What is the equivalent of Student's t-distribution on an interval?

Question

I have been using Student's t-distribution as a proxy for the Gaussian distribution estimated from a limited number of points in order to calculate alpha-intervals (for instance 95% confidence intervals).

However, lately I have been working on closed intervals with distributions that are close to beta-distribution family and couldn't find any distribution that could be useful to estimate alpha-intervals while taking in account the number of samples.

What distribution would be most appropriate for such application?

Answer 1

I suppose you are interested in obtaining a confidence interval for the true mean of a distribution with bounded support (although your post does not mention the mean at any point...)

If your sample size is not tiny, then Student intervals will provide usually an excellent approximation (Central Limit Theorem, no heavy tails).

To illustrate this, we can simulate from a beta distribution with known mean, compute the t-confidence interval and check in each run if the interval contains the mean $\mu$ (which is 1/3 in the example below).

n <- 20 # sample size
a <- 2  # parameters of beta distribution 
b <- 4
mu <- 1/(1 + b/a) # true mean of beta distribution

set.seed(1)
out <- replicate(10000, {
  x <- rbeta(n, a, b) 
  t.test(x)$conf.int
})

mean(out[1, ] <= mu & mu <= out[2, ]) # 0.948 

par(mfrow = 1:2)
hist(x)
boxplot(x)

So the real coverage probability is almost exactly identical to the nominal coverage of 0.95 even if the underlying is not normal but asymmetric beta. Histogram and boxplot in the last run of the simulation are looking as follows:

If we run above simulation with tiny sample sizes, we get the following coverage probabilities:

• 0.9399 ($n = 5$)
• 0.9365 ($n = 4$)
• 0.9403 ($n = 3$)
• 0.943 ($n = 2$)

Answer 2

This problem is very common in the Bayesian framework. Student's T distribution is the posterior predictive of the Gaussian model. Given $n$ points, having mean $\mu$ and variance $\sigma^2$, the posterior predictive distribution for subsequent points is (noncentral scaled) Student T distributed.

The exact same calculations can be done for your model, which is the Beta-binomial model. You can find details on the Wikipedia page. Like the Student T, it's a three parameter distribution with density:

\begin{align} f(k \mid n,\alpha,\beta) = \frac{\Gamma(n+1)}{\Gamma(k+1)\Gamma(n-k+1)} \frac{\Gamma(k+\alpha)\Gamma(n-k+\beta)}{\Gamma(n+\alpha+\beta)} \frac{\Gamma(\alpha+\beta)}{\Gamma(\alpha)\Gamma(\beta)}. \end{align}

This is over the domain $k\in [0,1]$. $\alpha-\frac12$ and $\beta-\frac12$ are your number of successes and failures, $n=1$ since your model is Beta-Bernoulli.

To find your $\alpha$-interval, I guess you'll have to integrate and invert the density, likely by numerical means.