Basic Examples (3) 

Two samples of size 20 drawn from the same uniform distribution:

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Simplest low-level call: no ties, exact rational p-value:

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Use the multiplicity vector to indicate how many pooled-sample values are tied at each distinct value; here m=120,n=150 with four groups of tied observations at q=11/60:

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Scope (2) 

For discrete circular data with ties, the Kuiper test is well-suited to circular data because its statistic is invariant to the choice of origin on the circle:

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Unequal sample sizes with ties from discrete distributions:

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Applications (11) 

Load the MTEX Forsterite EBSD (Electron Backscatter Diffraction) dataset. The dataset contains 178,403 crystallographic measurements from a polished peridotite thin section (Bachmann, Hielscher, Schaeben, 2010; GPL-2.0; available at https://github.com/mtex-toolbox/mtex/tree/develop/data/EBSD), including phase labels, spatial coordinates, Euler angles, and EBSD quality metrics (columns: Phase, x, y, Bands, Error, φ₁, φ₂, φ₃, MAD, BC, BS):

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Extract the φ₁ Euler angle for each mineral phase. Column 6 contains φ₁ in degrees. Separate the measurements into Forsterite (phase 1) and Enstatite (phase 2) for subsequent comparison of their angular distributions:

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Verify the number of measurements in each phase. The resulting subsets contain 152,345 Forsterite measurements and 26,058 Enstatite measurements:

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List the eight most frequently repeated φ₁ values in the Forsterite data. Maximum repeat count is 44, indicating moderate concentration around particular orientations:

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The full dataset (M = 152,345 Forsterite, N = 26,058 Enstatite) is too large to run the exact test directly. Draw a reproducible subsample of m = 1200 and n = 200, preserving the original ~6:1 phase ratio:

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Approximately 2.3% of the pooled subsample observations are tied, confirming that exact tie-handling is necessary:

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Compare the two circular φ₁ distributions using the two-sample Kuiper test, which is invariant to the choice of angular origin:

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Exact p-value evaluated to 30 significant digits:

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To assess the robustness of the exact tie-handling, round the subsample angles to 0.1° — within typical EBSD angular measurement precision — artificially increasing the proportion of tied observations while keeping the same data:

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The tie proportion increases from 2.3% to 44.7% after rounding:

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Apply the Kuiper test to the rounded subsample. The p-value is virtually unchanged despite the large increase in ties, confirming that exact tie-handling leaves the inference unaffected:

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Properties and Relations (3) 

The Kuiper statistic is invariant to cyclic shifts of the data. Shifting data₁ and data₂ by an arbitrary constant on the circle leaves the Kuiper statistic and p-value unchanged.

Four equally spaced mass points on[0, 2π) with unequal observation counts:

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Shift every observation by π/3 on the circle and retest:

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The Kuiper statistic and p-value are exactly preserved under the cyclic shift:

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Possible Issues (4) 

Random variables d2 and d3 are drawn from the same seed so they have nearly identical Kuiper statistics against d1; the timing difference is due to LCM[m,n] alone:

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Confirm the statistics are close:

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Complexity is O(mn*r), where r= LCM[m,n]. When m=n,LCM[m,n]=m so complexity is O(m³). When m and n are coprime (e.g. m=n+1), LCM[m,n]=mn and complexity is O(m²n²), which is much larger. For larger samples consider the R package KSgeneral which uses a compiled C++ implementation:

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Timing and p-values for both cases:

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