0,023562377 \)
\( \beta= 0,941738056 \)
This distribution has interesting variance which only depends on \( \beta \):
\( Variance= {(\pi^2/6)}*\beta= 1,64493*0,9417380= 1.458.... \)
and median
\( Median = \mu-\beta*\ln(\ln(2)) = 0.368721542 \)
Points of inflection:
\( X = \mu+-\beta*\ln(0.5*(3+5^{1/2}))= \mu+-0,96242*\beta=+0.877.. - 0.83.. \)
And fixed Skewness= 1.1395.. and Curtosis =3+ 12/5 = 5.4
The last 2 I can not match exactly from my data , I am few percent off, but I guess since it is continuous distribution more accurate calculations with much more points are needed to decide.
The other statistical parameters of this interesting distribution involve Gamma and digamma functions, while \( {\pi^2/6} \) obviosuly implies Zeta(2).
I find this exercise interesting, definitely worth iterating. I just wonder what continuous /negative iteration of it might mean.
Ivars