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I am reading a paper that uses the following equation to calculate the optimal bandwidth, however, I am confused about the position of "4" and "3" in the equation. is this a typo? or what does it mean?

the optimal value of h can be calculate (sic) from
$$h=\left(^4\sigma^5/_3L\right)^\frac15$$

where

$$\sigma=\sqrt{\frac{1}{L-1}\sum_{i=1}^L (g_i-\bar{g})^2}$$

Additionally $L$ is the number of cells and $g_i$ is the value for each cell.

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  • $\begingroup$ Without context it's a little hard to say much. Are $^{4}\sigma$ and $_{3}L$ defined earlier? Even if they aren't, looking for definitions might help you figure out what was intended (even in the presence of an error). What are you looking at? Is this in a book? $\endgroup$ – Glen_b -Reinstate Monica Dec 12 '19 at 2:15
  • $\begingroup$ it is in a paper, only $sigma$ and $L$ are defined. $\endgroup$ – flash_Meow Dec 12 '19 at 2:20
  • $\begingroup$ Can you give a full reference for the paper so we can see what you're looking at? $\endgroup$ – Glen_b -Reinstate Monica Dec 12 '19 at 2:21
  • $\begingroup$ That's a link rather than a complete reference to the paper, but in any case please edit it into your question. $\endgroup$ – Glen_b -Reinstate Monica Dec 12 '19 at 2:23
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It appears to be a mangled version of this formula:

$h = \left(\frac{4\hat{\sigma}^5}{3n}\right)^{\frac{1}{5}} \approx 1.06 \hat{\sigma} n^{-1/5}$

https://en.wikipedia.org/wiki/Kernel_density_estimation#A_rule-of-thumb_bandwidth_estimator

with $L$ for $n$ and $\sigma$ for $\hat{\sigma}$.

That is, the 3 and the 4 should not be sub- or superscripts.

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