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I came up with the following result, tested on many data sets, but I do not have a formal proof yet:

Theorem: The width $L$ of any confidence interval is asymptotically equal (as $n$ tends to infinity) to a power function of $n$, namely $L = A\cdot n^{-B}$ where $A$ and $B$ are two positive constants depending on the data set, and $n$ is the sample size.

See here and here for details. The $B$ exponent seems to be very similar to the Hurst exponent in time series, not only in terms of what it represents, but also in the values that it takes: $B=1/2$ corresponds to perfect data (no auto-correlation or undesirable features) and $B = 1$ corresponds to "bad data" typically with strong auto-correlations.

Note that $B = 1/2$ is what everyone uses nowadays, assuming observations are independently and identically distributed, with an underlying normal distribution. I also devised a method to make the interval width converges faster to zero: $O(n^{-1})$ rather than $O(n^{-1/2})$. This is also described in section 3.3. in my article on re-sampling (here) and my approach in this context seems very much related to what is called second-order accurate intervals (usually achieved with modern versions of bootstrapping, see here.)

My question is whether my theorem is original, ground-breaking, and correct, and how would someone prove it (or refute it.)

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    $\begingroup$ MathOverflow might also be a good place to ask. $\endgroup$ – 雨が好きな人 May 7 at 15:34

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