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Math Help - unbiased estimators

  1. #1
    Member Jason Bourne's Avatar
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    unbiased estimators

    Let X_1,X_2, . . . ,X_n be independent random variables, each with unknown mean \mu and unknown variance \sigma^2. Further, let Y_1, Y_2, . . . , Y_m be independent random
    variables (and independent of X_1,X_2, . . . ,X_n), also with mean \mu and variance \sigma^2


    (a) Show that W = a\bar{X} + (1-a)\bar{Y} is an unbiased estimator of \mu, where \bar{X} and  \bar{Y}
    are the respective sample means of the two samples.

    (b) Show that Var(W)=\sigma^2 \{\frac{a^{2}}{n} + \frac{(1-a)^2}{m}\}

    (c) Show that Var(W) is minimized when a=\frac{n}{n+m}

    (d) Show that

    S^2 = \frac{(n-1)S^2_X + (m-1)S^2_Y}{n+m-2}

    is an unbiased estimator of \sigma^2, where S^2_X and S^2_Y are the respective sample variances of the two samples.
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by Jason Bourne View Post
    Let X_1,X_2, . . . ,X_n be independent random variables, each with unknown mean \mu and unknown variance \sigma^2. Further, let Y_1, Y_2, . . . , Y_m be independent random
    variables (and independent of X_1,X_2, . . . ,X_n), also with mean \mu and variance \sigma^2


    (a) Show that W = a\bar{X} + (1-a)\bar{Y} is an unbiased estimator of \mu, where \bar{X} and  \bar{Y}
    are the respective sample means of the two samples.

    (b) Show that Var(W)=\sigma^2 \{\frac{a^{2}}{n} + \frac{(1-a)^2}{m}\}

    (c) Show that Var(W) is minimized when a=\frac{n}{n+m}

    (d) Show that

    S^2 = \frac{(n-1)S^2_X + (m-1)S^2_Y}{n+m-2}

    is an unbiased estimator of \sigma^2, where S^2_X and S^2_Y are the respective sample variances of the two samples.
    Are you sure that the X_i 's are not supposed to be iid RV's?
    (same for the Y_i 's)

    If they were then your problem is to show that E(W)=\mu given that E(\bar{X})=\mu, and E(\bar{Y})=\mu, and E(S^2)=\sigma^2, given E(S_X)=\sigma^2 and E(S_Y)=\sigma^2

    (though come to think of it this almost certainy works even if we drop the assumption of identical distributions)

    RonL
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  3. #3
    Member Jason Bourne's Avatar
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    Quote Originally Posted by CaptainBlack View Post
    Are you sure that the X_i 's are not supposed to be iid RV's?
    I don't know what you mean by "iid RV's". The problem is exactly as I have stated it.

    Quote Originally Posted by CaptainBlack View Post
    If they were then your problem is to show that E(W)=\mu given that E(\bar{X})=\mu, and E(\bar{Y})=\mu, and E(S^2)=\sigma^2, given E(S_X)=\sigma^2 and E(S_Y)=\sigma^2
    This seems like along the right lines for this part of the question thanks.
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  4. #4
    Grand Panjandrum
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    Quote Originally Posted by Jason Bourne View Post
    I don't know what you mean by "iid RV's". The problem is exactly as I have stated it.
    iid RV's: independent identicaly distributed random variables

    RonL
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