Given a sequence $\mathbf{x} = (x_1,x_2,\dots,x_n)$ which is sampled from some Gaussian process $GP(\mu_1,\Sigma_1)$ and a "target" sequence $\mathbf{y} = (y_1,y_2,\dots,y_n)$ sampled from another Gaussian process $GP(\mu_2,\Sigma_2)$, does anyone know of a test of whether $\mathbf{x}$ and $\mathbf{y}$ come from the same distribution? I guess I'd like to know the probability $\mathbf{x}$ and $\mathbf{y}$ are from the same distribution (is that even well posed?).
I am not assuming any of $\mu_1,\mu_2,\Sigma_1$ or $\Sigma_2$ are known. Also, I'm okay with assuming the GPs are stationary (but not centered) if that helps.
Best Answer
This sounds like an application for the two-sample KS test, which evaluates whether two samples were taken from the same continuous probability distribution. For more information, you might want to start here, which explains it in some detail: http://www.itl.nist.gov/div898/handbook/eda/section3/eda35g.htm.