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I'd like to calculate the derivative $\frac{\partial u}{\partial \beta}$, where

$$ u = [y - g^{-1}(X^T\beta)]^T \Sigma^{-1}[y - g^{-1}(X^T\beta)], $$

$y = (y_1, \ldots, y_n)^T$ and $\beta = (\beta_1, \ldots, \beta_p)^T$ are vectors and $x_{ji} = (x_{j1}, \ldots, x_{jn})$.

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Hints: (use chain rule, the following facts may help)

  • For symmetric matrix $S$, we have $\nabla_\beta (\beta^\top S \beta) = 2 S \beta$.
  • For a matrix $A$, we have $\nabla_\beta (A \beta) = A$.
  • You will have to take the derivative of $g^{-1}$ at some point.
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  • $\begingroup$ $\Sigma$ is symmetric. But I don't know how to apply the chain rule in this case. Without the $g$ function I know. $\endgroup$ Oct 30, 2018 at 23:23

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