Return to Answer

Denote $\varSigma_1$ and $\varSigma_2$ your matrices both of dimension $p$.

1. Cond number: $\log(\lambda_1)-\log(\lambda_p)$ where $\lambda_1$ ($\lambda_p$) is the largest (smallest) eigenvalue of $\varSigma^*$, where $\varSigma^*$ is defined as: $\varSigma^*:=\varSigma_1^{-1/2}\varSigma_2\varSigma_1^{-1/2}$

Edit: I edited out the second of the two proposals. I think I had misunderstood the question. The proposal based on condition numbers is used in robust statistics a lot to assess quality of fit. An old source I could find for it is:

Yohai, V.J. and Maronna, R.A. (1990). The Maximum Bias of Robust Covariances. Communications in Statistics–Theory and Methods, 19, 3925–2933.

I had originally included the Det ratio measure:

2. Det ratio: $\log(\det(\varSigma^{**})/\sqrt{\det(\varSigma_2)*\det(\varSigma_1)})$ where $\varSigma^{**}=(\varSigma_1+\varSigma_2)/2$.

which would be the Bhattacharyya distance between two Gaussian distributions having the same location vector. I must have originally read the question as pertaining to a setting where the two covariances where coming from samples from populations assumed to have equal means.

Denote $\varSigma_1$ and $\varSigma_2$ your matrices both of dimension $p$.

1. Cond number: $\log(\lambda_1)-\log(\lambda_p)$ where $\lambda_1$ ($\lambda_p$) is the largest (smallest) eigenvalue of $\varSigma^*$, where $\varSigma^*$ is defined as: $\varSigma^*:=\varSigma_1^{-1/2}\varSigma_2\varSigma_1^{-1/2}$

Edit: I edited out the second of the two proposals. I think I had misunderstood the question. The proposal based on condition numbers is used in robust statistics a lot to assess quality of fit. An old source I could find for it is:

Yohai, V.J. and Maronna, R.A. (1990). The Maximum Bias of Robust Covariances. Communications in Statistics–Theory and Methods, 19, 3925–2933.

I had originally included the Det ratio measure:

2. Det ratio: $\log(\det(\varSigma^{**})/\sqrt{\det(\varSigma_2)*\det(\varSigma_1)})$ where $\varSigma^{**}=(\varSigma_1+\varSigma_2)/2$.

which would be the Bhattacharyya distance between two Gaussian distributions having the same location vector. I must have originally read the question as pertaining to a setting where the two covariances were coming from samples from populations assumed to have equal means.

Denote $\varSigma_1$ and $\varSigma_2$ your matrices both of dimension $p$.

1. Cond number: $\log(\lambda_1)-\log(\lambda_p)$ where $\lambda_1$ ($\lambda_p$) is the largest (smallest) eigenvalue of $\varSigma^*$, where $\varSigma^*$ is defined as: $\varSigma^*:=\varSigma_1^{-1/2}\varSigma_2\varSigma_1^{-1/2}$

Edit: I edited out the second of the two proposals. I think I had misunderstood the question. The proposal based on condition numbers is used in robust statistics a lot to assess quality of fit. An old source I could find for it is:

Yohai, V.J. and Maronna, R.A. (1990). The Maximum Bias of Robust Covariances. Communications in Statistics–Theory and Methods, 19, 3925–2933.

I had originally included the Det ratio measure:

2. Det ratio: $\log(\det(\varSigma^{**})/\sqrt{\det(\varSigma_2)*\det(\varSigma_1)})$ where $\varSigma^{**}=(\varSigma_1+\varSigma_2)/2$.

which would be the Bhattacharyya distance between two Gaussian distributions having the same location vector. I must have originally read the question as pertaining to a setting where the two covariances where coming from samples from populations having equal means.

Denote $\varSigma_1$ and $\varSigma_2$ your matrices both of dimension $p$.

1. Cond number: $\log(\lambda_1)-\log(\lambda_p)$ where $\lambda_1$ ($\lambda_p$) is the largest (smallest) eigenvalue of $\varSigma^*$, where $\varSigma^*$ is defined as: $\varSigma^*:=\varSigma_1^{-1/2}\varSigma_2\varSigma_1^{-1/2}$

Edit: I edited out the second of the two proposals. I think I had misunderstood the question. The proposal based on condition numbers is used in robust statistics a lot to assess quality of fit. An old source I could find for it is:

Yohai, V.J. and Maronna, R.A. (1990). The Maximum Bias of Robust Covariances. Communications in Statistics–Theory and Methods, 19, 3925–2933.

I had originally included the Det ratio measure:

2. Det ratio: $\log(\det(\varSigma^{**})/\sqrt{\det(\varSigma_2)*\det(\varSigma_1)})$ where $\varSigma^{**}=(\varSigma_1+\varSigma_2)/2$.

which would be the Bhattacharyya distance between two Gaussian distributions having the same location vector. I must have originally read the question as pertaining to a setting where the two covariances where coming from samples from populations assumed to have equal means.

Denote $\varSigma_1$ and $\varSigma_2$ your matrices both of dimension $p$.

1. Cond number: $\log(\lambda_1)-\log(\lambda_p)$ where $\lambda_1$ ($\lambda_p$) is the largest (smallest) eigenvalue of $\varSigma^*$, where $\varSigma^*$ is defined as: $\varSigma^*:=\varSigma_1^{-1/2}\varSigma_2\varSigma_1^{-1/2}$

Edit: I edited out the second of the two proposals. I think I had misunderstood the question. The proposal based on condition numbers is used in robust statistics a lot to assess quality of fit. An old source I could find for it is:

Yohai, V.J. and Maronna, R.A. (1990). The Maximum Bias of Robust Covariances. Communications in Statistics–Theory and Methods, 19, 3925–2933.

Denote $\varSigma_1$ and $\varSigma_2$ your matrices both of dimension $p$.

1. Cond number: $\log(\lambda_1)-\log(\lambda_p)$ where $\lambda_1$ ($\lambda_p$) is the largest (smallest) eigenvalue of $\varSigma^*$, where $\varSigma^*$ is defined as: $\varSigma^*:=\varSigma_1^{-1/2}\varSigma_2\varSigma_1^{-1/2}$

Edit: I edited out the second of the two proposals. I think I had misunderstood the question. The proposal based on condition numbers is used in robust statistics a lot to assess quality of fit. An old source I could find for it is:

Yohai, V.J. and Maronna, R.A. (1990). The Maximum Bias of Robust Covariances. Communications in Statistics–Theory and Methods, 19, 3925–2933.

I had originally included the Det ratio measure:

2. Det ratio: $\log(\det(\varSigma^{**})/\sqrt{\det(\varSigma_2)*\det(\varSigma_1)})$ where $\varSigma^{**}=(\varSigma_1+\varSigma_2)/2$.

which would be the Bhattacharyya distance between two Gaussian distributions having the same location vector. I must have originally read the question as pertaining to a setting where the two covariances where coming from samples from populations having equal means.