Re: [boost] [math/staticstics/design] How besttonamestatisticalfunctions?

From: Topher Cooper (topher_at_[hidden])
Date: 2006-07-12 09:11:22

At 05:11 AM 7/12/2006, Paul A Bristow wrote:
> >>  In other words, 1 - P. Right? One response is why do you
> >>  need to define
> >>  it, given how easy it is to get from the cumulative density
> >> function?
> > Perhaps not really needed? Is there an accuracy reason for both?
>
>| It depends how accurate you want to be: calculating 1-P incurs
>cancellation
>| error if P is very near 1, where as for most (all?) distributions we can
>| calculate Q directly without the subraction from unity.
>
>| I think the "Boostified" name would be in all lower case: students_t or
>whatever.
>
>Agree with this.

This brings to mind another function that, though easily derived 
would be good to have to allow internal computations less subject to 
round-off error.  This is a two parameter function that is the 
cumulative probability between a lower and an upper 
bound.  Mathematically this can always be computed as "CDF(x[ub]) - 
CDF(x[lb])" (read the square brackets as mathematical subscript 
notation) but numerically with very small intervals, you can easily 
end up with 0 when you want something close to 
"PDF((x[ub]+x[lb])/2)*(x[ub]-x[lb])".  You don't need to make any 
general guarantee about precision and so could do initial 
implementations as the difference of the cumulative functions, but 
then go back and do better for individual distributions.

I don't know any standard term for this off the top of my head.  I 
would suggest just using a two argument version of whatever is 
decided on for the cumulative distribution.  So, using my suggested 
function name:

          standard_normal.cdf(-1.0, 1.0)

would return the probability that a random variate with a normal 
distribution is within one standard deviation of the mean.

The only problem I have with this is that if we look at the one 
parameter version as being the two parameter version with one 
parameter defaulted its the *first* parameter that is defaulted since:

          dist.cdf(x) = dsit.cdf(-INFINITY, x)

That would suggest using the complementary cdf instead, but that 
seems a lot less natural.

Topher