Similar weird things happen for the Cauchy distribution (whose probability density function is proportional to 1/(1+x^2)), which is symmetric around 0 but does not have mean 0 because the sum doesn’t converge.
Exercise: what do you expect to happen if you try to find the mean of the Cauchy distribution by simulation?
what do you expect to happen if you try to find the mean of the Cauchy distribution by simulation?
From the same source:
the distribution of the sample mean will be equal to the distribution of the samples themselves; i.e., the sample mean of a large sample is no better (or worse) an estimator of x0 than any single observation from the sample.
Now I’m wondering if there is a symmetric distribution where the sample mean is a strictly worse estimator of the median than a single observation.
Similar weird things happen for the Cauchy distribution (whose probability density function is proportional to 1/(1+x^2)), which is symmetric around 0 but does not have mean 0 because the sum doesn’t converge.
Exercise: what do you expect to happen if you try to find the mean of the Cauchy distribution by simulation?
From the same source:
Now I’m wondering if there is a symmetric distribution where the sample mean is a strictly worse estimator of the median than a single observation.
Off the cuff: it’s probably a random walk.
Edit: It’s now pretty clear to me that’s false, but plotting the ergodic means of several “chains” seems like a good way to figure it out.
Edit 2: In retrospect, I should have predicted that. If anyone is interested, I can post some R code so you can see what happens.