Re: [boost] [math/staticstics/design] Howbesttonamestatisticalfunctions?

From: John Maddock (john_at_[hidden])
Date: 2006-07-12 12:35:58

Paul A Bristow wrote:
>>  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])".
>
> An interesting suggestion.
>
> John Maddock has been muttering about using Boost.Interval with these
> functions.
> It's on his TODO list allegedly ;-)

Yeh, but it's a long list ;-)

> Would this help with the "CDF(x[ub]) - CDF(x[lb])"?
>
> And/or allow one to produce "PDF((x[ub]+x[lb])/2)*(x[ub]-x[lb])"
> using the density/mass/distribution?

No it's entirely different functionality.

Returning an interval guards against rounding error, or function 
sensitivity, leading you towards erroneous conclusions.

Calculating a probablity over an interval is the same as integrating the 
distribution function from x to y rather than -INF to x.  Doing it properly 
requires for example a four argument incomplete beta:

ibeta(a, b, x, y); // incomplete beta integral from x to y.

and a three argument incomplete gamma:

gamma_Q(a, x, y);  // incomplete gamma integral from x to y.

However, I don't know how to implement those:  I did have a very quick look 
into this when I did the incomplete gamma and didn't find any useful 
literature, so if anyone has any leads I'm all ears.  My inclination is to 
leave stuff like this for version 2 (or 3!) though :-)

John.