Trying to find the maximum of a log-likelihood, for a parameter in a covariance function.
I end up with the following problem, that should be concave if my calculations are correct,
\begin{align} &\max_{\kappa}\; -\sum_{i=1}^n \log \left( \lambda_i + \kappa \right) - \sum_{i=1}^n \frac{c_i}{(\lambda_i + \kappa)^2}, \\ &\mbox{subject to }\kappa>0, \end{align}
where $\lambda_i>0,c_i>0$ for all i.
Does there exists a closed form solution?, if not can one find an some bound on the solution?