Suppose all pedigrees from a sample have been amalgamated into a single pedigree. For a trait vector Y with E(Y ) = 0, consider the covariance components model YiYj = r k=1

σ2 kΓkij + eij , (8.16)

where the eij are independent, identically distributed random errors.

Let U be the matrix Y Y t

, Wk be the matrix Γk, and e be the matrix

(eij ), all written in lexicographical order as column vectors. Then the model (8.16) can be written as U = W σ2 +

e, (8.17)

where W = (W1,...,Wr). Show that the normal equations for estimating σ2 reduce to one step of scoring starting from (0,..., 0, 1)t

.

This result is due to Robert Jennrich.