Find an unbiased estimator with lower variance than given

Question

Let $X_1$ and $X_2$ be two independent observations from a distribution $F$ with $V ar(X_i) = σ^2 > 0$ and $0 < V ar(X_i^2 ) < ∞$. Prove that $T_1 = X_1^2 − X_1X_2$ is unbiased for $σ^2$. Find another unbiased estimator for $σ^2$ having variance strictly less than that of $T_1.$

first part i could prove but second part not being able to find a complete sufficient statistic on which i will calculate umvue

Answer 1

Hints:

• You are not being asked for the lowest variance unbiased estimator, just a lower variance unbiased estimator
• Can you show $T_2=X_2^2 − X_1X_2$ has the same distribution as $T_1$?
• Some suitable combination of $T_1$ and $T_2$ might still be unbiased but have a lower variance