X₁,…,X₁₀ are iid N₃(μ,Σ),μ and nonsingular Σ unknown. Split into groups of five. Let X̄₁,X̄₂ be the group means and S₁,S₂ their covariance matrices with divisor 4. Which statement is NOT true? Write Q=(X̄₁−μ)ᵀS₁⁻¹(X̄₁−μ).
(5/6)Q∼F_(3,2)
6/(5Q)∼F_(2,3)
4(S₁+S₂) is Wishart of order 3 with 8 degrees of freedom.
5(S₁+S₂) is Wishart of order 3 with 10 degrees of freedom.
Correct answer: (D) 5(S₁+S₂) is Wishart of order 3 with 10 degrees of freedom.
Explanation
Hotelling’s formula with n=5,p=3 gives [(5−3)/(3·4)]·5Q=(5/6)Q∼F₃,₂; reciprocation proves B. Each 4S_j∼Wishart₃(Σ,4),independently,so their sum is Wishart₃(Σ,8). Rescaling cannot turn the degrees of freedom into 10. **D**.
More Hotelling Tests and Correlation questions
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…
GATE 2021 ST Q19– Independent vectors X₁,X₂,X₃∼N₄(0,Σ₁) and Y₁,…,Y₄∼N₄(0,Σ₂) have positive definite…
GATE 2021 ST Q33– (Y,X₁,X₂) has mean (5,2,0)ᵀ and covariance rows (10,.5,−.5),(.5,7,1.5),(−.5,1.5,2). Find…
GATE 2021 ST Q34– Three iid observations from a bivariate normal population with unknown mean μ and positive…
GATE 2021 ST Q53– Y∼N₈(0,I₈). The quadratic forms YᵀΣ₁Y and YᵀΣ₂Y are independent central chi-square…