inequalitymeasures.jl

using Distributions, KernelDensity, QuadGK

function gini(x)
    n = length(x)
	xx = sort(x)
	2*(sum(collect(1:n).*xx))/(n*sum(xx))-1
end

gini(d::Exponential) = 0.5
gini(d::Beta) = (2/d.α)*beta(d.α+d.β, d.α+d.β)/(beta(d.α,d.α)*beta(d.β, d.β))
gini(d::Gamma) = gamma(d.α+1/2)/(d.α*gamma(d.α)*sqrt(π))
gini(d::LogNormal) = erf(d.σ/2)
gini(d::Pareto) = d.α ≥ 1 ? 1/(2*d.α - 1) : 1.0
gini(d::Uniform) = (d.b-d.a)/(3*(d.b-d.a))

function gini(d::Distribution) 
    ul = 0.1
    while cdf(d, ul)< 0.999
        ul*=2
    end
    mean(d)*QuadGK.quadgk(x->cdf(d, x)*(1-cdf(d, x)), eps(Float64), ul)[1]
end

function atkinson(x::Vector,e::Real)
  at =0.0
  mu = mean(x)
  N = length(x)
  if e<0
    error("e<0")
  elseif e ==1
    at = 1 -(1/mu)*(prod(x))^(1/N)
  else
    at = 1 -(1/mu)*(1/N*sum(x.^(1-e)))^(1/(1-e))
  end
  return at
end

function atkinson(d::Distribution, e)
    ul = 0.1
    while cdf(d, ul)< 0.999
        ul*=2
    end
    m = mean(d)
    1-1/m*QuadGK.quadgk(x->x^(1-e)*pdf(d, x), eps(Float64), ul)[1]^(1/(1-e))
end


theil(x) = 1.0/length(x)*sum((x/mean(x)).*log.(x/mean(x)))
theil(d::LogNormal) = d.σ^2/2

function atkinson(d::Distribution, e)
    ul = 0.1
    while cdf(d, ul)< 0.999
        ul*=2
    end
    m = mean(d)
    QuadGK.quadgk(x->x/m*log(x/m)*pdf(d,x), 0,ul)[1]
end