Assuming no typos Relationship between Pareto, Pareto(II)/Lomax and Generalized Pareto,GPD
import sympy as sy x = sy.Symbol('x') k = sy.Symbol('k') c = sy.Symbol('c') a = sy.Symbol('a') m = sy.Symbol('m') mgpd = sy.Symbol('mgpd')
gpd0 = (1 - c*x/k)**(1/c - 1)/k #JKB notation (c reversed sign) gpd = 1/k/(1 + c*(x-mgpd)/k)**(1/c + 1) #similar to Wikipedia par = a*k**a/(x-m)**(1+a) #JKB lom = a/k/(1+x/k)**(1+a)
lom.subs(k,1) #Pareto(II), Lomax (loc=0, scale=1) a*(1 + x)**(-1 - a) par.subs(k,1).subs(m,-1) #Pareto with loc=-1, scale=1 a*(1 + x)**(-1 - a) gpd.subs(c,1/a).subs(k,1/a).subs(mgpd,0) #GPD with loc=0, scale=1/a a*(1 + x)**(-1 - a)
par.subs(k,1).subs(m,0) # standard Pareto (loc=0, scale=1) a*x**(-1 - a) gpd.subs(c,1/a).subs(k,1/a).subs(mgpd,1) #GPD with loc=1, scale=1/a a*x**(-1 - a)
Josef