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Re: WeibullDistribution

• To: mathgroup at smc.vnet.net
• Subject: [mg42590] Re: WeibullDistribution
• From: Bill Rowe <listuser at earthlink.net>
• Date: Thu, 17 Jul 2003 03:45:09 -0400 (EDT)
• Sender: owner-wri-mathgroup at wolfram.com

On 7/16/03 at 9:13 AM, robert.nowak at ims.co.at (Robert Nowak) wrote:

> what about doing a NonlinearFit on the empirical CDF of the data
> against the CDF of the desired dsitribution. are there any
> tehoretical issues against such a fit ?

Yes, you can use NolinearFit to fit the empricial CDF of the data to the theorectical CDF. And yes, there are pros and cons. 

The primary disadvantage of using NonlinearFit is the difficulty in finding the true least squares fit, i.e., the set of paramerters that makes the summed square error globally minimal. It is often the case there are several local minina and it is easy for the non-linear algogrithm to get trapped in a local minima. And the real difficulty is there is no simple way of determining when this happens.

The primary advantage of using NonlinearFit is you truly are minimizing the least squares. This is particularly important when you want confidence limits on the fitted parameters. If you assume the usual model, i.e., the errors are normally distributed about the regression curve then you can use the usual normal distribution based statistics to compute the confidence limit. In fact, this is what NonlinearRegress does assume when it computes confidence limits.

It is possible to correctly compute confidence limits with the transformed problem. But this is much more difficult to do correctly.

> of course is see that linear fitting is much more elegant but isn't
> there a danger to get som bias in the estimated parameters due to the
> transformations isn't it neccessary to weight the data properly to
> take the transformations into account ?

>From a practical standpoint, no the linear fit to the transformed problem is good as is with out adjustments. Generally, the uncertainty in the fitted parameters is larger than the bias particularly when attempting to find parameters for a given distribution. Also it is very easy with the Weibull distribution to make point estimates based on two selected quantiles. These can easily be made free of bias and compared to the estimates from the transformed regression problem.

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  • Re: Re: WeibullDistribution
    • From: Dr Bob <drbob@bigfoot.com>