Results 1 to 3 of 3

Math Help - Find min sufficient statistics for a continuous random sample

  1. #1
    Super Member
    Joined
    Mar 2006
    Posts
    705
    Thanks
    2

    Find min sufficient statistics for a continuous random sample

    Let X_1,...,X_n be a random variable from an absolutely continuous distribution with density:

     f_ \theta (x) = \left \{ \begin {array}{rcl} \frac {2x}{ \theta ^2} & x \in (0, \theta ) \\ 0 & \mbox {otherwise} \end {array} \right.

    Find the minimum sufficient statistics T and its density.

    The solution is as of follows:

    Let  T(x) = \mbox {max} \{ X_i \} , M(x) = \mbox {min} \{ X_i \}

    Then  f_ \theta (x_1, \ldots ,x_n) = f _ \theta (x_1) \cdots f_ \theta (x_n)

    = \frac {2x_1}{ \theta ^2} \cdots  \frac {2x_n}{ \theta ^2} = \prod ^n_{i=1} \frac {2x_i}{ \theta ^2}

    = \frac {2^n}{ \theta ^{2n}} \cdot \prod ^n_{i=1}x_i

    = \frac {2^n}{ \theta ^{2n}} \cdot \prod ^n_{i=1} x_i \cdot 1 _{ \{ M(x)>0 \} } \cdot 1_{ \{ T(x)< \theta \} } , with  T = T(X)

    I understand that this density gives non zero values only when  M(X)>0 and  T(X) < \theta but I don't understand how that last line with the two indicator functions come into being. Thank you!
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor matheagle's Avatar
    Joined
    Feb 2009
    Posts
    2,763
    Thanks
    5

    Re: Find min sufficient statistics for a continuous random sample

    The suff stat is the max, because it cannot be separated from \theta
    that's the factorization theorem in use.
    NOW, for your answer.
    The marginal densities are ZERO if the X's are not in (0,\theta)
    HENCE, the joint density is ZERO if all the X's are not in (0,\theta)
    And, note that

    0< X_1,...,X_n <\theta is the same as 0< X_{(1)}<\cdots <X_{(n)} <\theta

    is the same as 0< X_{(1)} and X_{(n)} <\theta

    YOUR teacher should have had indicator functions on the previous lines as well, let me show you
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor matheagle's Avatar
    Joined
    Feb 2009
    Posts
    2,763
    Thanks
    5

    Re: Find min sufficient statistics for a continuous random sample

    Your teacher left out the indicator functions on the first few steps
    This should clear it up...

    Quote Originally Posted by tttcomrader View Post
    Let X_1,...,X_n be a random variable from an absolutely continuous distribution with density:

     f_ \theta (x) = \left \{ \begin {array}{rcl} \frac {2x}{ \theta ^2} & x \in (0, \theta ) \\ 0 & \mbox {otherwise} \end {array} \right.

    Find the minimum sufficient statistics T and its density.

    The solution is as of follows:

    Let  T(x) = \mbox {max} \{ X_i \} , M(x) = \mbox {min} \{ X_i \}

    Then  f_ \theta (x_1, \ldots ,x_n) = f _ \theta (x_1) \cdots f_ \theta (x_n)

    = \frac {2x_1}{ \theta ^2} I(0<x_1<\theta)\cdots \frac {2x_n}{ \theta ^2} I(0<x_n<\theta)

    = \prod ^n_{i=1} \frac {2x_i}{ \theta ^2}I(0<x_i<\theta)

    = \frac {2^n}{ \theta ^{2n}} \cdot \prod ^n_{i=1}x_i I(0<x_i<\theta)

    = \frac {2^n}{ \theta ^{2n}} \cdot \prod ^n_{i=1} x_i \cdot 1 _{ \{ M(x)>0 \} } \cdot 1_{ \{ T(x)< \theta \} } , with  T = T(X)

    I understand that this density gives non zero values only when  M(X)>0 and  T(X) < \theta but I don't understand how that last line with the two indicator functions come into being. Thank you!
    Whoever did this doesn't completely understand.
    The indicators were there from the beginning.
    Sticking them at the end means they don't understand functions as well.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 2
    Last Post: September 10th 2011, 10:50 AM
  2. sufficient statistics
    Posted in the Advanced Statistics Forum
    Replies: 1
    Last Post: August 26th 2011, 08:09 AM
  3. sufficient statistics and MVUE's
    Posted in the Advanced Statistics Forum
    Replies: 0
    Last Post: March 23rd 2010, 02:30 PM
  4. Random sample from a continuous type dist
    Posted in the Advanced Statistics Forum
    Replies: 1
    Last Post: March 6th 2009, 10:25 PM
  5. random statistics sample
    Posted in the Advanced Statistics Forum
    Replies: 5
    Last Post: May 21st 2008, 07:41 PM

Search Tags


/mathhelpforum @mathhelpforum