Correct answer: (C) Infinitely many unbiased estimators of θ² are functions of a minimal sufficient statistic.
Explanation
A minimal sufficient statistic is(ΣXi,ΣXi²). Both U=(ΣXi)²/[n(n+2)] and V=ΣXi²/(3n) are unbiased forθ², since E(ΣXi)²=n(n+2)θ² and EΣXi²=3nθ². They are distinct for n≥2, so every fixed linear combination aU+(1−a)V supplies an unbiased estimator. Therefore C; adding the unbiased sample mean to V also estimatesθ²+θ.
More Sufficiency and Completeness questions
GATE 2020 ST Q33– X₁,…,Xn have density e(−(x−2θ)) for x>2θ, zero otherwise, with θ>0 and n≥2. Let X(1)=min…
GATE 2020 ST Q44– X₁,…,Xn,n≥2, have density θ⁻¹exp[−(x−θ)/θ] for x>θ, θ>0. Which statement is true?
GATE 2026 ST Q19– A sample X₁,X₂ has density f α(x)=αx(α−1)e(−xα),x>0, zero otherwise,α>0 unknown. Let X…
GATE 2026 ST Q43– An iid sample of size 3 has P θ(X i=1)=θ,P θ(X i=2)=1−θ,0<θ<1. Which statistic is not…
GATE 2025 ST Q18– T is complete sufficient for a family of distributions, U is sufficient, and T≥0 almost…
GATE 2025 ST Q42– An iid sample n≥2 is N(−θ,θ), with θ>0 unknown. Which holds?