[R] cubic spline smoothers with heterogeneous variances

[Liaw, Andy]

Sorry for coming to this so late.  My understanding of the problem seems to
be different from Martin's.  Hopefully someone can set me straight.

In Martin's Step 1, why use a small df?  I thought if the objective is to
estimate the variance function first, then one would want estimate of the
regression function to have as small a bias as possible (so that the
residuals would consist of mostly error and very little bias).  I believe
that was the motivation behind those difference-based estimators of the
residual variance.

Also, I was under the impression that the estimation of derivatives has been
worked on quite a bit using local polynomials, so why not use those?  One
just need a variable bandwith smoother for heteroscedastic error.

Cheers,
Andy

-----Original Message-----
From: Martin Maechler [mailto:maechler at stat.math.ethz.ch]
>>>>> "Bill" == Bill Shipley <Bill.Shipley at Usherbrooke.ca>
>>>>>     on Tue, 22 Oct 2002 14:49:34 -0700 writes:

    Bill> Hello. I have data (plant weights over time) that are
    Bill> non-linear and in which the variance increases over
    Bill> time.  I have to estimate the first derivatives of
    Bill> plant weight given time (i.e. growth rate) and their
    Bill> se, using a regression smoother, and I have been
    Bill> considering cubic spline smoothers.  

fine. I do so too if I need derivatives.

    Bill> However, I do not know if this can be done given that
    Bill> the error variance would increase over time.  

I'd hope that a simple two-stage procedure (possibly iterated)
would be enough :

1. Smooth(x,y) with ``df = small'' (depend on your context),
   i.e. getting a smooth solution.
2. Get the residuals and  Smooth(x, abs(resid)) 
   to get an estimate proportional to sigma(x).
3. Smooth(x, y,  weights = 1 / sigma(x))

{now you could iterate "2." and "3." and hopefully see
 convergence (of some kind)}.

    Bill> Does anyone know what the effect of a non-constant error
    Bill> variance has on the estimates of the 1st derivative
    Bill> and its se?

"adverse" (effects), but hopefully you'd only look at the 1st
derivative after the above 2-stage solution.

Martin Maechler <maechler at stat.math.ethz.ch>
http://stat.ethz.ch/~maechler/
Seminar fuer Statistik, ETH-Zentrum  LEO C16	Leonhardstr. 27
ETH (Federal Inst. Technology)	8092 Zurich	SWITZERLAND
phone: x-41-1-632-3408		fax: ...-1228			<><
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