Coverage probability from fiducial/confidence distribution

I am trying to realize how to compute confidence intervals from a fiducial distribution/confidence distribution with possibly more than one parameter. But for now I would just like to understand if I am running correctly the procedure with only one parameter. My aim is to analyze the frequentist properties (i.e. coverage probability and interval length) of these CIs.

Following from the definition of confidence distribution, it is known that confidence distribution is a pivot.

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. It is known that 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?

• I would not be surprised that the fiducial distribution coincides with the Jeffreys posterior (but you have to check), and all the job you aim to do has been intensively studied: projecteuclid.org/euclid.aos/1015362189 Commented Jan 14, 2015 at 17:00

The same applies to confidence distributions. The fiducial distribution is the derivative of the CDF $$-\frac{\partial}{\partial \theta} F(x;\theta)$$ and the confidence distribution is, a bit more general, the derivative of any p-value $$\propto \frac{\partial}{\partial \theta} p(x;\theta)$$ (the factor of proportionality depends on the number of values $$\theta$$ that have the same p-values, and in the case of confidence regions instead of confidence intervals, this becomes a bit more complex)
If you would have fixed quantiles for the intervals then the coverage probability would coincide. But with discrete distributions the trouble is that $$p(x;\theta)$$ is not continuous in both the data $$x$$ and the parameter $$\theta$$.