Recently in an article, I stumble upon what seems to be a sum of multivariate heaviside functions: \begin{equation} \Gamma(\theta)=\frac{1}{k}\sum_{i=1}^nH(\theta_i\leq\theta,\rho_i\geq\rho_{z:n}) \end{equation} where $\rho_{z:n}$ is the z-th order statistic. On the same article, they proceed to evaluate the integral over d$\Gamma(\theta)$. What is the expression for d$\Gamma(\theta)$?
I suppose it should involve $\delta(\theta-\theta_i)$ but I haven't arrived any further.
I think I have figured out the answer. Given that it must apply that: \begin{equation} H(x,y)=H(x)H(y) \end{equation} The result is simply: \begin{equation} \text{d}\Gamma(\theta)=\frac{1}{k}\sum_{i=1}^n\delta(\theta-\theta_i)H(\rho_i-\rho_{z:n})\text{d}\theta \end{equation}