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Math Help - Unbiased Estimators

  1. #1
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    Unbiased Estimators

    Let Y_1, Y_2,...,Y_n be a random sample of size n from the pdf f_Y(y;theta)=(1/theta)*e^(-y/theta), y>0:

    a) Let theta hat=n*Y_min. Is theta hat unbiased for theta?

    For this question, I'm not sure how to generate the pdf for Y_min. Once I get that, I assume I multiply by n, multiply by the other pdf, and solve the integral.

    b) Is theta hat=(1/n)summation(from i=1 to n)Y_i unbiased for theta?
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  2. #2
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    Quote Originally Posted by eigenvector11 View Post
    Let Y_1, Y_2,...,Y_n be a random sample of size n from the pdf f_Y(y;theta)=(1/theta)*e^(-y/theta), y>0:

    a) Let theta hat=n*Y_min. Is theta hat unbiased for theta?

    For this question, I'm not sure how to generate the pdf for Y_min. Once I get that, I assume I multiply by n, multiply by the other pdf, and solve the integral.

    b) Is theta hat=(1/n)summation(from i=1 to n)Y_i unbiased for theta?
    (a) If the random variable Y has pdf f(y) then the pdf of Y_{(1)}= \text{min} \{ Y_1, \, Y_2, \, .... \, Y_n\} is found as follows:

    The cdf of Y_{(1)} is G(y) = \Pr(Y_{(1)} \leq y) = 1 - \Pr(Y_{(1)} > y).

    Since Y_{(1)} is the minimum of Y_1, \, Y_2, \, .... \, Y_n it follows that the event \Pr(Y_{(1)} > y) occurs if and only if the events \Pr(Y_i > y) occur for i = 1, 2, \, .... \, n. Since the Y_i are independent and \Pr(Y_i > y) = 1 - F(y) it follows that


    G(y) = \Pr(Y_{(1)} \leq y) = 1 - \Pr(Y_{(1)} > y) = 1 - \Pr(Y_1 > y, \, Y_2 > y, \, .... \, Y_n > y)

     = 1 - \Pr(Y_1 > y) \cdot \Pr(Y_2 > y) \cdot \, .... \, \cdot \Pr(Y_n > y) = 1 - [1 - F(y)]^n.


    The pdf of Y_{(1)} is given by g(y) = \frac{dG}{dy}: g(y) = n [1 - F(y)]^{n-1} f(y).


    Now you have to calculate E(n Y_{(1)}) and see if it's equal to \theta.
    --------------------------------------------------------------------------------

    (b) Calculate the expected value of the estimator and see whether or not you get \theta.
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