I have a constrained probit likelihood and I would like to know if it is globally concave.
My problem can be simplified as follows. Let $n\in\mathbb{N}^{\ast}$. Let $F$ be a function defined on $\mathbb{R}_+^{r_0 + 1}\times\mathbb{R}$ by $$F(\delta_1, \delta_2, \dots, \delta_{r_0}, \delta, \rho) =\sum_{r = 0}^{\infty} d_{r}\log(\Phi(a_r) - \Phi(a_{r + 1})),$$
where $d_r \in \{0,~1\}$, $a_r = \sum_{k = 1}^r \delta_r$, $\delta_r = (r - 1)^{\rho}\delta$ if $r > r_0$, and $\Phi$ is the distribution function of the standard normal distribution.
It is clear that $F < \infty$. I would like to know if it is globally concave in $(\delta_1, \delta_2, \dots, \delta_{r_0}, \delta, \rho)$.