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I have a statistical model whose parameters I would like to find a closed form maximum likelihood estimate for if possible. There are two parameters and I can solve for one, but the second is a bit messier. The first results in:

$$\lambda = \sum_{i=1}^{n}x_i^k$$

For the second, $k$, I'm stuck at:

$$\frac{1}{k}+\sum_{i=1}^{n}\left(1-\left(\frac{x_i}{\lambda}\right)^k\right)\ln\left(\frac{x_i}{\lambda}\right) = 0$$

I need to solve this system of equations for $k$ and $\lambda$, but I'm at a loss for how to proceed. Any ideas? I would prefer something exact, but I'm open to an approximation that holds exactly as $x_i\rightarrow\infty$ if there isn't an exact solution.

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I think your question can be better answered at Cross Validated. If you'd like I can migrate it over there for you. –  Willie Wong Jul 20 '12 at 7:34
Would you be satisfied with a computational method to approximate the answer to any give tolerance, or does it have to be an analytic formula? –  Nick Alger Jul 20 '12 at 7:58

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