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I want to understand the logic behind bootstrapping. I'm reading this book and on page 108, he started to discuss the bootstrapping technique:

So suppose, my population has $n=2000$ and $T_5$ is the $\text{median}(X_1,X_2,X_3,X_4,X_5)$, using the technique explained in the book I estimate this value.

What I don't understand is why we care about $T_n$? shouldn't our concern be the population parameter? in another words, suppose we don't have the CLT and we want to estimate $\mu$, why do I care about $\bar X_n=\frac{X_1+\ldots+X_n}{n}$?

If $T_n$ is an estimator for the population parameter, shouldn't he prove this?

user45523
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  • The idea behind the bootstrap is to use the available sample as your best "approximation" for the population, and then simulate the sampling distribution of the test statistic under this assumption (sampling with replacement). See also this thread: https://stats.stackexchange.com/q/26088/930. – chl Dec 04 '20 at 09:04
  • @chl yes, but shouldn't we prove this fact? I've seen the authors taking this for granted – user45523 Dec 04 '20 at 09:14
  • Maybe it is just for an exercise, letting the reader (student) be familiar with the idea of bootstrap. Otherwise, the authors should explain somewhere in the book, maybe next to this exercise or somewhere quite earlier. – TrungDung Dec 04 '20 at 09:38
  • @TrungDung do you know any place, site, book or pdf, etc. where the author shows why $T_n$ estimates the parameter of the entire population? I can't find any source proving that – user45523 Dec 04 '20 at 09:40
  • What is $T_n$ in your post? – TrungDung Dec 04 '20 at 09:51
  • @TrungDung it can be the median for example – user45523 Dec 04 '20 at 10:20
  • I mean what is $T_n$ in the exercise in the book? Because you ask "why we care about $T_n$". Who is the first one talk about $T_n$? You or the book? – TrungDung Dec 04 '20 at 10:51

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