Thread: 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.

I was able to solve this. Thanks.