more binomial MADness

I posted earlier about the mean absolute deviation (MAD) of a binomial variable S_n with parameters (n,p). Here’s a little follow-up with plots. This is a plot of \mathbb{E}|S_n - np| versus p for different values of n.

The first is for n = 10. Looks beautifully scalloped, no? As we’d expect, the MAD is symmetric about p = 1/2 and monotonically increasing for the first half of the unit interval. Unfortunately, it’s clearly not concave (although it is piecewise concave), which means I have to do a bit more algebra later on.

When $n = 100$ the scallops turn into a finely serrated dome.

By the time you get to $n = 1000$ the thing might as well be concave for all that your eye can tell. But you would be deceived. Like a shark’s skin, the tiny denticles can abrade your proof, damaging it beyond repair.

Why do I care about this? If you take n samples from a Bernoulli variable with parameter p, then the empirical distribution (unnormalized) is (n - S_n, S_n). So \frac{1}{n} \mathbb{E}|S_n - np| is the expected total variational distance between the empirical distribution and its mean. More generally, the expected total variational distance for finite-alphabet distributions is a sum of MAD terms.


2 thoughts on “more binomial MADness

Leave a Reply

Fill in your details below or click an icon to log in: Logo

You are commenting using your account. Log Out /  Change )

Facebook photo

You are commenting using your Facebook account. Log Out /  Change )

Connecting to %s

This site uses Akismet to reduce spam. Learn how your comment data is processed.