Arithmetic of confidence regions

The previous post defined confidence regions (CRs). As explained, the confidence condition is rather weak, and as a result CR algebra is not very rich. One thing that can be defined is CR arithmetic.

Let CA(X) be a level 1-alpha CR for a parameter A, and let B(X) be a level 1-beta CR for a parameter B, where both a and b are functions of the unknown distribution of X. Let C = C(A,B) be another parameter of the distribution of X – one which can be expressed as a function of the unknown A and B.

Then CC(X), a level 1-alpha-beta CR for the parameter C can be constructed as follows:

CC(x) = { C(a,b) : a in CA(x) and b in CB(x) }.

The proof is simple: in the event that both A in CA(X) and B in CB(X) then C must be in CC(X). The complement of this event is contained in the union of the event that A is not in CA(X) and the event that B is not in CB(X). Those events have probabilities alpha and beta, respectively. The union therefore has a probability of no more than alpha+beta. QED

Of course, while CC is a level 1-alpha-beta CR for C, it is not necessarily the only one.