Quiz 3 - Solutions

Quiz 3 - Solutions - STATISTICS 321 Spring 2009 Quiz 3(20...

This preview shows pages 1–2. Sign up to view the full content.

STATISTICS 321 – Spring 2009 Name: Quiz 3 (20 points) – This quiz has 2 sides. 1. Let X 1 ,X 2 ,...,X n be a random sample from a distribution with unknown mean μ and variance σ 2 = 2 . (a) We propose 2 point estimators ˆ μ 1 and ˆ μ 2 to estimate μ , where ˆ μ 1 = X 1 + X 2 + ... + X n n and ˆ μ 2 = X 1 + X 2 2 Which of the two: ˆ μ 1 or ˆ μ 2 will have the smaller Mean Squared Error (MSE), and why? (Recall that MSE = Bias 2 + Variance. Also, you don’t need to do any calculations. ) Bias(ˆ μ 1 ) = E μ 1 ) - μ = ( μ + ... + μ ) /n - μ = 0 Bias(ˆ μ 2 ) = E μ 2 ) - μ = ( μ + μ ) / 2 - μ = 0 Var(ˆ μ 1 ) = Var( X 1 + X 2 + ... + X n ) /n 2 = 2 n 2 = σ 2 n = 2 /n Var(ˆ μ 2 ) = Var( X 1 + X 2 ) / 2 2 = 2 σ 2 2 2 = σ 2 2 = 2 / 2 = 1 Thus, ˆ μ 1 has a smaller Mean Squared Error (MSE), provided n > 2 (b) When will the Mean Squared Error of a point estimator be negative? Why? Mean Squared Error (MSE) is never negative, since it is a sum of 2 non–negative terms.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

Page1 / 2

Quiz 3 - Solutions - STATISTICS 321 Spring 2009 Quiz 3(20...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online