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Math Help - Estimator question

  1. #1
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    Estimator question

    I'm having some problems understanding this question related to the bootstrap method:

    Generate a sample from X_1,X_2, \dots X_{25} from the uniform distrubition on [0,10]

    We pretend not to know \theta of [0,\theta] (which is in fact 10) and try two estimators for this parameter:

    T_1 = 2\bar{X}
    T_2 = \frac{(n+1)M}{n}

    To choose between T_1 and T_2 we are interested in the MSE E(T_1-\theta)^2 and E(T_2-\theta)^2. We can estimate these by inserting an estimator \theta in the expressions of their expectation value and variance. Determine a \theta based on the data to do this and compare the answers.
    ------------------------------

    I found the expectation value of both T_1 and T_2, but I have no idea what is meant by inserting an estimator \theta, and how to determine this from a given 25 draws from the distribution. Could someone help me by explaining this?
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  2. #2
    MHF Contributor matheagle's Avatar
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    1 I don't see where the bootstrap comes in
    2 I guess M is the largest order stat.
    3 Is n=25 or do you want this for any sample size?
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  3. #3
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    The question is under the header parametric bootstrap, because the next question is estimating the MSE by drawing from [0, \hat{\theta}]. That question is easy I think.
    M is indeed the largest order stat, and an explanation for n=25 or really any any would be great. I just don't understand the part where I decide an estimator from the data and fill it in in the expectation and variance of  T_1 and T_2.


    Err, well I though I had the expectation and variance right, but I don't think I got em. Could someone help me with these for both T_1[/tex] and T_2?

    The expectation of both would be simply  \theta but how do I get the variance? o_O
    Last edited by TiRune; October 24th 2009 at 05:22 AM.
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  4. #4
    MHF Contributor matheagle's Avatar
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     E(2\bar X)= 2E(\bar X)=2\biggl({\theta\over 2}\biggr)=\theta

     V(2\bar X)= 4V(\bar X) =4\biggl({V(X_1)\over n}\biggr) =4\biggl({\theta^2\over 12n}\biggr)={\theta^2\over 3n}

    show me your density for M and I'll look it over.
    I still don't know if you want n or 25.
    Both estimators are unbiased.
    The point is we want the one with the smaller variance.
    I still don't see the point of the bootstrap here.
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  5. #5
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    Well, the question is about n = 25, but for general n I need to know the variance.
    Is this correct?

    V(\frac{(n+1)M}{n}) = \frac{(n+1)^2}{n^2}V(M) = \frac{(n+1)^2}{n^2}*\frac{n}{(n+1)^2(n+2)} = \frac{1}{n(n+2)}

    The M is the maximum out of the n variables. For the uniform distribution on [0,1] The epectation value for this is \frac{n}{n+1} and the variance is \frac{n}{(n+1)^2(n+2)}
    For the uniform distribution with \theta = 10 instead of 1, I guess this translates to the expectation value times theta, although I don't know what to do with the variance.
    The propability function of M is the beta distribution with paramters a and 1 I think.

    What remains even more vague to me is how I deduce an estimator from a dataset with 25 X's, and how I should fill that in in the expectations and variances...

    The bootstrap comes in in the next question: determine the MSE of T_1 and T_2 by use of parametric bootstrap. First determine an estimator of \theta and then simulate E(T_1-\hat{\theta}) and E(T_2-\hat{\theta})
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  6. #6
    MHF Contributor matheagle's Avatar
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    the variance of M_n will have a factor of \theta^2

    You can use the uniform(0,1) and multiply it by theta

    then use V(aX+b)=a^2V(X)

    BUT I would derive the variance directly from the largest order stat of a U(0,\theta)
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