Mathematics · Multivariate Analysis
Suppose that X1, …, Xn, Xn+1 is a random sample of size n + 1, where p > 2 and n > p + 3, from a multivariate normal population, Np(μ, Σ); μ ∈ &mathbb;Rp and Σ > 0. Let X̄n = 1n ∑i=1n Xi and (n − 1)S = ∑i=1n (Xi − X̄n)(Xi − X̄n)T. Which of the following statements is correct?
Verified Solution4 OptionsUpdated: October 2026👁️1 View
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- 1(X̄n − Xn+1)T Σ−1 (X̄n − Xn+1) ∼ n + 1n χp2✓
- 2E(S−1) = n − 1n − p − 2 Σ−1
- 3E(X̄nT S X̄n) = trace(Σ2n) + μTΣμ
- 4(X̄n − Xn+1)T S−1 (X̄n − Xn+1) ∼ p(n2 − 1)n(n − p) Fp, n−p
Correct answer
(X̄n − Xn+1)T Σ−1 (X̄n − Xn+1) ∼ n + 1n χp2
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