# #StackBounty: #bootstrap #bias #unbiased-estimator #resampling #jackknife What's the point of reporting bootstrap bias?

### Bounty: 50

Suppose the data \$X = (X_1,X_2,cdots,X_n)\$ is a vector of iid observations \$X_i\$ where each \$X_i\$ has marginal distribution \$F(theta)\$. Suppose we observe \$x = (x_1,cdots,x_n)\$ and \$hat theta = hat theta(X,F)\$ and \$t = t(x)\$ are estimators and the estimate respectively. Define \$mathrm{Bias}(hat theta) := E(hat theta) – theta\$.

1. It seems to me that R bootstrap routine `boot` returns \$overline{t} – t\$ as the estimate for the bias. Is this true?
2. If this is true, due to the law of large numbers, it seems to me that \$overline{t} – t(x)\$ is an estimate of \$E(t) – t(x)\$, not \$E(hat theta) – theta\$ which is the \$mathrm{Bias}(hat theta)\$. Is this true or false?
3. If it is false, how can we see that \$overline{t} – t\$ indeed estimate \$mathrm{Bias}(hat theta)\$?

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