A random sample consists of 3 independent observations $𝑋_1, 𝑋_2
and 𝑋_3$
follow Normal distribution 𝑁(𝜇, $𝜎^2$) consider four estimators for population mean
𝜇 as follow:
$𝜇̂_1$ =
$\frac{X_1 + 3𝑋_2 − 2𝑋_3}
{2}$
, 𝜇̂2 =
$\frac{5𝑋_1 − 2𝑋_2}
{3}$
, 𝜇̂3 =
$\frac{1}{2}
𝑋_1 +
\frac{1}{2}
𝑋̅$, 𝜇̂4 =
$\frac{2𝑋_1 + 3𝑋_3 − 2𝑋̅}{3}$
Where 𝑋̅ is the sample mean of $𝑋_1, 𝑋_2, 𝑋_3$.
(a) Which of the above is/are the unbiased estimator(s) for 𝜇 ?
(b) Which of the above is the best unbiased estimator for 𝜇 ?
(c) Provide an estimator for 𝜇 which is better than aforementioned 𝜇̂1, 𝜇̂2, 𝜇̂3
and 𝜇̂4. Justify your answer.
For part a, I found all are unbiased and for part b 𝜇̂3 is the best. Are they correct?
But if they are correct, I think 𝜇̂3 is already the best one and I can't think of an estimator better than aforementioned. How can I provide a better estimator?
Best Answer
$$\mathbb{E}[\hat{\mu}_1]=\frac{1\mu+3\mu-2\mu}{2}=\mu, \ \ \mathbb{V}[\hat{\mu}_1]=\frac{\sigma^2+9\sigma^2+4\sigma^2}{4}=\frac{7}{2}\sigma^2,$$
$$\mathbb{E}[\hat{\mu}_2]=\frac{5\mu-2\mu}{3}=\mu, \ \ \mathbb{V}[\hat{\mu}_2]=\frac{25\sigma^2+4\sigma^2}{9}=\frac{29}{9}\sigma^2,$$
$$\mathbb{E}[\hat{\mu}_3]=\frac{1}{2}\mu+\frac{1}{2}\frac{3\mu}{3}=\mu, \ \ \mathbb{V}[\hat{\mu}_3]=\frac{1}{4}\sigma^2+\frac{1}{4}\frac{3\sigma^2}{9}=\frac{12}{4\cdot 9}\sigma^2=\frac{1}{3}\sigma^2$$
$$\mathbb{E}[\hat{\mu}_4]=\frac{2\mu+3\mu}{3}-\frac{2}{3}\frac{3\mu}{3}=\mu, \ \ \mathbb{V}[\hat{\mu}_4]=\frac{4\sigma^2+9\sigma^2}{9}+\frac{4}{9}\frac{3\sigma^2}{9}=\frac{13\sigma^2+\frac{4}{3}\sigma^2}{9}$$
Yes $\hat{\mu}_3$ is best among them and unbiased. However, $\hat{\mu}_3$ is such that $$\lim_{N\to \infty}\mathbb{V}[\hat{\mu}_3]=\frac{1}{4}\sigma^2+\frac{1}{4}\lim_{N \to \infty}\mathbb{V}[\bar{X}]=\frac{1}{4}\sigma^2+\frac{1}{4}\lim_{N \to \infty}\frac{\sigma^2}{N}=\frac{1}{4}\sigma^2$$ So it is not consistent. So a better estimator would be for example $\bar{X}$.