{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

lec22 - Lecture 22 22.1 One sided hypotheses continued It...

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

View Full Document Right Arrow Icon
Lecture 22 22.1 One sided hypotheses continued. It remains to prove the second part of the Theorem from last lecture. Namely, we have to show that for any δ K α Π( δ * , θ ) Π( δ, θ ) for θ > θ 0 . Let us take θ > θ 0 and consider two simple hypotheses h 1 : = θ 0 and h 2 : = θ . Let us find the most powerful test with error of type one equal to α. We know that if we can find a threshold b such that θ 0 f ( X | θ 0 ) f ( X | θ ) < b = α then the following test will be the most powerful test with error of type 1 equal to α : δ θ = ( h 1 : f ( X | θ 0 ) f ( X | θ ) b h 2 : f ( X | θ 0 ) f ( X | θ ) < b But the monotone likelihood ratio implies that f ( X | θ 0 ) f ( X | θ ) < b f ( X | θ ) f ( X | θ 0 ) > 1 b V ( T, θ, θ 0 ) > 1 b and, since now θ > θ 0 , the function V ( T, θ, θ 0 ) is strictly increasing in T. Therefore, we can solve this inequality for T and get that T > c b for some c b . This means that the error of type 1 for the test δ θ can be written as θ 0 f ( X | θ 0 ) f ( X | θ ) < b = θ 0 ( T > c b ) . 86
Image of page 1

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

View Full Document Right Arrow Icon
LECTURE 22. 87 But we chose this error to be equal to α = θ 0 ( T > c ) which means that c b should
Image of page 2
Image of page 3
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    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.

    Student Picture

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

  • Left Quote Icon

    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.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    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.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern