Question

Y 1

​
,Y 2
​
,…,Y n
​
represents an i.i.d. random sample from a normal distribution with a mean μ=0 and an unknown variance σ 2
. We wish to use a quantity K as an estimator for σ 2
. a) Suppose it is known that K∼Gamma(α=n−1,β= n
σ 2
​
). Compute the bias for K as an estimator of σ 2
. b) If K is a biased estimator for σ 2
, state the function of K that would make it an unbiased estimator for σ 2
.

Answer 1

The function that would make K an unbiased estimator for σ2 is K' = (n/n - 1)K.

a) We are given the Gamma distribution of K, that is, K ∼ Γ(α = n - 1, β = nσ2). Now, we have to compute the bias of K, i.e., B(K) = E(K) - σ2.Using the moments of Gamma distribution, we have,E(K) = α/β = (n - 1)/nσ2Now, B(K) = E(K) - σ2= (n - 1)/nσ2 - σ2= (n - 1 - nσ4)/nσ2b) To make K an unbiased estimator for σ2, we have to find a function of K that results in the expected value of K being equal to σ2. That is, E(K') = σ2.To find the required function, let K' = cK, where c is some constant. Then,E(K') = E(cK) = cE(K) = c(n - 1)/nσ2We want E(K') to be equal to σ2. So, we must have,c(n - 1)/nσ2 = σ2Solving for c, we get:c = n/n - 1Therefore, the function that would make K an unbiased estimator for σ2 is K' = (n/n - 1)K.

Learn more about Gamma distribution here,