Re: approximations for exponential, normal and gamma cdf


Reef Fish wrote:
herpers@xxxxxxxxxxxxxxxxx wrote:
Hello Bob,

I'll redo a longer one when I get homm in a few days.
thanks, I'm looking forward to it.

This was with reference to Koopman's suggestion and my response
on April 19:

Ray Koopman wrote:
A simple approximation to the standard normal cdf, with absolute
error < .000158, is 1/(1 + Exp[-.496937*x*Sqrt[x^2 + 10.28]]).


RF> I have a much longer follow-up which was eaten by the ISP of the
RF> Hilton in Manhattan NY in which I am staying when I tried to post
it.

RF> Here's a very short version. I'll redo a longer one when I get
homm

the same misspelling of "homm" was kept for proper id. :-)

RF> The approximations with absolute max error bounds on the entire
RF> range of the r.v. are called Tchebychev (or whatever way you
RF> spell it -- it's the same Cheby) approximations.

RF> For the standard normal, you can find simpler and better ones in
RF> the National Bureau of Standards Handbook of Mathematical
RF> Functions (1964 and later editions), as well as other exact and
RF> approximations for various continuous distributions.

I notice, three days later, on April 22, Wisenmius gave this

http://www.math.sfu.ca/~cbm/aands/page_932.htm from the above

reference. In particular, 26.2.16-20 are some of the simpler
Chebychev approximations with absolute bounds.

He also gave these from the same book:

in pages 262 and 263.
<http://www.math.sfu.ca/~cbm/aands/page_262.htm>
<http://www.math.sfu.ca/~cbm/aands/page_263.htm>

but those are for exact computations rather than for SIMPLE
approximations as you sought.


That was why I said,

Sascha, you're quite welcome, especially because you might have
found a reason to exhume some of my my journal articles on those
topics when I thought they are already obsolete.

Approximating probabilties are out of style, if not obsolete, today.
I guess because of better/faster computers, right?

By orders of magnitude!

Today, computation time is cheap, very cheap; and fast.

As I menitoned
above, I want the approximations for drawing diagramms in postscript.
The postscript interpreter has to do all the work when displaying the
graph or printing it. I don't think those drivers know how to integrate
functions. The set of commands is rather limited. Thus, approximation
are just what I need.

If you have access to JASA, you may want to take a look at my article
in 1978 (73, 274-283, "A Study of the Accuracies of Some
Approximations for ...".

I now have a copy of that article in front of me. The approximations
I
considered were simple alright, but not sure if they are simple enough
for you postscript interpreter because I don't know what it can or
cannot
do.

So, without a better description of your postscript interpreter needs,
I
can't offer much details than pointing to those in the NBS Handbook.

I did mention that some EXACT formulas are easier to compute than
their approximation counterparts. These were pointed out in my 1978
paper:

For t with 1-4 degrees of freedom, the right-tail probabilities are
exact:

Q(t|1) = 1 - 1/(pi * arctan(t))

Q(t|2) = 1/2 - (1/2) t/sqrt(t^2 + 2)

Q(t|4) = 1/2 - [1 + 2/(t^2+ 4)] *t/sqrt(t^2+4).

but then, you don't come across t with such small degrees of
freedom often in your stat computation.

In short, simple and accurate approximations of continuous
probabilities
are few and far between, and "exact" formulas are no longer prohibitive
in computing cost, so we are back to my original contention,

RF > Approximating probabilties are out of style, if not obsolete,
today.

Here's perhaps an exception, but I almost hate to mention it because
there are so many folks who abuse the Pearson's correlation
coefficient!

If you're good at doing square roots of large numbers in your head
you can estimate the critical value of R for large samples to be
2/sqrt(n) for alpha .05, and win a bar bet or impress some friends


-- Bob.

.

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