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GATE 2023 ST – Question 25

Multivariate Analysis · Multivariate Normal and Wishart Distributions · 1 mark · Multiple choice

X₁,…,X_n,Y₁,…,Y_n are iid N_p(μ,Σ),Σ nonsingular,p>1,n>1. Write their means X̄,Ȳ. Which claim is true?

  1. For some c>0,c(X̄−μ)ᵀΣ⁻¹(X̄−μ)∼χ²_p.
  2. For some c>0,c(X̄−Ȳ)ᵀΣ⁻¹(X̄−Ȳ)∼χ²_(p−1).
  3. For some c>0,cΣ_i(X_i−X̄)ᵀΣ⁻¹(X_i−X̄)∼χ²_p.
  4. For some c>0,cΣ_i(X_i−Y_i−X̄+Ȳ)ᵀΣ⁻¹(X_i−Y_i−X̄+Ȳ)∼χ²_p.

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Correct answer: (A) For some c>0,c(X̄−μ)ᵀΣ⁻¹(X̄−μ)∼χ²_p.

Explanation

Whitening √n(X̄−μ) gives p independent standard normals,so A holds with c=n. The difference-of-means quadratic also has p degrees of freedom,not p−1. The centered sums in C and D have p(n−1) degrees of freedom (D after scaling by 1/2),and a scalar cannot generally change that shape. The official answer is **A**. At the explicitly permitted special case n=2,C and D also have p degrees of freedom; the single-choice key implicitly treats general n rather than this exception.

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