The [tau-distribution](https://en.wikipedia.org/wiki/Studentized_residual#Distribution) is typically defined in terms of the [Student's t-distribution](https://en.wikipedia.org/wiki/Student%27s_t-distribution) as follows:
$$\tau_\nu \sim \sqrt{\frac{\nu \, t_{\nu - 1}^2}{\nu - 1 + t_{\nu - 1}^2}}.$$
I would be interested to compute the mean and variance of this random variable.
Because it seems daunting to directly compute expectations from this formula, I started looking for a Probability Density Function (PDF) for this distribution.
Unfortunately, I was not very successful because everyone seems to use the formulation above.
That made me wonder whether this formula (implicitly) defines the PDF and I am too stupid to see it, or if there simply is no proper density function.
Hence my question: what is the PDF for the tau distribution?
mr Tsjolder
·
2023-09-26T10:00:53Z (8 months ago)