Weighted Atkinson optimized

wtd_atkinson.jl

##### Atkinson index #####
"""
    wtd_atkinson(v, w, ϵ)
Computes the weighted Atkinson index given an inequality adversion parameter ϵ.
"""
function wtd_atkinson(v::Array{<:Real,1}, w::Array{<:Real,1}, ϵ::Real)::Float64
 
    norm_mean!(v)
    w = w/sum(w)
 
    if ϵ == 1
        w = w[v .!= 0]
        v = v[v .!= 0]
        
        return wtd_atk_eps_1_fast(v, w)
    elseif ϵ < 1
        return wtd_atk_eps_diff_1_fast(v, w, ϵ)
    else 
        w = w[v .!= 0]
        v = v[v .!= 0]
        
        return wtd_atk_eps_diff_1_fast(v, w, ϵ)
    end
end 


@inline wtd_atk_eps_1_fast(v, w)::Float64 = @fastmath 1 - (prod(exp.(w.*log.(v)))/sum(v .* w/sum(w)) )
@inline wtd_atk_eps_diff_1_fast(v, w, ϵ)::Float64 = @fastmath 1-(sum(((v/sum(v.*w/sum(w))).^(1-ϵ)).*w/sum(w))).^(1/(1-ϵ))

@inline function norm_mean!(x::Array{<:Real,1})::Vector{Float64} 
    x = convert(Vector{Float64}, x)
    μx = Statistics.mean(x)
    for i in 1:length(x)
        x[i] = x[i]/μx
    end
    x    
end