Let Y₁ < Y₂ < · · · < Y_n be the order statistics of a random sample from a N(θ, σ²) distribution, where σ² is fixed but arbitrary. Then ‾Y = ‾X is a complete sufficient statistic for θ. Consider another estimator T of θ, such as T = (Y_i + Y_n+1−i)/2, for i = 1, 2, . . . , [n/2], or T could be any weighted average of these latter statistics.
(a) Argue that T − ‾X and ‾X are independent random variables.
(b) Show that Var(T) = Var(‾X) + Var(T − ‾X).
(c) Since we know Var(‾X) = σ²/n, it might be more efficient to estimate Var(T )by estimating the Var(T − ‾X) by Monte Carlo methods rather than doing that with Var(T) directly, because Var(T) ≥ Var(T −‾X). This is often called the Monte Carlo Swindle.