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I'm confused on why anyone would appeal to asymptotic normality of mle,

$$\hat{\theta} - \theta_0 \rightarrow^D N(0,I^{-1}(\theta))$$

When we can simply invert the likelihood ratio test

$$L(\hat{\theta}) - L(\theta_0) \rightarrow^D \chi^2_1$$

to obtain a $(1-\alpha)$ CI. Is there a situation where this is not a good idea?

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    $\begingroup$ There are other options too, such as inverting a score test. $\endgroup$
    – Ben
    Jun 21 at 19:53
  • $\begingroup$ Agreed, but why not always use something (LRT, score etc) that converge to chi square to exploit asymmetric CIs? $\endgroup$
    – Casey
    Jun 22 at 0:27

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