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I'm trying to understand this answer: https://stats.stackexchange.com/a/151969/25186 which details Fisher's geometric derivation of the t-distribution.

There are some loose ends in my understand I'm trying to wrap up. For reference, we'll need the figure presented there.

Figure

In the description of the figure, the answerer says:

The figure depicts the upper hemisphere (with $Z \ge 0$) of $S^s$ in $\mathbb{R}^{s+1}$. The crossed axes span the $W$-hyperplane. The black dots are part of a random sample of a $s+1$-variate standard Normal distribution: they are the values projecting to a constant given latitude $\theta$, shown as the yellow band. The density of these dots is proportional to the $s-1$-dimensional volume of that band, which itself is an $S^{s-1}$ of radius $\theta$.

So far so good.

The cone over that band is drawn to terminate at a height of $\tan \theta$. Up to a factor of $\sqrt{s}$, the Student t distribution with $s$ degrees of freedom is the distribution of this height as weighted by the measure of the yellow band upon normalizing the area of the unit sphere $S^s$ to unity.

  1. Why is the cone drawn to terminate at $\tan \theta$? What significance does this have? Aren't all points projected on this space at whatever height eligible for projection to the sphere?

  2. It seems they use the notation $S^s$ to refer to the surface of an $s+1$ dimensional sphere. Is that correct? If so, I couldn't find other references to this notation anywhere. Can someone point to any links?

Next, about the normalizing constant:

Incidentally, the normalizing constant must be $1/\sqrt{s}$ (as previously mentioned) times the relative volumes of the spheres,

  1. Why is this?
ryu576
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    "$S^s$" is standard mathematical notation for the $s$-sphere in $\mathbb R^{s+1}.$ I am astonished you cannot find this. It's in the Wikipedia article on spheres of higher dimension. As far as terminating the image of the cone goes, how would you propose drawing the full cone which extends infinitely far upwards? The factor of $1/\sqrt s$ was derived in step 7. – whuber Jan 17 '23 at 17:59
  • For the image of the cone, why did you use $\tan \theta$ specifically? What's the significance of that? Why not any other height? – ryu576 Jan 17 '23 at 19:42
  • The factor $\frac{1}{\sqrt{s}}$ is fine and probably a consequence of how $t$ was defined, I was having trouble with why its the ratio of volumes of the spheres. Is it because we normalize the smaller unit sphere to have a volume of $1$ unit? – ryu576 Jan 17 '23 at 19:44
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    At this height, the diameter of the cone equals the diameter of the sphere. That yields a decent figure. The ratio of volumes indeed arises because of the normalization. Although only the ratio matters, note that a "unit sphere," by definition, has a radius of $1,$ not a volume of $1.$ – whuber Jan 17 '23 at 19:52

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