Hello i've a little problem:

i've to consider an estimation problem where the unknown parameter is a vector N dimensional and error is gaussian with zero mean and variance that changes for each measurement.

I've to prove that the estimation error covariance matrix P of the BLUE is a decreasing function of N.

In particolar i should use two matrix P1 e P2 that corrispond to N1 measurements and P2 to N1+N2 measurements.

Then using the information matrix (the inverse of the error covariance matrix) i've to find that I1<I2.

The model is Y=XU + E

where Y is the vector Nx1 of measurements

X the unknown vector nx1

U = Nx1 inputs used to find the unknows

E = Nx1 errors.

So my questions are:

1) how can i say that a matrix is lower than another one? since this is the situation of diagonal matrix could i use the trace?

2) if the trace is a right choise at the end i obtain that the trace of P is a sum between 1 and N in which at the nominator there's the variance of the noise while at the denominator there are the several inputs square for each measurement...

so how can i claim that my estimator is consistent? i know that my result depend on 1/N but at the nominator i have a variance noise that changes each time.. so if the variance increased when N grows i will have a constistent estimator?