Re: Calculating the Normal Distribution's CDF

On Mon, 01 Dec 2003 23:35:44 +0100, G Robin Edwards
<[EMAIL PROTECTED]> wrote:
[ snip, some]
> The question I have is why is extreme precision necessary in the real
> world?  We choose, presumably quite arbitrarily, to compute
> PROBNORM(2.00), and are I suppose pleased when the software produces a
> value with 14 digits, and particularly so if other software that we use
> provides the same set of digits.  We are comfortable that we're not
> making any kind of an unnoticed blunder in using the value so
> impressively computed.

Here is one reason that I have run into -- Somebody
might be using the numbers for something more 
subtle than you are.  They might be doing something 
that works out to the equivalent of having 'overflow' --
When you take the square of two 6-digit  numbers in
single-precision, you lose the final-digits of precision, 
and that is why the simple cumulation formula for 
variances can come up with a negative number.


Suppose, for my own simulation program's purposes,  I take 
Probnorm(2.00), Probnorm(2.01), and Probnorm(2.02).

Now:  I should *hope*  that these values are very close 
to each other, and that these have a particular relationship 
to each other.  It might be that my program will fail
or act rather odd, if the first derivative (difference) and 
second derivative (change in difference) are not correct.
 - If I were using the *difference*  between two evaluations
as an estimator of the density, I really do want a number
of digits of *relative*  accuracy.
 - There are several ways to describe the accuracy of the 
computation.  If the algorithm is great at all of them, then the
user can ignore the distinctions about which one ought to matter.
(This is sort of like Robin's assertion that they fix what they 
know how to fix; but I am glad that they do it.)

[ ... ]
> 
> I have similar thoughts about the arbitrarily chosen "conventional"
> values for many statements on probability levels.  Why do we choose
> "95%" or *99%" or "80%" (alpha .05, .01 or .20).  I hazard that they
> are an expression of simple integer betting odds.  
> 
> Is there a more erudite explanation?

Sure, that's why some people use them. 
For approximations, why use anything but integer ratios?

> 
> In a related vein, is "95%" a reasonable value to use when assessing
> the significance of an inference.  My industrial experience with design
> of experiments in new and imprecise technologies showed me that
> technologists just could seldom afford the time and amount of
[ snip, detail]

Other people (as Robin says about his own job), have good
reasons to use other numbers -- so, they *do*  use other numbers.

Sure, 95%  is reasonable for a lot of things.

I would say, '95%'  is a better rule of thumb, and
more widely used, than Cohen's  definitions for social
sciences of effect sizes that are small, medium and large.

When folks do mistake Cohen's  as being magical descriptors, 
automatically, *always*  relevant, that is a MISTAKE.  
However, how often does your own field need to depart? 
or, what part of the field?  

Similarly, for 95% -- It is not the end-all, but it has a lot
of good history.  Do you vary from your convention for some
*logical*  reason?  Was that planned, at the start?
You don't want to change convention merely because one 
experiment needs some excusing,  *after*  the fact. 
(As to the latter:  I once wrote a justification for using 1-tailed, 
10%  testing for a particular hypothesis, owing to the importance 
of the hypothesis and the difficulty of amassing the  N; it was
helpful, after the experiment, that I could point to the planning
before the experiment started.)

-- 
Rich Ulrich, [EMAIL PROTECTED]
http://www.pitt.edu/~wpilib/index.html
"Taxes are the price we pay for civilization."