S Autoregressive errors. Consider the model Xk D mC p

v k, k D 1; : : : ; n, for n realvalued observations with unknown expectation m 2 R and unknown v > 0. For the errors, suppose that k D k1 C k; here  > 0 is supposed to be known, 0 D 0, and 1; : : : ; n are uncorrelated and standardised random variables in L2. Show the following:

(a) Besides the sample mean M, S WD



X1 C .1/

Pn kD2.Xk  Xk1/

ı

1 C .n1/.1/2

is another unbiased estimator of m. Note that S ¤ M because  ¤ 0.

(b) For all m and v we have Vm;v.S/ < Vm;v.M/.

Hint: Write B D  for a suitable matrix B, and set up a linear model for Y WD BX.