On 22-Dec-01 Chaosheng Zhang wrote:
> Dear list,
>
> Happy Christmas! Many thanks to all those who replied my question about
> extreme values, especially Isobel Clark, Marcel Vall�e, Benjamin Warr,
> Claudio Cocheo, Martin Roseveare, Pierre Goovaerts, Jeff Myers.
Dear Chaosheng Xhang,
Thank you for your comprehensive summary (which now enables
me to delete all the interesting individual replies by others!).
I'd like to add one consideration which seems not to
have been mentioned by others.
Especially in a regulatory ("clean-up") context, the
regulator may want to have a determination of the total
quantity of contaminant on a site.
Your high sample values are typical of "hot-spot" values.
It is a useful formula (proof by integration by parts)
that
integral from 0 to inf (1 - F(x)) dx = expectation of X
where F(x) is the cumulative distribution function of X.
Applying this (somewhat crudely, numerically speaking)
to your data for Lead shows that the top second percentile
(98-100%) accounts for about 1/3 of the total content,
while the top 5th percentile (95-100%) accounts for well
over half the total.
It is therefore essential, for purposes such as the
above, both to take these extreme values very seriously,
and also to try to get estimates of the percentiles which
are as accurate as possible. The latter is not at all easy
(in fact I do not know of a satisfactory solution in
the context of "grid sampling" of a contaminated site,
where the amount and density of sampling which is feasible
in practice is usually quite insufficient -- how do you
know, for instance, that there are not much larger
extremes still, somewhere, remaining unobserved? their
probabilities of being sampled may be very small; but if
their values are very high they could dominate everything
else.)
However, as far as the data are concerned, you have
what you have got and you _must_ respect what it tells
you. From the above, for purposes of estimating the
total, rather than trying to ignore the high percentiles
you would even do better to ignore the lower percentiles,
since they contribute very little!
And, that being said,
Happy Christmas and Prosperous New Year to All!
Ted.
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E-Mail: (Ted Harding) <[EMAIL PROTECTED]>
Fax-to-email: +44 (0)870 167 1972
Date: 22-Dec-01 Time: 21:38:33
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