Question

Let X1 , X2 , ⋯, XN be a random sample of size n from normal

distribution with mean µ and variance σ 2 . a). Find the maximum

likelihood estimator of σ 2  2 . (2 points) b). Find the

asymptotic distribution of the maximum likelihood estimator of σ 2

 2 obtained in part (a).

Answer 1

(a)$variance \space v=v(\Sigma \frac{xi}{n})=\frac{1}{n^{2}}\Sigma v(xi)=\frac{n}{n^{2}}σ^{2}=\frac{σ^{2}}{n}$

therefore the likelihood estimator will be

$σ^{2}=s^{2} \frac{n}{n-1}=\frac{\Sigma (xi-x)^{2}}{n-1}$

s²=constant estimator

(b)The sample median estimator of the median Xn corresponding to p = 0.5, Xn is a then a normal distribution with parameters µ and σ2.