For estimation/simulation with a finite search neighborhood,
assuming an intrinsic hypothesis becomes quite tangential
to the main point of interest -- i.e., generating good estimates.
Assuming something is fBm is just that -- an assumption. It might even
be a valid one where variance increases without bounds, but still
other such concerns as the size of the search radius, number of
estimation control points, etc will still have an impact on overall
Syed Abdul Rahman
PS/- fBm is nonstationary but it's derivative -- fGn -- is. Easiest
simulation is to generate a Gaussian distribution using a random number
generator, then integrate to get fBm with Hurst exponent 0.5.
PS2/- _Sample_ covariance will never "fail" if calculated on
a simulated fBm trace. Does "fail" mean an inability to generate
"good" estimates, e.g. from a comparison of cross-validated
----- Original Message -----
From: Hillman RJT <R.J.T.Hillman@...>
Sent: Tuesday, November 30, 1999 10:02 PM
Subject: GEOSTATS: simulating intrinsically stationary processes
> Dear All,
> I hope you can help me. I'm an econometrician analysing high-frequency
> exchange rate data. We typically observe a transaction price at
> irregular intervals, between .01 of a second and three hours depending
> on the time of day. We can also observe other things like spreads,
> liquidity etc.
> I though it might be a good idea, given the irregular spacing of the
> data (on the time-scale) to use some geo-stat methods. I've got Cressie,
> plus some other papers, but there are still some outstanding questions.
> 1) Despite often reading claims that the variogram is defined for a
> wider class of processes than the covariance (i.e. intrinsically
> stationary processes), I haven't seen any convincing evidence that when
> we simulate a non-covariance-stationary process that IS intrinsically
> stationary, the variogram outperforms the covariance. I would imagine we
> could demonstrate this in terms of measuring the dependence and through
> kriging mean square erros. Do you know of any examples here people have
> demonstrated this?
> 2) I have read that Fractional Brownian motion is not stationary, but is
> intrinsically stationary. I thought FBM is stationary when -1/2<d<1/2.
> Could someone clarify this?
> What I am trying to so is simple.
> Generate a univariate time series X(t(1)),X(t(2)),X(t(3))...X(t(N))
> according to an intrinsically stationary process which isn't stationary
> in the usual sense. Then demonstrate that the covariance fails where the
> variogram succeeds.
> Can anyone suggest an easily simulatable process for X(t) that would do
> this for me?
> Ultimately I would like to argue that intrinsic stationarity is a useful
> concept for financial processes, but whilst it seems reasonable and
> apparently un-tested to a large degree in the geo-stats I've seen, I'd
> like some firm evidence of it's usefulness.
> Any suggestions...
> thanks in advance
> Robert J T Hillman
> Research Fellow
> Financial Econometrics Research Centre
> City University Business School
> Frobisher Crescent
> The Barbican
> EC2Y 8HB
> tel: +44 (0) 171 477 8734 Direct Line
> tel: +44 (0) 171 477 8611 Secretary
> fax: +44 (0) 171 477 8881
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