hw5_solution-1 - 131B HW#5 solution 7.8 Jointly Sucient...

Info icon This preview shows pages 1–2. Sign up to view the full content.

131B HW#5 solution 7.8 Jointly Sufficient Statistics 10. The p.d.f. of the uniform distribution is f ( x | θ ) = 1 θ 1 { 0 x θ } , so the likelihood is f n ( x | θ ) = n i =1 f ( x i | θ ) = 1 θ n 1 { max { x 1 ,...,x n }≤ θ } 1 { 0 min { x 1 ,...,x n }} It implies that the M.L.E. ˆ θ = max { X 1 , . . . , X n } . By the factorization criterion, ˆ θ is the sufficient statistic, so it is a minimal sufficient statistic. 12. The likelihood is f n ( x | θ ) = n i =1 f ( x | θ ) = 2 n i x i θ 2 n 1 { max { x 1 ,...,x n }≤ θ } 1 { 0 min { x 1 ,...,x n }} The M.L.E. of θ is ˆ θ = max { X 1 , . . . , X n } . ˆ θ is also a sufficient statistic by the factorization criterion. The median of the distribution is the value m such that m 0 f ( x | θ ) dx = 1 / 2, it implies that m = θ/ 2. By the invariance property, ˆ m = ˆ θ/ 2 is the M.L.E. of m . Note that ˆ m is also a sufficient statistic, so it is a minimal sufficient statistic. 16. It follows from Theorem 7.3.2 that the Bayes estimator of λ is ( α + n i =1 X i ) / ( β + n ).
Image of page 1

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

Image of page 2
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