Let Y have components Y i satisfying E (Y i) = i = ;i or,

Let Y have components Y i

satisfying E (Y i) = μ

i = ω

α;iβα

or, in matrix notation, E(Y) = Xβ, where X is n × q of rank q. Let ω

i,j

,

ω

i,j,k

,… be given tensors such that Y i − μ

i has cumulants κ2ω

i,j

, κ3,ω

i,j,k and so on. The first-order interaction matrix, X*, is obtained by appending to X, q(q +1)/2 additional columns having elements in the i throw given by

ω

i;r ω

i;s for 1 ≤ r & ≤ s ≤ q. Let H = X(X TWX)

−1X TW and let H*, defined analogously, have rank q* ≤ n. Show that are both

(i) unbiased for κ2

, (ii) invariant under the symmetric group (applied simultaneously to the rows of Y and X),

(iii) invariant under the general linear group applied to the columns of X

(i.e. such that the column space of X is preserved).

Show also that k2 is invariant under the general linear group (4.25) applied to the rows of X and Y, but that k*2 is not so invariant.