B4090.HW7

# B4090.HW7 - BTRY 4090 STSCI 4090 Spring 2011 Homework 7 Due...

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BTRY 4090 / STSCI 4090 Spring 2011 Homework 7 Due at 2:00pm on Friday, April 1 in 1198 Comstock Hall 1. We observe a random sample 1 2 , ,..., n X X X from the beta( θ ,1) distribution and an independent random sample 1 2 , ,..., m Y Y Y from the beta( λ ,1) distribution. (a) Find the generalized likelihood ratio test of H 0 : θ = λ versus H 1 : θ λ . Hint : Each step is routine: (1) Find the MLE of ( , )   over the parameter space {( , ) : 0, 0}     by writing the likelihood ( , ) L   and proceeding as usual. (2) Find the MLE of θ by writing the likelihood ( , ) L   , because {( , ) : , 0}   ; it is easier to recognize the resulting situation than to do the calculation, but either method works. (3) Compute the GLR. It looks pretty ugly initially – you wouldn’t want to bring it home to your mother – but this is OK, just write it down. We’ll see how well it cleans up in (b). (b) Define ln i i U X   for i = 1,2,…, n and ln j j V Y   for j = 1,2,…, m . Show that the

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• Spring '10
• -1
• Normal Distribution, probability density function, generalized likelihood ratio, Marx p.

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