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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?

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  • $\begingroup$ 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: stats.stackexchange.com/q/26088/930. $\endgroup$
    – chl
    Commented Dec 4, 2020 at 9:04
  • $\begingroup$ @chl yes, but shouldn't we prove this fact? I've seen the authors taking this for granted $\endgroup$
    – user45523
    Commented Dec 4, 2020 at 9:14
  • $\begingroup$ 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. $\endgroup$
    – TrungDung
    Commented Dec 4, 2020 at 9:38
  • $\begingroup$ @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 $\endgroup$
    – user45523
    Commented Dec 4, 2020 at 9:40
  • $\begingroup$ What is $T_n$ in your post? $\endgroup$
    – TrungDung
    Commented Dec 4, 2020 at 9:51

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