Let X1,...,Xn be an independent and identically distributed (IID) sample from Np(μ,Σ). Denote its sample mean and sample covariance matrix as X̄n = (1/n) Σ Xi and Sn = (1/n) Σ (Xi - X̄n)(Xi - X̄n)ᵀ, respectively. Let X* be a random variable that is independent of X1,...,Xn.
(a) Show that (X* - X̄n) ~ Np(0, (n+1)Σ).
Hint: You can use Theorem 1 below without giving a proof.
Theorem 1: Suppose that Y ~ Np(0,Σ), U ~ Wk(Σ), where Wk(Σ) is a Wishart distribution with k degrees of freedom and positive-definite covariance matrix Σ ∈ â„^(p×p), and Y is independent of U. Then
p
p(k - p + 1)
Fp,k-p+1 denotes a random variable having the F distribution with p and k - p + 1 degrees of freedom.
