Calculate the bias of the estimator

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In summary, the conversation discusses the pdf for the minimum order statistic in a random sample and the problem of finding the bias. The integral \int_x^\infty {y n 2 x^2 y^{-3} ( x^2 y^{-2})^{n-1} dy is mentioned and it is suggested that the difficulty may be due to an indeterminate form. However, the speaker was able to solve the integral and find the answer.
  • #1
dynas7y
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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, in fact, using the limits of integration I have listed, I'm getting an indeterminant form so I'm guessing that this might be the problem. Can anyone point me in the right direction as to what I'm doing wrong? Thanks.
 
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  • #2
Are you asking about the integral

[tex] \int_x^\infty {y n 2 x^2 y^{-3} ( x^2 y^{-2})^{n-1} dy [/tex]

[tex] = 2 n x^{2n} \int_x^\infty y^{-2n} dy [/tex]
 
  • #3
Stephen Tashi said:
Are you asking about the integral

[tex] \int_x^\infty {y n 2 x^2 y^{-3} ( x^2 y^{-2})^{n-1} dy [/tex]

[tex] = 2 n x^{2n} \int_x^\infty y^{-2n} dy [/tex]

I was, but I actually was able to get the answer. Thank you.
 

What is the definition of "bias of the estimator"?

The bias of an estimator is a measure of how far the expected value of the estimator is from the true value of the parameter it is estimating.

How do you calculate the bias of an estimator?

The bias of an estimator can be calculated by taking the expected value of the estimator and subtracting the true value of the parameter it is estimating. The resulting value is the bias of the estimator.

Why is it important to calculate the bias of an estimator?

Calculating the bias of an estimator allows us to assess the accuracy of the estimator. A low bias indicates that the estimator is close to the true value, while a high bias indicates that the estimator is systematically overestimating or underestimating the true value.

What are some common causes of bias in an estimator?

Some common causes of bias in an estimator include using a small sample size, using biased data, and using an incorrect model or assumptions in the estimation process.

How can bias in an estimator be reduced?

Bias in an estimator can be reduced by increasing the sample size, using unbiased data, and using appropriate models and assumptions in the estimation process. Regularly checking and adjusting for bias can also help to reduce it.

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