In trying to figure out if this problem I’m working on has been addressed by the statistics community, I found myself forming Google queries on “semiparametric Gaussian estimation.” The problem I’m looking at is the following. Suppose X(t) is an iid Gaussian vector. I get to observe Y(t) = A(t) X(t) + W(t) where W(t) is an iid Gaussian vector of noise and A(t) is a matrix-valued random variables taking values in a finite set, iid across time. I want to form a minimum mean-squared error (MMSE) estimate of X(t) from Y(t). If A(t) is known at all times t then this is easy since I can make a linear estimator for each value that A(t) can take. Instead, I’ll make the crazy assumption that the estimator has to be linear and designed off-line (i.e. not data dependent), and that the distribution p(A) is not known. What’s the best estimator and worst-case error?

In my searching the web, however, I turned up some crazy things on long memory parameters in nonstationary time series. I also came across the term kriging, which looked like a typo for something. Instead, it really means Gaussian process regression — yet another instance of jargon standing in the way of understanding. Unfortunately, I don’t think it’s quite what I want. Back to the ol’ search engine…