Maximum Likelihood of Wishart parameters

I'm having some difficulty in deriving the ML estimation of the parameters of a Wishart distribution.

Given a set of matrices $\{W_1, W_2,\dots,W_N\} \in \mathbb{C}^{k\times k}$ for which $W_i \sim Wishart_C(\nu,\Sigma)$ is assumed (and $\nu > k$), I want to estimate $\Sigma$ and $\nu$.

Defining the log-likelihood gives $$\mathcal{L} = C - \sum_{j=1}^k\ln{\Gamma(\nu - j +1)}+ \nu \ln\det(\Sigma^{-1})\\ \qquad+(\nu - k)\sum_i \ln\det(W_i) -\sum_i\operatorname{tr}(\Sigma^{-1}W_i),$$ where $C$ is a constant normalizer. Finding the MLE for $\Sigma$ is fairly straightforward (although it depends on $\nu$), but I'm having difficulty with the degree of freedom $\nu$.

Taking the derivative with respect to $\nu$, $$\frac{\partial \mathcal{L}}{\partial \nu} = \ln\det(\Sigma^{-1}) + \sum_i\ln\det(W_i) -\sum_{j=1}^k \frac{\partial}{\partial \nu}\ln \Gamma(\nu-j+1) \overset{set}{=} 0.$$ This is where I have been stumped. I realize that $$\frac{\partial}{\partial \nu}\ln \Gamma(\nu-j+1) = \psi(\nu-j+1),$$ where $\psi(x)$ is the digamma function, but this still doesn't present a tractable method to solve for $\nu$.

Are there approximations or linearizations (or tricks that I haven't found) that I could use?

• It's a one-dimensional root finding problem with a continuous, monotonic function in the parameter of interest ($\nu$) and as such is easily solved using any number of numeric techniques. – jbowman Oct 28 '17 at 16:18
• I was wondering if that was the way to go, but hoping that there was something analytical that I had missed. – aepound Oct 28 '17 at 16:22
• Yeah, I know how that is. Some distributions just aren't helpful in that regard :) – jbowman Oct 28 '17 at 16:24