{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

# hw3sol - 36-226 Summer 2010 Homework 3 Solutions 1 X1 Xn...

This preview shows pages 1–3. Sign up to view the full content.

36-226 Summer 2010 Homework 3 Solutions 1. X 1 , . . . , X n are iid ∼ U (0 , 2 θ ). Then f X ( x ) = 1 2 θ I [0 , 2 θ ] ( x ) L ( θ ) = 1 (2 θ ) n I [0 , 2 θ ] ( x 1 , . . . , x n ) Note that you can not find the mle just by direct calculation in this case (which you can usually realize if you actually try solving it that way). So, you have to find the mle by inspection, like we did in class for the case of the U (0 , θ ) distribution. First, you note that in general L ( θ ) decreases as θ increases ( θ ↑ ⇒ (2 θ ) n 1 (2 θ ) n ). Also, we need to look at the values for which θ is non-zero. Since L ( θ ) = 1 (2 θ ) n I [0 , 2 θ ] ( x 1 , . . . , x n ) = 1 (2 θ ) n I [0 , 2 θ ] ( x ( n ) ) L ( θ ) is non-zero if x i 2 θ, i = 1 , . . . , n ⇐⇒ 2 θ x ( n ) (the maximum) ⇐⇒ θ x ( n ) 2 . So, the likelihood L ( θ ) looks like this: X ( n ) 2 is the MLE. Now, we also need to find the expectation of the mle. 1

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

View Full Document
E [ b θ ] = E X ( n ) 2 = 1 2 E [ X ( n ) ] = To find the expectation of the maximum, we first find its pdf using the result about the distribution of order statistics.
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern