I am new to statistics especially in the topic of estimators and sufficient statistic.

I am reading a note which says "unbiasedness is a desirable (but not necessary) property of a good estimator". Then it gives an example where unbiased estimators don't exist. Since there is no further explanation in the example, I am struggling to understand it.

The example says:

If $X\sim B(\varphi(\theta))$ for some function $\varphi$, then $\theta^*\in K_0\iff\mathbb{E}_\theta\theta^*\equiv\theta^*(0)(1-\varphi(\theta))+\theta^*(1)\varphi(\theta)=\theta,\forall\theta\in\Theta)$, (where $K_0$ is the class of all unbiased estimators).

This is just for reference:
An estimator $\theta_0^*=\theta_0^*(X)$ from a class $K$ of estimators of $\theta$ is called efficient in $K$ if, for any $\theta^*\in K$, $$\mathbb{E_\theta}(\theta_0^*-\theta)^2\le\mathbb{E_\theta}(\theta*-\theta)^2,\forall\theta\in\Theta.$$

For a function $b=b(\theta),\theta\in\Theta$, let $$K_b=\{\theta^*:\mathbb{E}_\theta\theta^*=\theta+b(\theta),\forall\theta\in\Theta\}$$ be the class of all estimators with the bias $b(\theta)$.

I understand the basic idea of estimators, the property for being biased and unbiased. But I have no clue why unbiased estimators don't exist in the given example above. May anyone kindly explain it to me please?

Many thanks!


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.