Question

Let X₁, X₂, X₃, and X₄ be a random sample of observations from a population with mean μ and variance σ². The observations are independent because they were randomly drawn. Consider the following two point estimators of the population mean μ:

₁ = 0.10 X₁ + 0.40 X₂ + 0.40 X₃ + 0.10 X₄ and
₂ = 0.20 X₁ + 0.30 X₂ + 0.30 X₃ + 0.20 X₄

Which of the following statements is true?

HINT: Use the definition of an unbiased estimator and the expected value rules to figure this question out. See example in Review notes.

• A.

  ₁ is biased, but ₂ is an unbiased estimator of μ.

• B.

  Both ₁ and ₂ are unbiased estimators of μ.

• C.

  ₁ is unbiased, but ₂ is a biased estimator of μ.

• D.

  Both ₁ and ₂ are biased estimators of μ.

Answer 1

Solution

D. Both 1 and 2 are unbiased estimators of μ

E(1) = E(0.1X1+0.4X2+0.4X3+0.1X4) = 0.1 μ+0.4 μ+0.4 μ+0.1 μ =  μ

E(2) = E(0.2X1+0.3X2+0.3X3+0.2X4) = 0.2 μ+ 0.3 μ+0.3 μ+0.0 μ =  μ

Therefore both the estimators are unbiased