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Thread: Estimators question

  1. #1
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    Estimators question

    Estimators question-statistics.jpg

    Dear All, I am having some trouble with this question. Note that the random variable X has a population mean of
    μ.
    Thank you for your help!

    THe markscheme gives α (alpha) to be 1/2 but why?
    Last edited by bbear123; Mar 1st 2018 at 09:15 AM.
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  2. #2
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    Re: Estimators question

    1) should be easy enough. What is the definition of an unbiased estimator? Therefore what must $E[U]$ equal? Given the expression for $U$ in terms of $\alpha$ what must $\alpha$ equal?

    we'll work through this 1 bit at a time.
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    Re: Estimators question

    After giving it more time, I see how to do this question now! OK the only part I don't understand is the last part of the question, (v), where it asks me for a more efficient estimator of μ. The markscheme isn't particularly helpful. I have included it.
    Estimators question-statistics-2.jpg
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    Re: Estimators question

    Quote Originally Posted by bbear123 View Post
    After giving it more time, I see how to do this question now! OK the only part I don't understand is the last part of the question, (v), where it asks me for a more efficient estimator of μ. The markscheme isn't particularly helpful. I have included it.
    Click image for larger version. 

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    markscheme seems clear enough.

    3 iid rvs, you should almost intuitively know the best unbiased estimator of the mean is the sample average, i.e.

    $\bar{\mu} = \dfrac 1 3 (X_1+X_2+X_3)$

    the variance of $\bar{\mu}$ is

    $V[\bar{\mu}] = \dfrac 1 9 \left(\sigma^2 + \sigma^2 + \sigma^2\right) = \dfrac {\sigma^2}{3}$

    $\dfrac 1 3 < \dfrac 3 8$

    What isn't clear?
    Thanks from bbear123
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  5. #5
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    Re: Estimators question

    Ah OK, I see how it is done now. Thank you!
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    Re: Estimators question

    Quote Originally Posted by bbear123 View Post
    Ah OK, I see how it is done now. Thank you!
    as an aside you might try reworking the problem with

    $U = \alpha X_1 + \beta X_1 + (1- \alpha - \beta) X_3$

    I believe in this case you'll derive that $\alpha = \beta = \dfrac 1 3$
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  7. #7
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    Re: Estimators question

    Hey bbear123.

    Hint - You know E[Xn] = mu [from your first post] and if U is unbiased estimator for mu then E[U] = mu as well.

    This should help summarize what you need to do to find the unknowns.

    Hint - you will need two equations in two unknowns to get alpha and beta and I would recommend looking at the Variance operator to do so.
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