Find out which exponential distribution the data belongs to

Question

I'm doing a GLM homework, and I'm stuck with the following problem:

Suppose that data ($Y_i$; $\mathbf{X}_i$); $i = 1, . . . , n$ are observed, where $\mathbf{X}_i$ is a p-dimensional vector for predictors of patient i and $Y_1 , . . . , Y_n$ are given by the following table:

Y	0	1	2	3	4	5	6	7	8
Number of Subjects	35	26	12	9	7	4	4	2	1

Given the nature of the response variable, what (exponential family) distribution is most appropriate for Y?

I just don't know where to start; I observed that the number of subjects is decreasing as Y increases, and I computed the sample mean = 1.71 and sample variance = 3.8443 (which doesn't seem to be helpful), but how can they help me determine which family is Y from? Could someone give me some hints? Thanks

Answer 1

This question seems to indicate that the question-writer has a misconception about GLMs. In particular, GLMs deal with the conditional distribution of the outcome; however, you are being asked this question about the marginal distribution.

With that out of the way, three possibilities come to mind.

1. Poisson

2. Negative Binomial

3. Binomial

Poisson is kind of the first thought for count data, meaning that the Poisson distribution gets on the list. However, the Poisson distribution is restrictive in that the mean and variance are equal. Negative binomial is an method to loosen that restriction. Depending on the level of sophistication of the class, I could believe either of these to be the full-credit answer: Poisson if the assignment just wants you to identify this as a count model, and negative binomial if the assignment wants you to identify this as a count model but also identify Poisson as restrictive.

However, my guess is binomial, and my justification is that there seems to be a cap on how high the count is. A binomial distribution with eight trials has its maximum value capped at eight, which is your maximum, while Poisson and negative binomial have no upper bound. If the conditional distribution were Poisson or negative binomial, I might expect to see some kind of outlier-type of point where there is a stray high value like twelve, which a binomial distribution on eight trials does not allow.

Nonetheless, maybe the upper bound of eight is just because of the particular feature values, and with different feature values, counts like twelve would be possible (which would not be true for a binomial distribution with eight trials). Just looking at the marginal distribution of $Y$, we simply cannot distinguish between what happens because of the feature values and what happens because of the conditional distribution.