Yes, you nailed it, Gareth. —Dan
On Nov 4, 2015, at 3:26 PM, Gareth McCaughan <gareth.mccaughan@pobox.com> wrote:
On 03/11/2015 21:01, Dan Asimov wrote:
Suppose X, Y, Z, W are real random variables with a joint distribution such that each one has a finite mean and standard deviation.
Suppose that all pairs of these random variables have the same correlation coefficient:*
R = rho(X,Y) = rho(X,Z) = rho(X,W) = rho(Y,Z) = rho(Y,W) = rho(Z,W) .
Find the minimum possible value of R.
WLOG X,Y,Z,W all have mean 0 and stddev 1 so E(XX)=E(YY)=...=1. We are given that E(XY)=E(XZ)=... and we want to know how small that can be. (That value will be the correlation coefficient.)
We can think of X,Y,Z,W as vectors in some Hilbert space, with E(XY) etc. being their inner products. And then of course we can look at just the space spanned by X,Y,Z,W, which looks just like R^4. So the question is: given four real 4-dimensional vectors, each of norm 1 and with all their pairwise inner products equal, how small (i.e., large and negative) can the inner products be?
Put our vectors in a single array A; then the matrix of inner products is At.A which we require to have the form I+kM where I is the identity and M is the 4x4 matrix that's all ones, and we want k as negative as possible. Well, At.A has to be positive semidefinite, which happens iff k >= -1/3. (For those values the matrix is diagonally dominant. Otherwise the all-1s vector is an eigenvector with negative eigenvalue.)
And any positive semidefinite matrix has a factorization of the form At.A, so in fact the value -1/3 is attained.
I therefore claim that R=-1/3 is the best possible.
-- g
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