# maximize $-\sum_{i=1}^n \log \left( \lambda_i + \kappa \right) - \sum_{i=1}^n \frac{c_i}{\lambda_i + \kappa}$

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?

-

The derivative of this function is positive for all $\kappa>0$, so the maximum is taken for whatever maximum you have for your range. Given that you have no upper bound for $\kappa$, this is $+\infty$ (although that isn't a "maximum" so much as a supremum.)