Tests for normed vector

I have many vectors $a_i\in\mathbb{R}^p:||a||=1,1\leq i \leq n$. I would like to test whether the $a_i$'s are randomly spread out on $S^{p-1}$ (the $p$ variate unit circle). Can anyone point to a test for that?

• How is what you're wanting to do different from a goodness-of-fit test that your data come from the uniform measure on the $S^{p-1}$ sphere? – cardinal Jul 1 '12 at 23:59
• Could work. How would you test this in practice? – user603 Jul 2 '12 at 7:41

You could transform your data points to spherical coordinates so that you get $p-1$ angles. Your null hypothesis is equivalent to the fact that those angles are independent and uniformly distributed. So you can do a goodness of fit test.
Now there is the complication that you have a $p-1$-dimensional distribution. This paper shows a multidimensional version of the Smirnov statistic that may come in handy. Otherwise, another idea that comes to my mind woud be to scale the angles to $(0,1)$, apply the inverse erf function and do a test for multivariate normality.
Erratum: it is not true that the angles will be uniform as shown in this answer. One angle is uniform between $0$ and $2\pi$, and the cosine of the others is uniform between $-1$ and $1$.
• Depends on your parametrization. For example in 3D, spherical coordinates are usually given as an angle between $-\pi$ and $\pi$ (longitude) and another one between $-\pi/2$ and $\pi/2$ (latitude). For longitude add $\pi$ and divide by $2\pi$ and for latitude add $\pi/2$ and divide by $\pi$. – gui11aume Jun 3 '12 at 10:42