I have this Maximum Likelihood (ML) problem, which gives after simplification: $${x_{\hat{\eta}}}^T y_{\hat{\eta}} \times {\mathbb{1}}^T \mathbb{1} - {x_{\hat{\eta}}}^T \mathbb{1} \times {y_{\hat{\eta}}}^T \mathbb{1} = 0$$ where $x_{\hat{\eta}}$ and $y_{\hat{\eta}}$ are two vectors that depends only on the scalar $\hat{\eta}$ (the ML estimator of $\eta$) and observed data. These two vectors are not collinear.

I feel like this could give me a closed form or at least a geometric explanation but I'm not sure. Any idea?

For now, I solve $\text{argmin}{f^2(\hat{\eta})}$ (where $f(\hat{\eta})$ is the left-hand side of the first equation) with R function optimize. I feel like this is safer than using a root-finder such as uniroot. Can anyone confirm?

  • $\begingroup$ This is a one dimensional equation. Unless $\eta$ is also of dimension one, this sounds insufficient to identify $\hat\eta$. $\endgroup$ – Xi'an Nov 4 '16 at 9:36
  • $\begingroup$ Yes, $\eta$ is of dimension one (scalar). $\endgroup$ – F. Privé Nov 4 '16 at 9:41
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    $\begingroup$ Your equation looks like an empirical version of $\mathbb{E}[XY]=\mathbb{E}[X]\mathbb{E}[Y]$, hence it picks the $\eta$ that makes $X$ and $Y$ (empirically) uncorrelated. $\endgroup$ – Xi'an Nov 4 '16 at 9:59

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