Understanding the distribution of the sample standard deviation can be done using the delta method. By applying the multivariate central limit theorem and a Taylor series expansion, an asymptotic approximation is derived. While this first-order approach works under most conditions, it fails for edge cases like Bernoulli variables with , where zero gradients necessitate a second-order expansion, yielding a negative chi-squared distribution instead of a normal one.
In portfolio optimization, Mean-Variance Optimization (MVO) relies on estimated inputs for expected returns and covariance , introducing uncertainty. While portfolios generated using true parameters meet return constraints, those based on estimates may not. Robust MVO addresses this risk by incorporating uncertainty into the model, improving reliability under estimation errors.