MSE of sample standard deviation (Lean by AI)

Wed, 21 Oct 2026 00:00:00 +0000

Back in 2023, I wrote a post using SymPy to compute the second order expansion of the mean squared error (MSE) of the sample standard deviation up to $O(n^{-2})$. Earlier this year, I revisited the problem in MSE of sample standard deviation (Gen AI) and showed that a modern LLM can derive the entire second order expansion directly from a prompt.

When an AI produces a long asymptotic derivation with dozens of polynomial moment terms, how can we trust that every coefficient is right and that no terms of order $n^{-2}$ were dropped? To check the derivation rigorously, I formalized and verified the entire proof in Lean 4 with Mathlib.

MSE of sample standard deviation (Gen AI)

Tue, 20 Jan 2026 00:00:00 +0000

In the era of gen AI, I am trying to use LLM to compute the second order expansion of $MSE(S)$ that I did in 2023.

The standard sample s.d. is given by S=sqrt(\sum_{i=1}^n(x_i - \bar x)^2/ (n-1)). Without assumption the underlying distribution, compute the MSE(S) of it in terms of the first few moments of x_i up. Expand the terms up to 1/n^2.

And it just nails it: https://gemini.google.com/share/fa35ab43436f

AI on Randy Lai

MSE of sample standard deviation (Lean by AI)

MSE of sample standard deviation (Gen AI)