X₁,…,Xn are iid Np(0,Σ), Σ positive definite and1≤p≤n−1. Define X̄=n⁻¹ΣXi and $(n-1)S=\sum(X_i-\bar X)(X_i-\bar X)^T$. Find the distribution of T²=nX̄ᵀS⁻¹X̄.
Hotelling’s one-sample statistic satisfies[(n−p)/(p(n−1))]T²∼Fp,n−p. Rearranging gives T²∼[p(n−1)/(n−p)]Fp,n−p, C. The sample covariance contributes the n−1 scaling and denominator degrees of freedom n−p.
More Hotelling Tests and Correlation questions
GATE 2020 ST Q43– Two bivariate-normal populations have means(0,0),(1,1), common covariance…
GATE 2020 ST Q58– A centred trivariate normal has variances1 and all pairwise correlations1/3. Find partial…
GATE 2025 ST Q64– X N 3(0,pmatrix1&.4&0\\.4&1&.6\\0&.6&1pmatrix). Find the partial correlation of X₁,X₂…
GATE 2023 ST Q48– X₁,…,X₁₀ are iid N₃(μ,Σ),μ and nonsingular Σ unknown. Split into groups of five. Let…
GATE 2022 ST Q17– X₁,…,X₂₀ are iid N₆(μ,Σ),Σ nonsingular,μ and Σ unknown. Let X̄=ΣX i/20 and S=Σ(X i−X̄)(X…
GATE 2022 ST Q39– X∼N₃(μ,Σ),μ=(2,−3,2)ᵀ,Σ has rows (4,−1,1),(−1,2,a),(1,a,2). The partial correlation of X₂…