Question

Suppose that X_1,...,X_n is a random sample of size n, from a normal distribution N(0,theta) with zero mean and variance theta.
State, justifying your answer, whether any of the following estimators of theta are unbiased. If you state any are biased, calculate their bias.

1) (thetahat)_1 = sigma(i=1 to n) (((X_i)^2/(n-1))

2) (thetahat)_2 = sigma(i=1 to n) ((X_i)^2/(n))

3) (thetahat)_3 = n((x)^2) where (x)= sigma(i=1 to n) ((X_i)/n)

Answer 1

2 is biased, 1 is not

Answer 2

don't ask me why

Answer 3

check this out http://en.wikipedia.org/wiki/Unbiased_estimation_of_standard_deviation