HW11_2007

# HW11_2007 - PROF HOHG FALL 2007 ECONOMICS 619 PROBLEM SET...

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PROF HOHG FALL 2007 ECONOMICS 619 PROBLEM SET # 11 1. [# 7.1, p.355] One observation in taken on a discrete random variable X with pmf f ( x; ° ) , where ° 2 f 1 ; 2 ; 3 g . Find the MLE of ° x f ( x j 1) f ( x j 2) f ( x j 3) 0 1 3 1 4 0 1 1 3 1 4 0 2 1 6 1 4 1 2 3 1 6 0 1 4 2. [# 7.7, p.355] Let X 1 ; :::; X n be i.i.d. with one of two pdfs. If ° = 0 , then f ( x; ° ) = ( 1 0 < x < 1 0 otherwise while if ° = 1 , then f ( x; ° ) = ( 1 = (2 p x ) 0 < x < 1 0 otherwise Find the MLE of ° . 3. [# 7.8, p.356] One observation, X , is taken from a N (0 ; ± 2 ) population. (a) Find an unbiased estimation of ± 2 : (b) Find the MLE of ±: (c) Discuss how the method of moments estimator of ± might be founded. 4. [# 7.19, p.358] Suppose that the random variables Y 1 ; :::; Y n satisfy Y i = ²x i + " i ; i = 1 ; :::; n; where x 1 ; :::; x n are °xed constants, and are i.i.d. N (0 ; ± 2 ) , ± 2 are unknown. (a) Find a two-dimensional su¢ cient statistic for ( ²; ± 2 ) : (b) Find the MLE of ² , and show that it is an unbiased estimator of ²: (c) Find the distribution of the MLE of ²: 5. [# 7.40, p.362] Let X 1 ; :::; X n be i.i.d. Bernoulli( p ). Show that the variance of ° X n attains

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