Created with Wolfram Mathematica 9.0
Stable 2:3 Order Statistic Distribution
The updating of the graphs after moving a slider may take several seconds. There are no explicit formulas for the functions to compute the graphs; three numerically integrated functions are multiplied together.
This is the distribution generated by selecting the median of three stable random variables. It has a tail exponent, α, which has a range (0, 4], but when it is 4 it is no longer a heavy tailed distribution but is generated from a normal distribution. α is twice that of the parent stable distribution, and because of that it conveniently covers the tail behavior found in stock market log returns.
β, [-1, 1], is a skewness parameter like the stable distribution the effect diminishes as α → 4, where it disappears.
γ is the positive scale factor, in the module it can range from 0.1 to 3, but its full domain is positive real numbers. For daily stock market log returns it is in the neighborhood of 0.01.
δ is the location parameter, which has the domain of real numbers, in the graphic it has a range [-5, 5].
© Copyright 2013 Robert H. Rimmer, Jr. Thu 28 Nov 2013