The bull’s-eye can be taken to represent the true value of the parameter, the rifle the estimator, and each shot a particular estimate.” Use this analogy to discuss small and large sample properties of estimators. How do you think the author approached the n → ∞ condition? (Dependent on your view of the world, feel free to substitute guns with bow and arrow, or missile.)
What will be an ideal response?
Answer:
Unbiasedness: the shots produce a scatter, but the center of the scatter is the bulls-eye. If the riffle produces a scatter of shots that is centered on another point, then the gun is biased.
Efficiency: Requires comparison with other unbiased guns. Looking at the scatters produced by the shots, the smallest scatter is the one from the efficient gun.
BLUE: Remove all guns which are not linear and/or biased. The gun among these remaining ones which produces the smallest scatter is the BLUE gun.
Consistency: n → ∞ is the condition as you march towards the bulls-eye, i.e., the distance becomes shorter as n → ∞. A shot fired from a consistent gun hits the bull’s-eye with increasing probability as you get closer to the bull’s-eye. Or, perhaps even better, you might want to substitute “being very close to the bull’s-eye” for “hitting the bull’s-eye.”