This is an exercise from Computer Age Statistical Inference by Efron and Hastie, student edition of the book.

Draw a schematic graph of $\dot{l}_{x}(\theta)$ versus $\theta$. Use it to justify (4.25) $\left[\hat{\theta}\dot{=}\theta + \frac{\dot{l}_{x}(\theta)/n}{\ddot{l}_{x}(\theta)/n}\right]$

I am unsure of what is being asked here. The score $\dot{l}_{x}(\theta)$ has been defined in several ways across the chapter. For example, in equation 4.14 as $\frac{\partial}{\partial\theta}\log f_{\theta}(x)$, equation 4.16 with $\dot{l}_{x}(\theta)\sim (0, \mathcal{I}_{\theta})$ (no distribution in here), equation 4.26 with $\dot{l}_{x}(\theta)/n \dot{\sim} \mathcal{N}(0, \mathcal{I}_{\theta}/n)$, and in the specific case of the Gaussian distribution, equation 4.30 gives $\dot{l}_{x}(\theta)=\frac{1}{\sigma^{2}}\sum_{i=1}^{n}(x_{i}-\theta)$.

I made plots for equation 4.26 and equation 4.30 (with a single observation):

schematic figure of equation 4.26 schematic figure of equation 4.30

Additionally, what is there to justify equation 4.25? I think the point is that if we find the $\hat{\theta}$ so that $\dot{l}_{x}(\theta)=0$, then the estimate is unbiased. If we don't find the exact value, some form of spread on the bias is given in inverse proportion to $\ddot{l}_{x}(\theta)$. Is this interpretation correct? Is it the justification that the book is asking for?



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