       Re: How to numerically estimate an asymptotic equivalent?

• To: mathgroup at smc.vnet.net
• Subject: [mg46441] Re: How to numerically estimate an asymptotic equivalent?
• Date: Wed, 18 Feb 2004 00:37:12 -0500 (EST)
• Sender: owner-wri-mathgroup at wolfram.com

```On 2/16/04 at 11:41 PM, Franck.Michel at wanadoo.fr (Franck Michel)
wrote:

> >But you don't have errors in the same sense in your problem. You
> >know for several values of n the precise value of the sequence since
> >you have computed it. There is no error term. Consequently, you
> >cannot expect Regress to produce meaningful confidence limits. In
> >fact, the normal meaning of confidence limits doesn't apply to your
> >problem.

>Yes, you're right. In fact, I thought about that, but I was
>wondering if the informations provided by Regress could be used to
>give some insight about the quality of estimations, even if the
>exact values of SE are not pertinent.

The information provided by Regress certainly tells you something about the quality of the approximation. The problem is finding a meaningful way to interpret this information. It is clear the smaller the error sum of squares the better the approximation. But it is not clear how to relate the error sum of squares to the parameters of the approximation.

>Also, this points out another difference between the usual regression
>problem and the problem you are trying to solve. In the normal
>regression problem you expect the residuals to be independent of the
>predictor variables. This should not be the case for your problem.

>Yes, I've plotted the residuals; you can see them at
>http://www.medicis.polytechnique.fr/~fmichel/figure.gif

>The shape of this figure is independant of the range of n, and
>independant of the function estimated (I've always had the same
>shape in all my trials, in particular with the divergent expansions
>coming from classical special functions like erf or gamma). There
>is no really decrease in the values of the residuals, but the
>function estimated is rapidly increasing.

The double hump shown in these plots are typical when there is a mismatch between the data being fitted and the model the data is fitted to. Since you are looking for an asymptotic approximation to the sequence, it is reasonable to expect this behavior. All it is telling you is your approximation alternates between being high and low when compared to the actual sequence. As long as the error term is decreasing and is sufficiently small, I wouldn't worry.
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