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?