I have a problem very similar to the one presented chapter 6 of BDA, the speed of light example. You use the distribution of the minimum scores from the posterior predictive distribution, show that it’s not realistic given the data, and suggest that an asymmetric contaminated normal distribution or a symmetric long-tailed distribution would be better.
How does one use such a distribution?
You can actually use a symmetric long-tailed distribution such as t with low degrees of freedom. One striking feature of symmetric long-tailed distributions is that a small random sample from such a distribution can have outliers on one side or the other and look asymmetric.
Just to see this, try the following in R:
par (mfrow=c(3,3), mar=c(1,1,1,1))
for (i in 1:9) hist (rt (100, 2), xlab="", ylab="", main="")
You’ll see some skewed distributions. So that’s the message (which I learned from an offhand comment of Rubin, actually): if you want to model an asymmetric distribution with outliers, you can use a symmetric long-tailed model.
Of course, if you really want to blow your mind, try this:
If you’re like me, you’ll have to think a little bit about this before you see what’s happening.