Is population growth rate t-distributed

What happens when you make measurements over different sizes of time interval?

random variation across years (see also Pielou)

• years are variable.
• all plots get the same r each year
• plots have different starting amounts
• maybe even different starting years!

give each a random starting size check after an sampled number of years

That’s not quite what I expected – why isn’t it flatter in the sample?

so maybe the right way to think about it is as a compound distribution . Just as the gamma-poisson happens, when you have a poisson count but the rate parameter varies – here you have a normal value but the variance parameter varies. and we just don’t know which is which, and the result is a t-distribution

just as in the neg-binomial case, there is another solution – fine control, with random effects for sites, species, years, etc.

but i still find it unsatisfying that the nu parameter, the degrees of freedon, doesn’t factor into this model in the same way!

is the inverse gamma a distribution of sample standard deviations?

The mean of the inverse gamma rows with $\nu$ like this:

$$\mu = \frac{\beta}{\alpha + 1} = \frac{\nu\sigma^2/2}{\nu/2 + 1} = \frac{\nu\sigma^2}{\nu + 2}$$

Of course that makes sense – why would the sample standard deviation be biased in any way?

inspired by here: https://www.sumsar.net/blog/2013/12/t-as-a-mixture-of-normals/

But is it related to sample size?

oh wow it just the sampling distribution of the mean according to the central limit theorem

it looks like – mayyybe this is it. definitely something to think about more!