#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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