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I'm trying to obtain the maximum likelihood estimate of the parameters for a model I'm building. I have constants $\sigma$, $\mu$, and $q_0$; a boolean matrix $\alpha$; and vectors $A, \beta, r, d,$ and $Q$, all quantities non-negative. I set the derivative of the log likelihood function equal to zero with

$$0=\sum_j \alpha_{kj} \Bigg[ \beta_j r_j \frac{\ln (A_j - r_j\sum_i \alpha_{ij} Q_i) + \sigma^2 - \mu}{\sigma^2 (A_j - r_j\sum_i \alpha_{ij} Q_i)} + (1-\beta_j)\frac{\ln(1-d_j)}{q_0} \Bigg]$$

How can I solve for $Q$?

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    $\begingroup$ Even in the case where $Q$ is a scalar you will need a numerical solver because the basic form of the equation is $\log(Q)=aQ+b$ for constants $a$ and $b$. Although conceivably a closed formula could be developed in terms of the Lambert W function it likely would not be of much help. $\endgroup$
    – whuber ♦
    Commented Mar 8, 2014 at 22:17
  • $\begingroup$ Using a numerical solver is fine by me -- do you know how I can set up the expression though? (Is it a linear program? Can it be stated as an optimization of a vector given constraints?) $\endgroup$
    – rhombidodecahedron
    Commented Mar 8, 2014 at 22:46

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