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When we do maximum likelihood estimation of a (let's say scalar) parameter $\theta$, we get a point estimate wherever the likelihood function is maximized. However, if we want an interval estimate, the points giving nearly maximum likelihoods seem like good candidates for points to include in the interval estimate.

  1. (Main question) Is there a way to relate something like confidence level (I think I mean inverting the interval to give a hypothesis test) to give an interval estimate based on some criteria like, "I want all points that give likelihood within $\text{x}\%$ of the maximum likelihood"?

  2. If we form a confidence interval based on inverting an $\alpha$-level likelihood ratio test, is that basically what we're doing?

It would be nice to have decent conditions that will assure the interval estimate is indeed a connected interval, but that is of secondary importance.

EDIT

Here is a sketch of the situation. I mean the interval between the red vertical lines., where the red horizontal line is $\text{x}\%$ of the maximum value of the purple likelihood function.

enter image description here

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Under some conditions you can show that the MLE of a parameter has the following distribution: $\sqrt n (\hat\theta-\theta)\sim\mathcal N(0,\hat\sigma^2)$, where $\hat\sigma^2$ is the corresponding estimate of the variance. The variance estimate $\hat\sigma^2/n$ can be used to estimate the confidence interval of the parameter estimate $\hat\theta$. The variance estimate is related to the Fisher information of the distribution (likelihood) function.

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  • $\begingroup$ How does this help get the points within $\text{x}\%$ of the maximum likelihood? $\endgroup$ – Dave Jun 2 '20 at 19:25
  • $\begingroup$ If you agree that $(\hat\theta-\theta)\sim\mathcal N(0,\hat\sigma^2/n)$ then you can get all the percentile in usual way from the normal distribution $\endgroup$ – Aksakal Jun 2 '20 at 19:43
  • $\begingroup$ I don't think I'm talking about the probability of the MLE. I think I mean the y-axis values. If at the MLE the likelihood function has a value of 2, I am looking to get all inputs to the likelihood function that return values of at least 1.8, for instance. $\endgroup$ – Dave Jun 2 '20 at 20:04
  • $\begingroup$ Are you looking for inverse problem? The domain of x within likelihood interval? $\endgroup$ – Aksakal Jun 2 '20 at 20:07
  • $\begingroup$ Maybe. I've edited the original question with a sketch. $\endgroup$ – Dave Jun 2 '20 at 20:13

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