Characterization of negative binomial extended to the reals

[boxtown]

See 6b2c726#diff-8f4094da35079696cbcdd4a269333ad0R101 for a little more context.

We're currently allowing the negative binomial distribution to accept positive real r values as it seems to be a common use case. However, this causes the CDF to behave continuously rather than like in other discrete distributions which breaks our internal check_discrete_distribution test.

I'm not familiar enough with this particular distribution to provide much guidance but there are a couple options:

1. simply treat the distribution as continuous, allowing real value inputs for the "probability mass functions"
2. Create a special case test for negative binomial (not sure how often this case will arise in practice)
3. Create a special case distribution type

Option 1 seems a vastly superior choice but possibly semantically wrong (although also possibly to the point of pedantry)

[dsaxton]

Hi @boxtown, what was the original reasoning behind making r real-valued (i.e., what were the use cases)? I had only ever seen r interpreted as a count. I'm assuming a similar treatment isn't given to say n for the ordinary binomial distribution?

[WarrenWeckesser]

statrs is still version 0.x, so wouldn't it be acceptable to change the type of r to an integer? I agree with @dsaxton; every reference that I've seen refers to the negative binomial distribution as a discrete distribution, so allowing that parameter to be continuous is unusual.

If it turns out later that there is sufficient demand for a continuous version, then perhaps a new distribution could be added then, with a name such as ContinuousNegativeBinomial, that makes it explicit that it is not the traditional discrete distribution. This would avoid the need to shoehorn an inherently discrete distribution into a continuous API, or vice versa.

[WarrenWeckesser]

On the other hand, r is just a parameter of the distribution; whether that parameter is float or integer shouldn't change the "discreteness" of the distribution.

However, this causes the CDF to behave continuously rather than like in other discrete distributions which breaks our internal check_discrete_distribution test.

Could you explain what goes wrong? How does r being a float make the CDF "behave continuously"? The two tests at the end of negative_binomial.rs that use check_discrete_distribution (currently commented out) pass when I uncomment them. (I'll experiment further, but some details on the nature of the problem might be helpful.)