# How to determine mse of estimate from correlation matrix of estimate error?

I have a model of an information transmission system Y = XH + N, where X is a diagonal matrix with the transmitted "symbols" (known), H is a column vector which distorts the transmitted symbols and N is a "noise (column) vector" which is sampled from a zero mean Gaussian distribution of known variance. An estimator of the distortion H would be: $$\hat{H} = (X^H X)^{-1} X^H Y$$. Now, if we apply this estimator to the model, we obtain

$$(X^H X)^{-1} X^HX H + (X^H X)^{-1} X^HN = H + (X^H X)^{-1} X^HN$$. This implies $$(X^H X)^{-1} X^HN$$ is the error in the estimator. I am interested to get the mse of this estimator. I was thinking of proceeding as follows: $$C_{\tilde{N}} = E[((X^H X)^{-1} X^HN)((X^H X)^{-1} X^HN)^H]$$

$$= (X^H X)^{-1} X^HE[NN^H]\ X (X^H X)^{-1}$$

$$= (X^H X)^{-1} X^H (\sigma^2 I) \ X (X^H X)^{-1}$$

$$= (X^H X)^{-1} X^H (\sigma^2 I) \ X (X^H X)^{-1} = \sigma^2 (X^H X)^{-1}$$

We now have, I believe, the covariance matrix of the estimator noise (since X is a matrix here). Can I obtain the MSE of the estimator from this? And, in general, if we have a covariance matrix of the form $$\sigma^2 P$$, where P is guarenteed to be positive definite (as a covariance matrix should be) but no other information is known (P is not diagonal), how could I calculate the MSE?

• Any citation (reference books?) to such a result would be appreciated. Mar 25, 2019 at 3:05

So it turns out this was rather simple, when I thought about it. $$MSE = E[|H-\hat{H}|^2] = E[(H-\hat{H})^H (H-\hat{H})]$$
$$=E[((X^H X)^{-1} X^HN)^H (X^H X)^{-1} X^HN]$$
What I had calculated was $$E[(H-\hat{H}) (H-\hat{H})^H]$$, which is a rank 1 matrix. To get the MSE, we just need to sum the diagonal elements of this correlation matrix, which can be derived using basic linear algebra.