Results 1 to 5 of 5

Math Help - statistical estimators

  1. #1
    Newbie
    Joined
    Feb 2010
    Posts
    4

    statistical estimators

    I would really appreciate some help!

    Suppose that a certain drug is to be administered to two different types of patients A & B. It is known that the mean response of patient of type A is the same as the mean response of patients of type B, but the common value Q of this mean is unknown and must be estimated.

    It is also known that the variance of the response of patients of type A is four times as large as variance of the response of patients of type B.

    Let X1...Xm be responses of random sample of m patients of type A.
    Let Y1...Yn be responses of random sample of n patients of type B.

    Estimator Qhat = aX(bar-m_ + (1-a)Y(bar-n)

    a) For what values of a, m, and n is Qhat an unbiased estimator of Q?
    b) For fixed values of m and n, what value of a yields an unbiased estimator with minimum variance?

    Thank you (sorry the question is so long).
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor matheagle's Avatar
    Joined
    Feb 2009
    Posts
    2,763
    Thanks
    5
    I don't understand what ...'common value Q of this mean' means.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Feb 2010
    Posts
    4
    Quote Originally Posted by matheagle View Post
    I don't understand what ...'common value Q of this mean' means.
    in the problem the Q is a theta. It's supposed to be the mean of type A & B together.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor matheagle's Avatar
    Joined
    Feb 2009
    Posts
    2,763
    Thanks
    5
    theta is just a greek letter, your highness.
    do you mean that the expected value from each distribution is Q?
    As in E(X)=E(Y)=Q?
    Follow Math Help Forum on Facebook and Google+

  5. #5
    MHF Contributor matheagle's Avatar
    Joined
    Feb 2009
    Posts
    2,763
    Thanks
    5
    IF E(X_i)=Q=E(Y_j) then

    E\left(a\bar X_n+(1-a)\bar Y_m\right)=aQ+(1-a)Q=Q for all a,n,m.

    V\left(a\bar X_n+(1-a)\bar Y_m\right)=a^2{V(X)\over n}+(1-a)^2{V(Y)\over m}

    =a^2{4V(Y)\over n}+(1-a)^2{V(Y)\over m}

    =V(Y)\left({4a^2\over n}+{(1-a)^2\over m}\right)

    So, minimize \left({4a^2\over n}+{(1-a)^2\over m}\right) wrt a.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Estimators
    Posted in the Advanced Statistics Forum
    Replies: 1
    Last Post: November 8th 2010, 04:34 PM
  2. estimators
    Posted in the Advanced Statistics Forum
    Replies: 0
    Last Post: October 31st 2010, 12:36 PM
  3. estimators
    Posted in the Advanced Statistics Forum
    Replies: 2
    Last Post: May 1st 2009, 12:26 AM
  4. estimators
    Posted in the Advanced Statistics Forum
    Replies: 4
    Last Post: April 30th 2008, 07:30 AM
  5. Estimators
    Posted in the Advanced Statistics Forum
    Replies: 0
    Last Post: February 1st 2007, 07:44 AM

Search Tags


/mathhelpforum @mathhelpforum