Commit f97a72d2 authored by Michael Wimmer's avatar Michael Wimmer
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fix typos

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......@@ -122,8 +122,8 @@ which simply is the estimated standard deviation of the set of $\langle r^2(L)\r
#### Approximate analytical formulas
Apart from the numerical bootstrap procedure , there are also approximate analytical formulas.
When computing the error of the estimate in Eq. (2) we must remember that both $w_k$ and $r_k^2$ are random variables. Following [this stackexchange post](https://stats.stackexchange.com/questions/25895/computing-standard-error-in-weighted-mean-estimation/33959#33959) the best approximation comes from theory that deals with approximating the error of the expectation value of a ratio of random variables. In a compact form (equivalent to the formula in stackexchange post), the error can be written as
$$s(\langle r^2(L)\rangle) = \sqrt{\frac{N}{(N-1)} \frac{\sum_{k=1}^N \left(w_k^{(L)})^2 \left(r^2_k(L) - \langle r^2(L)\rangle\right)^2}{\left(\sum_{k=1}^N w_k^{(L)}\right)^2} }\tag{5}$$
Note that this is an approximation (derived for example in [W.G. Cochrane, Sampling Techniques](https://books.google.com/books/about/Sampling_Techniques.html?id=8Y4QAQAAIAAJ)). [Gatz and Smith](https://doi.org/10.1016/1352-2310(94)00210-C) compared various approximate formulas to the result of bootstrapping for some weather data and found that Eq. (5) gave the best agreement. We confirmed this also for the polymer project.
$$s(\langle r^2(L)\rangle) = \sqrt{\frac{N}{(N-1)} \frac{\sum_{k=1}^N \left(w_k^{(L)}\right)^2 \left(r^2_k(L) - \langle r^2(L)\rangle\right)^2}{\left(\sum_{k=1}^N w_k^{(L)}\right)^2} }\tag{5}$$
Note that this is an approximation (derived for example in [W.G. Cochran, Sampling Techniques](https://books.google.com/books/about/Sampling_Techniques.html?id=8Y4QAQAAIAAJ)). [Gatz and Smith](https://doi.org/10.1016/1352-2310(94)00210-C) compared various approximate formulas to the result of bootstrapping for some weather data and found that Eq. (5) gave the best agreement. We confirmed this also for the polymer project.
### The problem with Rosenbluth
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