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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!

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