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So I am reading "In All Likelihood" by Pawitan. I love the book, but I am not sure what happened to one of the derivations. It is probably something simple, but I can't figure it out.

Some Definitions

$logL(\theta)$ is the log likelihood

$S(\theta)$ is the score function which is the first derivative of the log Likelihood

$I(\theta)$ is the negative second derivative of the log Likelihood.

The problem:

Using a second-orders Taylor's expansion around $\hat{\theta}$

$$logL(\theta)\approx logL(\hat{\theta})+S(\hat{\theta})(\theta-\hat{\theta})-\frac{1}{2}I(\hat{\theta})(\theta-\hat{\theta})^2$$ we get $$log\frac{L(\theta)}{L(\hat{\theta})} \approx -\frac{1}{2}I(\hat{\theta})(\theta-\hat{\theta})^2$$

providing a quadratic approximation of the normalized log-likelihood around $\hat{\theta}$.

My question:

What happened to the $S(\hat{\theta})(\theta-\hat{\theta})$ term?

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By definition, $\hat{\theta}$ is chosen as an extremum of the log-likelihood, so that $S(\hat{\theta})=0$: the first-order term just cancels out.

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  • $\begingroup$ Thank you! That is all I needed! $\endgroup$
    – Alex
    Feb 29, 2016 at 1:13

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