To obtain the distribution of s, unconditioned on λ, we see that we can write s as where D = diag(λ 1, λ 2,… λm). After an orthogonal transformation where X is our Wishart matrix and b is uniform (isotropic) on the unit sphere. By properties of the Gaussian distribution, the distribution of s as given by Equation 20 is independent of b. We choose b to be (1,0,0,…). It follows that s/σ 2 is distributed as a variate so that s = 2σ 2 G where G is Γ(n/2)-distributed. Thus, Comparing Equations 17 and 21, this proves: and comparing Equations 19 and 22, we find: From Equations 23 and 24: so that a natural estimator for n is: We then obtain as an estimate for σ: If we set: then Equation 25 simplifies to: This completes the proof of Theorem 1.
