# Did Fisher consider a joint fiducial distribution for the Gaussian model?

Consider the Gaussian model $$y_i \sim_{\text{iid}} \mathcal{N}(\mu,\sigma^2)$$, $$i = 1, \ldots, n$$, with unknown mean $$\mu$$ and unknown standard deviation $$\sigma$$.

The random variable $$t = \tfrac{\overline{y}-\mu}{\text{sd}(y)}$$ has a Student-$$t$$ distribution with degrees of freedom parameter $$n-1$$. This distribution is free of unknown parameters.

One has $$\color{blue}{\boxed{\mu = \overline{y} - \text{sd}(y) \cdot t}}$$ The somehow esoteric fiducial argument consists in "switching the roles of the data and the parameters", and from the above boxed formula, the fiducial distribution of $$\mu$$ is the distribution of $$\overline{y} - \text{sd}(y) \cdot t$$ considering $$\overline{y}$$ and $$\text{sd}(y)$$ as fixed constants and $$t \sim \text{Student}_{n-1}$$ as the random variable.

With a similar approach, the fiducial distribution of $$\sigma^2$$ is the distribution of $$(n-1)\tfrac{\text{sd}(y)}{\chi}$$ where $$\chi \sim \text{Chi}^2_{n-1}$$ is the random variable.

These are, I think (not a master in this topic), the fiducial distributions considered by Fisher.

My question is: did Fisher consider a joint fiducial distribution of $$(\mu,\sigma)$$, and if yes, what is this distribution?

• Does $\text{sd}(y).t$ mean $\text{sd}(y)\cdot t$? Nov 28 '20 at 10:05
• @RichardHardy It means multiplication. The traditional notation is \cdot? I forgot that. Nov 28 '20 at 11:33
• OK. I do not think I have ever seen $.$ as the symbol for multiplication. $\cdot$ and (less frequently seen) $\times$ seem to be the standard. Nov 28 '20 at 13:37
• Ok, I edited my post. Nov 30 '20 at 15:47

I see. One has the bivariate pivot $$(z, w) = \left(\frac{\sqrt{n}(\overline{y}-\mu)}{\sigma}, \frac{\text{sd}^2(y)}{\sigma^2} \right) \sim \mathcal{N}(0,1) \otimes \frac{\chi^2_{n-1}}{n-1}.$$ By "inverting", $$\mu = \overline{y} - \frac{\text{sd}(y)\cdot z}{\sqrt{n w}} \qquad \sigma^2 = \frac{\text{sd}^2(y)}{w}.$$ This leads to the joint fiducial distribution of $$(\mu,\sigma)$$.