Assess the Cornish–Fisher validity check and linear-time counting-sort optimization

Determine whether enforcing the Cornish–Fisher expansion’s domain of validity on each interval and implementing a genuinely linear-time counting sort can further tighten the worst-case behavior of MomentQuant’s approximate and exact quantile-computation modes.

Background

MomentQuant approximates interval quantiles with a fourth-moment Cornish–Fisher expansion, but the paper does not enforce the expansion’s domain of validity separately for each interval. Consequently, the approximation can potentially produce non-monotone or otherwise invalid quantile estimates for highly non-normal intervals.

For the exact mode, the paper also investigates an optimization that globally sorts each representation once and reuses that ordering to recover interval-specific sorted values. The authors explain that a linear-time bucket-assignment step could reduce the asymptotic cost, but their experiments do not establish whether a genuinely linear-time counting-sort implementation would eliminate the remaining performance gap. These two unresolved issues are identified as targets for improving worst-case behavior.