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GATE 2024 ST – Question 55

Estimation · Unbiased and Minimum Variance Estimation · 2 marks · Multiple select

For iid Bernoulli(p) X₁,X₂,X₃, define T₁(X_i,X_j,X_k)=X_i−X_j(1−X_k) and T₂(X_i,X_j,X_k)=(X_iX_j+X_jX_k)/2 for distinct indices. Which hold?

  1. Cyclic T₁ versions have equal distributions but differ for some sample realizations.
  2. T₂(X₁,X₂,X₃) and T₂(X₃,X₂,X₁) are unbiased for p².
  3. Cyclic T₁ versions are unbiased for p² and equal on every realization.
  4. T₂(x₁,x₂,x₃)=T₂(x₃,x₂,x₁) for all realizations.

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Show answer and explanation

Correct answer: (A) Cyclic T₁ versions have equal distributions but differ for some sample realizations.; (B) T₂(X₁,X₂,X₃) and T₂(X₃,X₂,X₁) are unbiased for p².; (D) T₂(x₁,x₂,x₃)=T₂(x₃,x₂,x₁) for all realizations.

Explanation

IID exchangeability gives equal distributions of permuted T₁, but at (1,0,0), T₁(1,0,0)=1 and T₁(0,0,1)=0, proving A and refuting C’s pointwise equality. E T₁=p−p(1−p)=p². Also E T₂=(p²+p²)/2=p², proving B, and T₂=X_j(X_i+X_k)/2 is unchanged by exchanging its first and third arguments, proving D. **A,B,D**.

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