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Frequentist statistics textbooks typically consider point and interval estimation but not density estimation of a parameter. Since the density (of the sampling distribution) of the estimator is involved in deriving the interval estimator, why not consider the density itself? Is there something wrong with or uninteresting about it so that it is not considered?

Edit: a/the relevant term seems to be confidence distribution (thanks to @KjetilHalvorsen) as in

I wonder why confidence distributions are not covered right after interval estimation in typical textbooks.

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  • $\begingroup$ It is considered even at the most elementary level. You would be hard pressed to find any intro textbook that does not explain, for instance, how to estimate the density of a distribution that is assumed to be Normal. $\endgroup$
    – whuber
    Jan 10 '20 at 19:32
  • $\begingroup$ @whuber, I am talking about the density of a parameter estimator, not of raw data. I noticed that most textbooks start with point and end with interval estimation of parameters, and the density does not get analyzed. $\endgroup$ Jan 10 '20 at 19:41
  • $\begingroup$ Then I am even more baffled, because consideration of an estimator's distribution is fundamental to all probability-based statistical techniques. I can't imagine any book on the subject that doesn't cover this. Going back to the previous example: even elementary textbooks explain how the usual estimator of the mean has a Normal distribution (and how to estimate its parameters) and the usual estimator of the variance has a distribution proportional to a chi-squared distribution (and how to estimate its parameters). $\endgroup$
    – whuber
    Jan 10 '20 at 19:44
  • $\begingroup$ @whuber, this is true. However, this is still not what I am getting at. I am looking at the progression: point estimation, interval estimation, density estimation, all targeting the same parameter. I do see posterior densities of parameters in Bayesian statistics but not the "corresponding" densities in frequentist statistics (while point estimattion and interval estimation have such "correspondences" between frequentist and Bayesian approaches). $\endgroup$ Jan 10 '20 at 19:51
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    $\begingroup$ @jbowman, not many other uses than the ones you say. Nevertheless, I think it would be convenient to have a density reported so that any user could take his/her favourite loss function and calculate the appropriate point estimate or choose a favourite confidence level to obtain an appropriate confidence interval. The density is so much more informative than a single point or a single interval. So it is kind of surprising to see it missing from the textbooks. Perhaps this is a matter of time; in the forecasting literature, density prediction is much more common now than it was some years ago. $\endgroup$ Jan 10 '20 at 20:03

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