Summary: Computational Statistics and Statistical Modelling:
Mathematical Tripos, Part IIA, questions from 1997 onwards
P.M.E.Altham, Statistical Laboratory, University of Cambridge.
July 5, 2005
i) Assume that the n-dimensional observation vector Y may be written
Y = X + ,
where X is a given n × p matrix of rank p, is an unknown vector, and
Let Q() = (Y - X)T
(Y - X). Find ^, the least-squares estimator of , and show that
Q(^) = Y T
(I - H)Y
where H is a matrix that you should define.
If now X is written as X = X11 + X22, where X = (X1 : X2), T
1 : T
2 ), and 2 is
of dimension p2, state without proof the form of the F-test for testing H0 : 2 = 0.