Synopsis

One-Sample Estimators

Two-Sample Estimators

Randomization

$r \leftarrow \operatorname{Rng}(s)$ — random number generator with seed $s$
$r.\operatorname{UniformFloat}()$ — uniform random value in $[0, 1)$
$r.\operatorname{UniformInt}(a, b)$ — uniform random integer in $[a, b)$
$r.\operatorname{Sample}(\mathbf{x}, k)$ — select $k$ elements without replacement
$r.\operatorname{Resample}(\mathbf{x}, k)$ — select $k$ elements with replacement
$r.\operatorname{Shuffle}(\mathbf{x})$ — uniformly random permutation

The table below maps each toolkit function to the underlying algorithm and its complexity.

One-Sample Estimators

Function	Algorithm	Complexity
$\operatorname{Center}$	Monahan’s implicit-matrix selection	$O(n \log n)$
$\operatorname{CenterBounds}$	Binary search over pairwise averages + SignedRankMargin	$O(n \log n)$
$\operatorname{Spread}$	Monahan’s selection adapted for differences	$O(n \log n)$
$\operatorname{SpreadBounds}$	Disjoint-pair sign-test inversion	$O(n \log n)$
$\operatorname{Compare1}$	Composition: CenterBounds/SpreadBounds + threshold validation + verdict logic	Depends on bounds complexity

Two-Sample Estimators

Function	Algorithm	Complexity
$\operatorname{Shift}$	Value-space binary search over pairwise differences	$O((n+m) \log L)$
$\operatorname{ShiftBounds}$	PairwiseMargin + Shift quantile selection	$O((n+m) \log L)$
$\operatorname{Ratio}$	Log-exp transform + Shift	$O((n+m) \log L)$
$\operatorname{RatioBounds}$	Log-exp transform + ShiftBounds	$O((n+m) \log L)$
$\operatorname{Disparity}$	Composition: $\operatorname{Shift} / \operatorname{AvgSpread}$	$O((n+m) \log L + n \log n + m \log m)$
$\operatorname{DisparityBounds}$	Bonferroni split: ShiftBounds + AvgSpreadBounds	$O((n+m) \log L + n \log n + m \log m)$
$\operatorname{Compare2}$	Composition: ShiftBounds/RatioBounds/DisparityBounds + threshold validation + verdict logic	Depends on bounds complexity

Randomization

Function	Algorithm	Complexity
$\operatorname{UniformFloat}$	53-bit extraction from xoshiro256++ output	$O(1)$ per draw
$\operatorname{UniformInt}$	Modulo reduction of raw 64-bit output	$O(1)$ per draw
$\operatorname{Sample}$	Fan–Muller–Rezucha selection sampling	$O(n)$
$\operatorname{Resample}$	Uniform integer sampling with replacement	$O(k)$
$\operatorname{Shuffle}$	Fisher–Yates (Knuth shuffle)	$O(n)$

Auxiliary

Function	Algorithm	Complexity
$\operatorname{AvgSpread}$	Weighted average of two Spread calls	$O(n \log n + m \log m)$
$\operatorname{AvgSpreadBounds}$	Bonferroni combination of two SpreadBounds	$O(n \log n + m \log m)$
$\operatorname{Median}$	Sort + pick middle	$O(n \log n)$
$\operatorname{SignMargin}$	Binomial CDF inversion + randomized cutoff	$O(n)$
$\operatorname{PairwiseMargin}$	Löffler recurrence (exact) / Edgeworth (approx)	$O((c^)^2 + c^(n+m))$ (worst case $O((n m)^2)$ ) / $O(\log(n m))$
$\operatorname{SignedRankMargin}$	Dynamic programming (exact) / Edgeworth (approx)	$O(n^3)$ / $O(\log n)$

Here $c^*$ denotes the first dominance index where the exact CDF reaches $\mathrm{misrate}/2$ .