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I am going through this paper. I am unable to understand the definition of a confidence distribution.

A statistic C is an exact confidence distribution for a real parameter ρ if:

  1. ρ → C(ρ; y) is a cumulative distribution function for all y in the sample space of the data Y.
  2. C(ρ; Y ) ∼ U(0, 1)

[Wikipedia] Θ is the parameter space of the unknown parameter of interest θ, and χ is the sample space corresponding to data Xn={X1, ..., Xn}. A function Hn(•) = Hn(Xn, •) on χ × Θ → [0, 1] is called a confidence distribution (CD) for a parameter θ, if it follows two requirements:

(R1) For each given Xn ∈ χ, Hn(•) = Hn(Xn, •) is a continuous cumulative distribution function on Θ;
(R2) At the true parameter value θ = θ0, Hn(θ0) ≡ Hn(Xn, θ0), as a function of the sample Xn, follows the uniform distribution U[0, 1].

I have been studying from this textbook also.

The things that confuse me are that it's not a probability distribution (it gives 'confidence') but it is a distribution function of the parameter, it is a random variable when conditioned on the true parameter value, and unlike a distribution which is coupled with a random variable, it is not unique. I also find a mismatch between mathematical rigor and intuition.

Please explain the definition and assumptions in as much detail as possible (including the arguments of the function and its output) and point me prerequisites or to sources that give more details and illustrative examples. Please explain the motive behind the development of this concept.

Mewbacca
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    Welcome to CV, Mewbacca. You will find your question better received if, instead of asking readers to go to another site are read something there (e.g., your "in this paper," etc.) you quote the relevant information so that everything readers need to understand your question is on this page. – Alexis Nov 28 '21 at 17:47
  • What is it about your Wikipedia reference that appears to lack mathematical rigor? – whuber Nov 28 '21 at 17:51
  • A book-length treatment with many interesting examples is at [Confidence, Likelihood, Probability: Statistical Inference with Confidence Distributions](https://bookshop.org/books/confidence-likelihood-probability-statistical-inference-with-confidence-distributions/9780521861601) – kjetil b halvorsen Nov 28 '21 at 18:00
  • @kjetilbhalvorsen, the OP is not happy with the book. (Look at the link in the OP.) – Richard Hardy Nov 28 '21 at 18:02
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    @Richard Hardy: Ooops ... then he will be dissatisfied with most we can provide, that book is very good! especially, very few theory books have so many interesting real data examples ... – kjetil b halvorsen Nov 28 '21 at 18:05
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    @kjetilbhalvorsen, I agree about the real data examples. I have tried reading the book (found it heavily discounted in the campus bookstore and could not resist to buy it) but did not have enough time to get very far, so I cannot comment on its pedagogical quality. But the posts on the FocuStat blog (OP links to it, too) are usually pretty good. And if the OP did not find them good enough, then finding something better might be a challenge. – Richard Hardy Nov 28 '21 at 18:39
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    @Alexis Thanks for the suggestion. – Mewbacca Nov 29 '21 at 07:04
  • @Whuber, I meant the blog post lacks rigor and the others lack intuition, I am unable to connect them. – Mewbacca Nov 29 '21 at 07:05
  • @RichardHardy I think I should study again more carefully to understand the topic and my exact problem. However, I welcome all guidance ("he will be dissatisfied with most we can provide"). Thank you! – Mewbacca Nov 29 '21 at 07:08

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