# Theoretical bootstrap estimate versus simulated bootstrap estimate of the bias of estimated parameter.

Let $$T(F) = \theta$$. Then $$R(\chi, F) = T(\hat{F}) - T(F) = \hat{\theta} - \theta$$. The mean of $$R(\chi, F)$$ is $$E_F[\hat{\theta} - \theta] = E_F[\hat{\theta}] - \theta$$. It is the bias of $$\theta$$.

Then theoretical bootstrap estimate is $$E_{\hat{F}}(\hat{\theta}^*) - \hat{\theta}$$. The simulated bootstrap estimate is obtained by drawing B bootstrap sample:

$$\frac{1}{B} \sum_{b=1}^B \hat{\theta}^*_b - \hat{\theta} = \overline{\theta}^* - \hat{\theta}.$$

I don't understand how theoretical bootstrap estimate of bias differs from simulated. Can anyone assist?