Unbiased Estimator.

Cynderella

New Member
the radius of a circle is measured with an error of measurement which is distributed normal with mean $$0$$ and variance $$\sigma^2$$,$$\sigma^2$$ unknown.Given $$n$$ independent measurements of the radius , find an unbiased estimator of the area of the circle.

By using *Maximum Likelihood Estimator* I found

$$\hat\sigma^2=\frac{\sum r_i^2}{n}$$

where $$r$$ is the radius of the circle and $$r\sim N(0,\sigma^2)$$.

Then i am stucked to find the unbiased estimator of the area of the circle

Dragan

Super Moderator
the radius of a circle is measured with an error of measurement which is distributed normal with mean $$0$$ and variance $$\sigma^2$$,$$\sigma^2$$ unknown.Given $$n$$ independent measurements of the radius , find an unbiased estimator of the area of the circle.

By using *Maximum Likelihood Estimator* I found

$$\hat\sigma^2=\frac{\sum r_i^2}{n}$$

where $$r$$ is the radius of the circle and $$r\sim N(0,\sigma^2)$$.

Then i am stucked to find the unbiased estimator of the area of the circle

What is the expected value of: $$\pi\frac{\sum r_i^2}{n}$$,
while noting that the sample variances of the n radius measures will have expected value of sigma^2.