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
- Schweder & Hjort "Confidence, Likelihood, Probability: Statistical Inference with Confidence Distributions" (2016) and threads
- Changing the size of a confidence interval in order to emphasize results and
- How to compute confidence interval from a confidence distribution.
I wonder why confidence distributions are not covered right after interval estimation in typical textbooks.