Proving consistency for an estimator.

Question

Let $Y_1,Y_2,...,Y_n$ be a random sample of size $n$ from a normal pdf having $\mu=0$. Show that $S_{n}^{2} = \frac{1}{n} \sum_{i=1}^n {Y_{i}^{2}}$ is a consistent estimator for $\sigma^2= Var(Y)$.

I guess the easiest way is to create an example and show that the estimator your example is not consistent. — Le Chifre, Mar 22 '13 at 14:53

score 2 · Answer 1 · answered Mar 21 '13 at 16:21

2

Hints:

write the definition of the consistent estimator
compute the mean and the variance of $S^2_n$
apply Chebyshev's inequality to $X = S_n^2$

answered Mar 21 '13 at 16:21

SBF

36,041

Thank you! This was very helpful! – user67866 Apr 01 '13 at 20:32

score 1 · Answer 2 · answered Mar 21 '13 at 16:20

1

Hint: Apply the law of large numbers of the sequence $Y_1^2,Y_2^2,\ldots,Y_n^2$.

answered Mar 21 '13 at 16:20

Stefan Hansen

25,582
7
59
91

Proving consistency for an estimator.

2 Answers2