Coverage probability from fiducial/confidence distribution

Question

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?

Answer 1

Does the actual coverage probability coincide with the nominal, since I am computing the confidence interval from a continuous distribution such as a Beta?

The coverage probability does not coincide with the nominal probability when you let the quantiles for the computation of the interval depend on the observation. An example is given here with the computation of a highest density interval for a fiducial distribution.

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)

Does it solve the problem of the discrepancy between actual and nominal coverage probability of the discrete distributions?

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 $\theta$.

The logic behind confidence and fiducial inverse "probability" is that it considers the distribution of the data conditional on the parameter. If the data is not continuous distributed and are discrete instead, then the intervals based on it may have glitches.