# HW2sol - SOLUTIONS FOR HWK 2 Problem A (a) and (b) follow...

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

SOLUTIONS FOR HWK 2 Problem A (a) and (b) follow from theorems proved in class since μ = θ,σ 2 = θ 2 . (d) E ( ˆ Θ) = 1 15 μ + 2 15 μ + 3 15 μ + 1 15 μ + 8 15 μ = μ . Hence ˆ θ is unbiased. (f) ) = ± 1 15 2 θ 2 + ± 2 15 2 θ 2 + ± 3 15 2 θ 2 + ± 1 15 2 θ 2 + ± 8 15 2 θ 2 = 0 . 351 θ 2 . (g) Since n = 5, we have V ar ( ¯ X ) = θ 2 / 5 = 0 . 2 θ 2 . Therefore: re ( ¯ X : ˆ Θ) = 0 . 351 θ 2 0 . 2 θ 2 = 1 . 755 > 1. h) ¯ X is much more eﬃcient than ˆ Θ. Problem B Let r > 0 and let X be a random variable. Assume that E ( | X | r ) < . The moment of order r , μ r def = E ( X r ) (of course, the moment of order 1 μ 1 = μ = E ( X ), is the expectation of X ). Consider a random sample X 1 ,...,X n where X k ’s have the same distribution as X . The empirical moment of order r M r def = 1 n n k =1 X r k . 1) Prove that M r is an unbiased estimator of μ r where r > 0 . E ( M r ) = E ( 1 n n X k =1 X r k ) = 1 n E ( n X k =1 X r k ) = 1 n n X k =1 E ( X r k ) = 1 n n X k =1 μ r = 1 n r = μ r . 2) Prove that the sequence of the empirical moments M ( n ) r = 1 n n k =1 X r k is a strongly consistent sequence of estimators of the moment μ r . We apply the Strong Law of Large Numbers:

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.

## This note was uploaded on 02/24/2009 for the course STAT 415 taught by Professor Senturk,damla during the Spring '06 term at Pennsylvania State University, University Park.

### Page1 / 4

HW2sol - SOLUTIONS FOR HWK 2 Problem A (a) and (b) follow...

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

View Full Document
Ask a homework question - tutors are online