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Math Help - Finding the expectation of a constant bias

  1. #1
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    Finding the estimate of a constant bias on measurements

    Someone is making four measurements:
    y_1, y_2 of \lambda, \mu respectively,  y_3 and y_4 both of \lambda + \mu.
    The measurements are subjected to independent, normally distributed random errors with known variance {\sigma}^2.
    He suspects that there is a constant bias \beta
    How do you show that this bias is estimated as y_1+y_2-\tfrac{1}{2}(y_3+y_4) ?
    Last edited by mongrel73; April 10th 2009 at 09:57 AM. Reason: Wrong title
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by mongrel73 View Post
    Someone is making four measurements:
    y_1, y_2 of \lambda, \mu respectively,  y_3 and y_4 both of \lambda + \mu.
    The measurements are subjected to independent, normally distributed random errors with known variance {\sigma}^2.
    He suspects that there is a constant bias \beta
    How do you show that this bias is estimated as y_1+y_2-\tfrac{1}{2}(y_3+y_4) ?
    Suppose:

    y_1=\lambda+\beta+\varepsilon_1

    y_2=\mu+\beta+\varepsilon_2

    y_3=\lambda+ \mu +\beta+\varepsilon_3

    y_3=\lambda+ \mu +\beta+\varepsilon_4

    where \varepsilon_1,\ \varepsilon_2,\ \varepsilon_3,\ \varepsilon_4 are normally distributed with zero mean.

    Now write:

    \theta= y_1+y_2+(y_3+y_4)/2

    Now find E(\theta) and show that this is \beta

    CB
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  3. #3
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    write:

    \theta= y_1+y_2-(y_3+y_4)/2

    Now find E(\theta) and show that this is \beta
    Doesn't that just show that the estimate is unbiased?
    I mean, if you write \theta=y_1+y_2-y_3, then E(\theta)=\beta as well, so that can't be sufficient.
    How do you show that the one given in the question is the best estimate to use?
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  4. #4
    Grand Panjandrum
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    Quote Originally Posted by mongrel73 View Post
    Doesn't that just show that the estimate is unbiased?
    I mean, if you write \theta=y_1+y_2-y_3, then E(\theta)=\beta as well, so that can't be sufficient.
    How do you show that the one given in the question is the best estimate to use?
    1. E(\theta)=\beta is the condition that \theta is an (unbiased ) estimator of the bias.

    2. The question did not ask that you show it is the best estimate, nor specify in what sense it should be "best".

    CB
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