Question

0 and an Let X1, X2, ..., Xn be a random sample where each X; follows a normal distribution with mean u unknown standard deviation o. Let K (n-1)s2 = n 202 (a) [2 points] Assume K ~ Gamma(a = n71,8 bias for K. *). We wish to use K as an estimator of o2. Compute the n

Answer 1

(a)

First we find the expected value of K:
$\begin{align*} E(K) &= \alpha \beta \\ &\text{[Using the formula for mean of a Gamma distribution]} \\ &= \frac{n-1}{2}*\frac{2\sigma^2}{n} \\ &= \frac{(n-1)\sigma^2}{n} \end{align*}$

Thus, the bias for K is given by:

Bias(K) = E(K) - 02 (n-1)02 n 102 =(4-1) -,-"). n = 02 0² = - ANSWER n

(b)

In part (a), we found that K is a biased estimator for $\sigma^2$ . Moreover, we found the expected value of K to be:
$\begin{align*} & \ \ E(K) &&= \frac{(n-1)}{n}\sigma^2 \\ &\Rightarrow \frac{n}{(n-1)}E(K) &&= \sigma^2 \\ &\Rightarrow E\left(\frac{nK}{(n-1)} \right) &&= \sigma^2 \end{align*}$

Thus, we can conclude that the function of K which is an unbiased estimator of $\sigma^2$ is $\bf \frac{nK}{n-1}$ . [ANSWER]

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