[R] assessing goodness of variance prediction

[Vadim Ogranovich]

Dear R-Helpers,

I am looking for ways to assess quality of a predictor of variance of a
random variable. Here a two related, but yet distinct, setups.

1. I observe y_t, t=1,...,T which is normally distributed with unknown
variance v_t (note that the variance is time-dependent). I have two
"predictors" for v_t, dubbed v1_t and v2_t, and I want to tell which
predictor is better. Here better is to be defined, but intuitively it is
thought to be analogous to R^2 of an ordinary regression.
I was thinking along the lines of fitting a GLM of the form log(abs(y)) ~
log(v1) with some link function, but couldn't figure out which link function
would be appropriate.


2. I observe y_t, t=1,...,T which is multivariate normal iid with unknown
covariance matrix C (which is constant here). I have two estimations of C,
dubbed C1 and C2, and I want to tell which estimation is better. Here again
better is to be defined.
I could of course compute the sample covariance matrix of y_t and then the
L2 norm of the difference (C1 - sampleC), but I don't know if this is a
meaningful measure of a distance between two covariance matrices.


Any lead will be highly appreciated.

Thanks,
Vadim

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