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I am a bit confused about how to get a 95% confidence interval from a confidence distribution. My question is based on the paper "Confidence distribution, the frequentist distribution estimator of a Parameter: A review" written by Xie & Singh (2013)

I think I am getting lost in something really naive. This is the confidence distribution in Example 7:

\begin{align} H_n(\theta) &=\int^\infty_{n\bar{x}}\left(\frac{t}{\theta}\right)^{np_0}\frac {e^\frac{t}{\theta}}{\Gamma\left(np_0\right)t}dt \\ &= 1-F_{\Gamma\left(np_0,\theta\right)}(n\bar{x}) \end{align}

where $F_{\Gamma\left(np_0,\theta\right)}(\cdot)$ is the cumulative distribution function of a ${\rm Gamma}(np_0,\theta)$

From that confidence distribution, I just need to take the inverse of the CD with respect to the parameter of interest (in this case $\theta$) in order to get the level $100\left(1-\alpha \right)\%$ confidence interval, for the parameter $\theta$, defined in the paper as $\left(H^{-1}_n\left(\frac{\alpha}{2}\right),H^{-1}_n\left(1-\frac{\alpha}{2}\right)\right)$. As a complementary source, here is the Wikipedia link.

Now, my questions is: is it possible to compute the confidence intervals algebraically? I find it difficult to invert a CDF to get a confidence interval with the tools and methods I have been studying so far. I just started learning about the confidence distribution and I am looking for some insights about making inference.

EDIT: I have a doubt about the whole process I'm going through, and it also involves coverage probability. From a statistical model I obtained a fiducial distribution that it's also a confidence distribution. Saying, from a random variable X distributed as a Binomial, I obtained the fiducial distribution of the parameter p which is a Beta. And I know that this fiducial distribution is also a confidence distribution.

I want to make inference about p, in particular I would like to compute a 95% confidence interval. As I stated above, I can compute the inverse of the CD and obtain the confidence interval.

Since I don't need any approximation, I suppose I can compute an exact confidence interval for p, right? Does the actual coverage probability coincide with the nominal, since I am computing the confidence interval from a continuous distribution such as a Beta? Does it solve the problem of the discrepancy between actual and nominal coverage probability of the discrete distributions?

I would also like to implement this procedure in R, computing the quantile of the beta distribution with the q function and then plotting the coverage probability graph with different sample size. Which function should I use?

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    $\begingroup$ In general you'd do it numerically. In some cases you could do it algebraically. $\endgroup$ – Glen_b Jan 4 '15 at 2:04
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    $\begingroup$ So in this case how should I proceed? This part is creating me troubles. Because I would like to figure out how to find confidence intervals for confidence distributions such as Betaas or Gammas and I find it difficult to reach a closed-form solution for θ. $\endgroup$ – Thiago Jan 4 '15 at 2:27
  • $\begingroup$ That edit is both different enough and long enough that you might be better posting a new question. Also, what's a conference distribution? While fiducial intervals aren't used much any more, at least in the univariate case they should always correspond to CI's derived from pivotal quantities, which many people should be happy to discuss without wondering why you're using fiducial inference. $\endgroup$ – Glen_b Jan 14 '15 at 5:00
  • $\begingroup$ Ok, I'll post a new question. Sorry, conference distribution was just a typo, I meant confidence distribution. $\endgroup$ – Thiago Jan 14 '15 at 11:36
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In general you'd do it numerically. In some cases you could do it algebraically, but often there will be no explicit closed form algebraic solution.

I would like to figure out how to find confidence intervals for confidence distributions such as Betas or Gammas and I find it difficult to reach a closed-form solution for $θ$.

The inverse cdf for a number of common distributions (including the beta and gamma) is readily available in software.

However, even if all you can do is evaluate a cdf, decent root-finding software can solve an equation in the cdf for you (there's always at least binary section!).

R, for example has many inverse cdfs built in (many more in packages), and has root-finding functionality.

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  • $\begingroup$ Thank you very much, that was what I was looking for, since I am starting to "play" with R and I wanted to know if I could find some closed form solution even without a software. But thanks to your answer, I will look for solutions through R. $\endgroup$ – Thiago Jan 4 '15 at 3:10
  • $\begingroup$ If you have a specific issue you could try asking about it (as an addendum to your question and nudging me with a comment under my answer), and I could perhaps say a little more. $\endgroup$ – Glen_b Jan 4 '15 at 3:14
  • $\begingroup$ I just edited my question, since a new doubt arose in my mind. $\endgroup$ – Thiago Jan 14 '15 at 0:32

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