411_hw05

411_hw05 - T = T X 1,X n is suﬃcient for θ if the...

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Stat 411 – Homework 05 Due: Wednesday 02/22 Undergraduates may solve the “Graduate only” problem(s) for possible extra credit. 1. Let X 1 ,...,X n be an iid sample from a log-normal distribution with PDF f θ ( x ) = 1 x 2 πθ 2 e - (log x - θ 1 ) 2 / 2 θ 2 , x > 0 , θ = ( θ 1 2 ) R × R + . (a) Find the MLE of θ . (b) Calculate the Fisher information matrix I ( θ ). (Hint: if X f θ ( x ), then log X N ( θ 1 2 ); that’s why it’s called “log-normal.”) (c) State the asymptotic distribution of ˆ θ n . 2. Problem 7.2.4 on page 379. 3. Problem 7.2.6 on page 380. 4. Problem 7.2.8 on page 380. 5. By deﬁnition, a statistic
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Unformatted text preview: T = T ( X 1 ,...,X n ) is suﬃcient for θ if the conditional distribution of X 1 ,...,X n given T = t does not depend on θ . Suppose that X 1 ,X 2 are independent Pois ( θ ) observations. Use the deﬁnition to show that T = X 1 + X 2 is suﬃcient for θ . Does the conditional distribution look familiar? 6. (Graduate only) Problem 6.4.4 on page 350. (Two things to note: ﬁrst, you’re given the CDF not the PDF; second, the support depends on θ 1 .) 1...
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This note was uploaded on 03/12/2012 for the course STAT 411 taught by Professor Staff during the Spring '08 term at Ill. Chicago.

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