# arbitrariness in bootstrap bias estimation

The bootstrap estimates bias by applying the "plug-in" principle to $$E(\hat{\theta}_n) - \theta$$ I got this knowledge from p.124 of Efron, Tibshirani, 1994.
equation(10.1) $$\text{bias}_F=E_F[s(\mathbf{x})] -t(F)$$ and equation(10.2) $$\text{bias}_\hat{F} = E_\hat{F}[s(\mathbf{x}^*)] - t(\hat{F})$$

the "plug-in" of the first term has definitive meaning, since $$\hat\theta_n$$ is a prescribed statistics on the sample. We simply need to take the expectation of it with respect to the empirical distribution.

the "plug-in" of the second term however is rather confusing. Since there are an infinite number of ways to write a given distribution parameter as a functional of the distribution. For example, take $$\theta$$ to be the decay rate parameter of the exponential distribution, thus $$f(X) = \theta e^{-\theta X}$$, and one could get $$\theta = 1/E(X)$$ as well as $$\theta = 1/\sqrt{D^2(X)}$$, these are different functionals of $$f(X)$$ and would lead to different plug-in estimate of $$\theta$$ on finite sample.

This might be a naive question but I hope I've clarified my confusion.

Since I didn't get an answer after half a year, I though maybe I mis-stated the question, So I'll restate it: What's the definition of a "parameter" as repeatedly used in Efron's book? Is it a functional by definition? Or is there a standardized way to write every "parameter" as a functional? Can you give some more examples of a "parameter" (other than the "mean" and "variance")?

Quoted from page 124 of "A introduction to the bootstrap":

... . We want to estimate a real-valued parameter $$\theta = t(F)$$. For now we will take the estimator to be any statistic $$\hat{\theta}=s(x)$$

This is best explained using the figure from the Second Thoughts on the Bootstrap paper by Bradley Efron. In real world, there is a distribution $$P$$ (denoted as $$F$$ in the book), where the data $$\mathbf{x}$$ comes from the distribution. We also have the estimator $$t$$, using it we can obtain population parameter $$\theta = t(P)$$. We also can use statistic $$s$$ to estimate $$\theta$$ from the sample obtaining $$\hat{\theta} = s(\mathbf{x})$$.

In bootstrap world, we obtain a sample $$\mathbf{x}^*$$ from bootstrap distribution $$\hat{P}$$, we can as well obtain the population estimate of the bootstrap distribution $$t(\hat{P})$$, or sample statistic from the bootstrap sample $$s(\mathbf{x}^*)$$.

The plug-in principle means that you substitute the distribution $$P$$ with bootstrap distribution $$\hat{P}$$, and sample $$\mathbf{x}$$ with bootstrap sample $$\mathbf{x}^*$$. We are allowed to do this because bootstrap imitates sampling $$\mathbf{x}$$ from $$P$$, by applying equivalent sampling procedure to sample $$\mathbf{x}^*$$ from $$\hat{P}$$.

What follows, if we have the definition of bias

$$\text{bias}_P(\hat{\theta}, \theta) = \text{bias}_P = E_P[s(\mathbf{x})] - t(P)$$

then we substitute

$$\text{bias}_\hat{P} = E_\hat{P}[s(\mathbf{x}^*)] - t(\hat{P})$$

To give an example, say that you want to assess the bias of sample mean $$s(\mathbf{x})$$ as an estimator of population mean $$t(P)$$, then you use bootstrap to sample from the distribution $$\hat{P}$$ (i.e. sample with replacement from $$\mathbf{x}$$), calculate the sample means on the bootstrap samples $$s(\mathbf{x}^*)$$, and compare their expected value (mean), with the population mean of the bootstrap distribution $$t(\hat{P})$$. The "population mean" from bootstrap distribution is equivalent to sample mean of $$\mathbf{x}$$, since distribution $$\hat{P}$$ is created by sampling from $$\mathbf{x}$$, so $$\hat{\theta} = s(\mathbf{x}) = t(\hat{P})$$.

• p. 125 Efron 1994, equation (10.2), what then does the $t(\hat F)$ mean? – Xavier Xing Dec 12 '18 at 11:04
• $\text{bias}_F=E_F[s(\mathbf{x})] -t(F)$ – Xavier Xing Dec 12 '18 at 12:43
• I've copied the two equations in Efron's book, it seems there's a plug-in for $\theta$ ( or $t(F)$ in Efron's notation ) after all – Xavier Xing Dec 12 '18 at 12:49
• So we don't estimate bias of things we don't know, so does this mean that Efron's $t(\hat F)$ is not generally accepted by statisticians? – Xavier Xing Dec 13 '18 at 5:52
• you said "the $θ=t(F)$ value that we know", you didn't mention any $t(\hat F)$ as appeared in Efron's book. – Xavier Xing Dec 13 '18 at 7:02