# Question about a new type of confidence interval

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.)

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