Solution
Doubling the Mean Versus the Maximum
Show the problem again
A sample of n = 10 independent observations is drawn from the uniform distribution on [0, θ]. Consider two unbiased estimators of θ: A = 2X̄ (twice the sample mean) and B = ((n + 1)/n) · max(X₁, ..., Xₙ). What is the ratio Var(A)/Var(B)?
Worked solution
For A: Var(Xᵢ) = θ²/12, so Var(X̄) = θ²/(12n) and Var(A) = 4θ²/(12n) = θ²/(3n). For B: the maximum M has density n mⁿ⁻¹/θⁿ, E[M] = nθ/(n + 1), E[M²] = nθ²/(n + 2), so Var(M) = nθ²/((n + 1)²(n + 2)). Therefore Var(B) = ((n + 1)/n)² Var(M) = θ²/(n(n + 2)). The ratio is n(n + 2)/(3n) = (n + 2)/3, which for n = 10 is 4. The variance of A shrinks like 1/n while that of B shrinks like 1/n², because the maximum is sufficient and A ignores that information. The answer is 4.
Source: Classic comparison of estimators for the endpoint of a uniform distribution; textbook material. Statement written for AxiomIQ.