Synopsis

One-Sample Estimators

$\operatorname{Center}(\mathbf{x})$ — robust average
$\operatorname{CenterBounds}(\mathbf{x}, \mathrm{misrate})$ — confidence interval for center
$\operatorname{Spread}(\mathbf{x})$ — robust spread
$\operatorname{SpreadBounds}(\mathbf{x}, \mathrm{misrate})$ — confidence interval for spread

Two-Sample Estimators

$\operatorname{Shift}(\mathbf{x}, \mathbf{y})$ — robust difference
$\operatorname{ShiftBounds}(\mathbf{x}, \mathbf{y}, \mathrm{misrate})$ — confidence interval for shift
$\operatorname{Ratio}(\mathbf{x}, \mathbf{y})$ — robust ratio
$\operatorname{RatioBounds}(\mathbf{x}, \mathbf{y}, \mathrm{misrate})$ — confidence interval for ratio
$\operatorname{Disparity}(\mathbf{x}, \mathbf{y})$ — robust effect size $= \operatorname{Shift}\text{/}\operatorname{AvgSpread}$
$\operatorname{DisparityBounds}(\mathbf{x}, \mathbf{y}, \mathrm{misrate})$ — confidence interval for disparity

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
$\operatorname{Center}$	Monahans implicit-matrix selection	$O(n \log n)$
$\operatorname{CenterBounds}$	Binary search over pairwise averages + SignedRankMargin	$O(n \log n)$
$\operatorname{Spread}$	Monahans selection adapted for differences	$O(n \log n)$
$\operatorname{SpreadBounds}$	Disjoint-pair sign-test inversion	$O(n \log n)$

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: $\frac{\operatorname{Shift}}{\operatorname{AvgSpread}}$	$O((n+m) \log L + n \log n + m \log m)$
$\operatorname{DisparityBounds}$	Bonferroni split: ShiftBounds + AvgSpreadBounds	$O((n+m) \log L + n \log n + m \log m)$

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)	$O(n m)$ / $O(\log(n m))$
$\operatorname{SignedRankMargin}$	Dynamic programming (exact) / Edgeworth (approx)	$O(n^3)$ / $O(\log n)$