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?


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