# How to Make This Estimator Unbiased?

Could someone clarify if I'm interpreting this question correctly?

As the sample size increases and approaches infinity, then the expected value of the estimator would approach $0.$ The estimator is biased because $θ$ is not equal to $θ/n.$

But, if $n = 1,$ then wouldn't the expected value${} = θ$? Is this how the statistician's estimator would become unbiased?

Clarification appreciated!

## 1 Answer

We don't have to take limit of $n$ to $\infty$, currently we have

$$\mathbb{E}[\hat{\theta}]=\frac{\theta}{n}$$

We want to find $k$ such that

$$\mathbb{E}[k\hat{\theta}]=\frac{k\theta}{n}=\theta$$

Can you solve for $k$?

• So that would solve for k = n? – MathsHelp May 8 '18 at 17:49
• yup, can you an unbiased estimator $\tilde{\theta}$ such that $\mathbb{E}[\tilde{\theta}]=\theta$ now? – Siong Thye Goh May 8 '18 at 17:51
• Wouldn't that just be any number =/= 0? Unless k represents something specifically? – MathsHelp May 8 '18 at 17:54
• i thought you have conclude that $k=n$, that is $\mathbb{E}[k\hat{\theta}]=\theta$. can you find $\tilde{\theta}$ such that $\mathbb{E}[\tilde{\theta}]=\theta$? – Siong Thye Goh May 8 '18 at 17:56
• $$V(n \hat{\theta})=n^2 V(\hat{\theta})=\frac{n^3\sigma^2}{n-1}$$ – Siong Thye Goh May 8 '18 at 19:12