Now, note that these two families are disjoint subsets of the exponential power family. When p=1, you're in a double-exponential family (Laplace); when p=2 you're in a normal family.
Michael Sherman - Comparing the Sample Mean and the Sample Median: An Exploration in the Exponential Power Family shows that when p=1.407, the mean and the median are equally good.
Note that the mean and median are examples of linear combinations of the order statistics. We can imagine how different linear combinations of the order statistics would be optimal for different values of p.
This is what I saw in class:
Definition: an equivariant estimator T(X) is one that satisfies T(X + ε 1) = T(X) + ε. i.e. if you shift all the data by some amount ε, the estimator changes by ε. Examples: sample mean, sample median, sample max, sample min.
Definition: an invariant estimator T(X) is one that satisfies T(X + ε 1) = T(X). i.e. if you shift all the data by the same amount, the estimator does not change. Examples: sample variance, interquartile range.
Theorem: if T1 is equivariant and T2 is invariant, then T1 + T2 is equivariant.
Definition: the maximum invariant Y(X) is the (n-1)-dimensional vector (X2 - X1, X3 - X1, ..., Xn - X1).
Theorem: every invariant estimator is a function of the maximum invariant.
This is how I made sense of it all.
For the record, I am beginning to use LyX. It is nice, but when I adjust the displayed font size, the math stays the same size, and so it looks comparatively tiny. This post was made by exporting to XHTML, copy-pasting onto DreamWidth, and then deleting the silly "magicparlabel" A-tags that were making everything look green. mirror of this post