Hey there! Sign in to join this conversationNew here? Join for free

Need help with estimators (Statistics) Watch

Announcements
    • Thread Starter
    Offline

    2
    ReputationRep:
    We have that X1 to Xn are i.i.d Uniform (0, theta) random variables, where theta is greater than zero but unknown. The question is to explain why (2/n)(Sum from i=1 to i=n of Xi) is not a sufficient statistic. Any ideas? I have no clue to be honest.

    Sorry for the poor notation, hope it's legible.
    Offline

    13
    ReputationRep:
    (Original post by pineapplechemist)
    We have that X1 to Xn are i.i.d Uniform (0, theta) random variables, where theta is greater than zero but unknown. The question is to explain why (2/n)(Sum from i=1 to i=n of Xi) is not a sufficient statistic. Any ideas? I have no clue to be honest.

    Sorry for the poor notation, hope it's legible.
    I'll try and get you started! So,

    (i) What is the definition of sufficiency and what does it mean?

    (ii) Can you think of a big theorem that might help us here?
    • Thread Starter
    Offline

    2
    ReputationRep:
    (Original post by Gregorius)
    I'll try and get you started! So,

    (i) What is the definition of sufficiency and what does it mean?

    (ii) Can you think of a big theorem that might help us here?
    (i) It's to do the the conditional distribution depending on the unknown parameter or not

    (ii) I'm not entirely certain: I was thinking along the lines of the factorization theorem but I'm not 100 percent sure. I think the only other theorem I'm aware of to do with sufficiency is Rao-Blackwell: would this help?
    Offline

    13
    ReputationRep:
    (Original post by pineapplechemist)
    (i) It's to do the the conditional distribution depending on the unknown parameter or not

    (ii) I'm not entirely certain: I was thinking along the lines of the factorization theorem but I'm not 100 percent sure. I think the only other theorem I'm aware of to do with sufficiency is Rao-Blackwell: would this help?
    You're getting close. A statistic is sufficient for a parameter \theta if the conditional distribution of the sample given the value of the statistic is independent of \theta. What this means informally is that the statistic contains all the information about \theta that is present in the sample.

    The factorization theorem is a key tool here (not Rao-Blackwell, as that is about improving estimators by conditioning on a sufficient statistic). It's key because it's an "if and only if" theorem.

    So, can you write down the joint distribution of the sample given \theta? Then have a good think and see if it can factor in the way that the theorem requires.
    • Thread Starter
    Offline

    2
    ReputationRep:
    (Original post by Gregorius)
    You're getting close. A statistic is sufficient for a parameter \theta if the conditional distribution of the sample given the value of the statistic is independent of \theta. What this means informally is that the statistic contains all the information about \theta that is present in the sample.

    The factorization theorem is a key tool here (not Rao-Blackwell, as that is about improving estimators by conditioning on a sufficient statistic). It's key because it's an "if and only if" theorem.

    So, can you write down the joint distribution of the sample given \theta? Then have a good think and see if it can factor in the way that the theorem requires.
    Well the joint distribution given theta is 1/(theta)^n right? (by independence). I'm still unsure as to what to do with my T(X) though.
    Offline

    13
    ReputationRep:
    (Original post by pineapplechemist)
    Well the joint distribution given theta is 1/(theta)^n right? (by independence). I'm still unsure as to what to do with my T(X) though.
    You have nearly got the joint distribution. But notice that it is non zero on a restricted domain that you can write in terms of the x_i. That is, you need an indicator function in there.

    Apologies in advance but I am going to be away from computers for much of the rest of the day, so won't be able to help again (if you need it) until tomorrow. A google search on sufficient statistic uniform should give you some useful stuff if you are still stuck!
    • Thread Starter
    Offline

    2
    ReputationRep:
    (Original post by Gregorius)
    You have nearly got the joint distribution. But notice that it is non zero on a restricted domain that you can write in terms of the x_i. That is, you need an indicator function in there.

    Apologies in advance but I am going to be away from computers for much of the rest of the day, so won't be able to help again (if you need it) until tomorrow. A google search on sufficient statistic uniform should give you some useful stuff if you are still stuck!
    Thanks for your help!
 
 
 
  • See more of what you like on The Student Room

    You can personalise what you see on TSR. Tell us a little about yourself to get started.

  • Poll
    Should Spain allow Catalonia to declare independence?
    Useful resources

    Make your revision easier

    Maths

    Maths Forum posting guidelines

    Not sure where to post? Read the updated guidelines here

    Equations

    How to use LaTex

    Writing equations the easy way

    Student revising

    Study habits of A* students

    Top tips from students who have already aced their exams

    Study Planner

    Create your own Study Planner

    Never miss a deadline again

    Polling station sign

    Thinking about a maths degree?

    Chat with other maths applicants

    Can you help? Study help unanswered threads

    Groups associated with this forum:

    View associated groups
  • See more of what you like on The Student Room

    You can personalise what you see on TSR. Tell us a little about yourself to get started.

  • The Student Room, Get Revising and Marked by Teachers are trading names of The Student Room Group Ltd.

    Register Number: 04666380 (England and Wales), VAT No. 806 8067 22 Registered Office: International House, Queens Road, Brighton, BN1 3XE

    Quick reply
    Reputation gems: You get these gems as you gain rep from other members for making good contributions and giving helpful advice.