Properties of frequentist confidence levels derivatives

In high energy physics, results from searches for new particles or rare processes are often reported using a modified frequentist approach, known as $\rm{CL_s}$ method. In this paper, we study the properties of the derivatives of $\rm{CL_s}$ and $\rm{CL_{s+b}}$ as signal strength estimators if the confidence levels are interpreted as credible intervals. Our approach allows obtaining best fit points and $\chi^2$ functions which can be used for phenomenology studies. In addition, this approach can be used to incorporate $\rm{CL_s}$ results into Bayesian combinations.