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GATE 2020 ST – Question 47

Multivariate Analysis · Multivariate Normal and Wishart Distributions · 2 marks · Multiple choice

Independent X₁,X₂,X₃∼N₄(0,Σ), Σ positive definite. Let X be the3×4 matrix whose rows are Xᵢᵀ. Wm(n,V) denotes order-m Wishart with n degrees of freedom. Which is true?

  1. $\Sigma^{-1/2}X^TX\Sigma^{-1/2}\sim W_4(3,I_4)$
  2. $\Sigma^{-1/2}X^TX\Sigma^{-1/2}\sim W_3(4,I_3)$
  3. $\operatorname{tr}(X\Sigma^{-1}X^T)\sim\chi_4^2$
  4. $X^TX\sim W_3(4,\Sigma)$

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Correct answer: (A) $\Sigma^{-1/2}X^TX\Sigma^{-1/2}\sim W_4(3,I_4)$

Explanation

Whitening each four-vector gives independent N₄(0,I₄) rows. Their outer-product sum isΣ⁻¹/²XᵀXΣ⁻¹/², an order4 Wishart with3 degrees of freedom and identity scale. Thus A. Its trace has12, not4, chi-square degrees of freedom.

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