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Math Help - Calculate the bias of the estimator

  1. #1
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    Calculate the bias of the estimator

    Suppose that Y_1,...,Y_n is a random sample where the density of each random variable Y_i is f(y) = 2*x^2*y^(-3), y >= x for some parameter x > 1. Let x^hat := min{Y_1,...,Y_n}.

    I figured out that the pdf for the minimum order statistic is n*[f(y)]*[1-F(y)]^(n-1).

    Also I think that 1-F(y) = Integrate[2*x^2*t^(-3), t, y, Infinity] = x^2*y^(-2)

    Plugging this into the pdf for the first order statistic, we have n*[2*x^2*y^(-3)]*[x^2*y^(-2)]^(n-1).

    Now to find the bias we have that B(x^hat) = E(x^hat) - x

    So I think E(x^hat) = Integrate[y*n*[2*x^2*y^(-3)]*[x^2*y^(-2)]^(n-1), y, x, Infinity].

    This is where I am running into problems because I am finding it very difficult to get a "nice" integral here. Can anyone help to show me where I'm going wrong? Thanks.
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  2. #2
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    I was able to solve this. Thanks.
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