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Sébastien Brisard commented on MATH692:

Hi Christian,
I've started looking into this issue. As I said, you will have to be patient with me ;).
I can see there already is a default implementation of {{AbstractContinuousDistribution.inverseCumulativeProbability}}.
So what exactly would you like me to do? Is this implementation fragile? Would you like me
to improve robustness? Provide full testing?
I think there might be issues when the PDF falls down to zero in a range (in which case the
cum exhibits a plateau). The returned value might differ from the mathematical definition
you proposed. Is this what you want me to work on? Have you already identified other issues?
Best regards,
Sébastien
> Cumulative probability and inverse cumulative probability inconsistencies
> 
>
> Key: MATH692
> URL: https://issues.apache.org/jira/browse/MATH692
> Project: Commons Math
> Issue Type: Bug
> Affects Versions: 1.0, 1.1, 1.2, 1.3, 2.0, 2.1, 2.2, 2.2.1, 3.0
> Reporter: Christian Winter
> Priority: Minor
> Fix For: 3.0
>
>
> There are some inconsistencies in the documentation and implementation of functions regarding
cumulative probabilities and inverse cumulative probabilities. More precisely, '<' and
'<=' are not used in a consistent way.
> Besides I would move the function inverseCumulativeProbability(double) to the interface
Distribution. A true inverse of the distribution function does neither exist for Distribution
nor for ContinuosDistribution. Thus we need to define the inverse in terms of quantiles anyway,
and this can already be done for Distribution.
> On the whole I would declare the (inverse) cumulative probability functions in the basic
distribution interfaces as follows:
> Distribution:
>  cumulativeProbability(double x): returns P(X <= x)
>  cumulativeProbability(double x0, double x1): returns P(x0 < X <= x1) [see also
1)]
>  inverseCumulativeProbability(double p):
> returns the quantile function inf{x in R  P(X<=x) >= p} [see also 2), 3), and
4)]
> 1) An aternative definition could be P(x0 <= X <= x1). But this requires to put
the function probability(double x) or another cumulative probability function into the interface
Distribution in order be able to calculate P(x0 <= X <= x1) in AbstractDistribution.
> 2) This definition is stricter than the definition in ContinuousDistribution, because
the definition there does not specify what to do if there are multiple x satisfying P(X<=x)
= p.
> 3) A modification could be defined for p=0: Returning sup{x in R  P(X<=x) = 0} would
yield the infimum of the distribution's support instead of a mandatory infinity.
> 4) This affects issue MATH540. I'd prefere the definition from above for the following
reasons:
>  This definition simplifies inverse transform sampling (as mentioned in the other issue).
>  It is the standard textbook definition for the quantile function.
>  For integer distributions it has the advantage that the result doesn't change when
switching to "x in Z", i.e. the result is independent of considering the intergers as sole
set or as part of the reals.
> ContinuousDistribution:
> nothing to be added regarding (inverse) cumulative probability functions
> IntegerDistribution:
>  cumulativeProbability(int x): returns P(X <= x)
>  cumulativeProbability(int x0, int x1): returns P(x0 < X <= x1) [see also 1) above]

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