ENH: Add exact Cramér-von Mises two-sample p-value gufunc

include/xsf/config.h

@@ -60,6 +60,7 @@
 #include <cuda/std/cstddef>
 #include <cuda/std/cstdint>
 #include <cuda/std/limits>
+#include <cuda/std/numeric>
 #include <cuda/std/tuple>
 #include <cuda/std/type_traits>
 #include <cuda/std/utility>
@@ -156,6 +157,9 @@ XSF_HOST_DEVICE constexpr T clamp(T &v, T &lo, T &hi) {
 template <typename T>
 using numeric_limits = cuda::std::numeric_limits<T>;
 
+using cuda::std::gcd;
+using cuda::std::lcm;
+
 // Must use thrust for complex types in order to support CuPy
 template <typename T>
 using complex = thrust::complex<T>;
@@ -244,6 +248,7 @@ using cuda::std::uint64_t;
 #include <iterator>
 #include <limits>
 #include <math.h>
+#include <numeric>
 #include <tuple>
 #include <type_traits>
 #include <utility>

include/xsf/stats.h

@@ -1,6 +1,7 @@
 #pragma once
 
 #include "xsf/bessel.h"
+#include "xsf/binom.h"
 #include "xsf/cephes/bdtr.h"
 #include "xsf/cephes/chdtr.h"
 #include "xsf/cephes/fdtr.h"
@@ -284,6 +285,126 @@ inline double pdtrc(double k, double m) { return cephes::pdtrc(k, m); }
 
 inline double pdtri(int k, double y) { return cephes::pdtri(k, y); }
 
+namespace detail {
+
+    template <typename FreqTable2D>
+    XSF_HOST_DEVICE inline void
+    cvm_freq_table_all(int64_t m, int64_t n, int64_t a, int64_t b, FreqTable2D gs, FreqTable2D next_gs) {
+        using T = typename FreqTable2D::value_type;
+        int64_t K = static_cast<int64_t>(gs.extent(1));
+
+        // initialize gs to 0
+        for (int64_t v = 0; v < m + 1; ++v)
+            for (int64_t k = 0; k < K; ++k)
+                gs(v, k) = T(0);
+        // base case: gs(0, 0) = 1
+        gs(0, 0) = T(1);
+
+        for (int64_t u = 0; u < n + 1; ++u) {
+            // v = 0: no next_gs(v-1, ...) term
+            {
+                int64_t d = -b * u;
+                int64_t d2 = d * d;
+                int64_t kstart = (d2 < K) ? d2 : K;
+                // next_gs(0, k) = gs(0, k - d2) for k >= d2, else 0
+                for (int64_t k = 0; k < kstart; ++k) {
+                    next_gs(0, k) = T(0);
+                }
+                for (int64_t k = kstart; k < K; ++k) {
+                    next_gs(0, k) = gs(0, k - d2);
+                }
+            }
+            // v > 0: both terms contribute
+            for (int64_t v = 1; v < m + 1; ++v) {
+                int64_t d = a * v - b * u;
+                int64_t d2 = d * d; // d^2 = (a*v - b*u)^2
+                int64_t kstart = (d2 < K) ? d2 : K;
+                for (int64_t k = 0; k < kstart; ++k) {
+                    next_gs(v, k) = T(0);
+                }
+                for (int64_t k = kstart; k < K; ++k) {
+                    next_gs(v, k) = next_gs(v - 1, k - d2) + gs(v, k - d2);
+                }
+            }
+            FreqTable2D tmp = gs;
+            gs = next_gs;
+            next_gs = tmp;
+        }
+        // We swap `gs` and `next_gs` at each u-step, so buffer parity depends on n.
+        // If n is even, the final table ends up in the original `next_gs` buffer;
+        // copy it back so the caller can always read results from the original `gs`.
+        if (n % 2 == 0) {
+            for (int64_t v = 0; v < m + 1; ++v) {
+                for (int64_t k = 0; k < K; ++k) {
+                    next_gs(v, k) = gs(v, k);
+                }
+            }
+        }
+    }
+
+} // namespace detail
+
+template <typename FreqTable2D>
+XSF_HOST_DEVICE inline void cvm_2samp_freq_table(int64_t m, int64_t n, FreqTable2D freq_table, FreqTable2D workspace) {
+    /*
+     * Generate the exact Cramér-von Mises two-sample frequency table for
+     * sample sizes m and n. The table is independent of the scalar statistic.
+     */
+    if (m <= 0 || n <= 0) {
+        set_error("cvm_2samp_freq_table", SF_ERROR_DOMAIN, "m and n must be positive");
+        return;
+    }
+    // [1, p. 3]
+    int64_t lcm = std::lcm(m, n);
+    // [1, p. 4], below eq. 3
+    int64_t a = lcm / m;
+    int64_t b = lcm / n;
+
+    detail::cvm_freq_table_all(m, n, a, b, freq_table, workspace);
+}
+
+template <typename FreqTable2D>
+XSF_HOST_DEVICE inline double pval_cvm_2samp_exact(double s, int64_t m, int64_t n, FreqTable2D freq_table) {
+    /*
+     * Compute the exact p-value of the Cramér-von Mises two-sample test
+     * for a given value s of the test statistic and where m and n are the sizes
+     * of the samples.
+     *
+     * [1] Y. Xiao, A. Gordon, and A. Yakovlev, "A C++ Program for
+     *     the Cramér-Von Mises Two-Sample Test", J. Stat. Soft.,
+     *     vol. 17, no. 8, pp. 1-15, Dec. 2006.
+     * [2] T. W. Anderson "On the Distribution of the Two-Sample Cramér-von Mises
+     *     Criterion," The Annals of Mathematical Statistics, Ann. Math. Statist.
+     *     33(3), 1148-1159, (September, 1962)
+     */
+    if (m <= 0 || n <= 0) {
+        set_error("pval_cvm_2samp_exact", SF_ERROR_DOMAIN, "m and n must be positive");
+        return std::numeric_limits<double>::quiet_NaN();
+    }
+    // [1, p. 3]
+    int64_t lcm = std::lcm(m, n);
+    // Combine Eq. 9 in [2] with Eq. 2 in [1] and solve for $\zeta$
+    // Hint: `s` is $U$ in [2], and $T_2$ in [1] is $T$ in [2]
+    int64_t mn = m * n;
+
+    // Uses double floor division since s is double
+    int64_t zeta =
+        static_cast<int64_t>(std::floor((lcm * lcm * (m + n) * (6.0 * s - mn * (4.0 * mn - 1))) / (6.0 * mn * mn)));
+
+    int64_t K = static_cast<int64_t>(freq_table.extent(1));
+
+    // Clamp to prevent negative indexing when zeta < 0.
+    int64_t k0 = std::max<int64_t>(0, zeta);
+
+    int64_t sum_freq = 0;
+    for (int64_t k = k0; k < K; ++k) {
+        sum_freq += freq_table(m, k);
+    }
+
+    double combinations = xsf::binom(static_cast<double>(m + n), static_cast<double>(m));
+    return sum_freq / combinations;
+}
+
 inline double smirnov(int n, double x) { return cephes::smirnov(n, x); }
 
 inline double smirnovc(int n, double x) { return cephes::smirnovc(n, x); }

tests/xsf_tests/test_pval_cvm_2samp_exact.cpp

@@ -0,0 +1,53 @@
+#include "../testing_utils.h"
+#define MDSPAN_USE_PAREN_OPERATOR 1
+#include <xsf/stats.h>
+#include <xsf/third_party/kokkos/mdspan.hpp>
+
+/*
+// Reference values computed with scipy.stats._hypotests._pval_cvm_2samp_exact
+
+import numpy as np
+from scipy import stats
+
+rng = np.random.default_rng(seed=42)
+
+list_m = rng.integers(3, 30, size=5)
+list_n = rng.integers(3, 30, size=5)
+rtol = 1e-10
+
+for m, n in zip(list_m, list_n):
+    x = rng.standard_normal(m)
+    y = rng.standard_normal(n)
+    res = stats.cramervonmises_2samp(x, y, method="exact")
+    T = res.statistic
+    # Convert normalized statistic T to the unnormalized U
+    U = m * n * (m + n) * T + m * n * (4 * m * n - 1) / 6
+    p_value = stats._hypotests._pval_cvm_2samp_exact(U, m, n)
+    assert np.isclose(res.pvalue, p_value, rtol=rtol), "The p-values do not match!"
+    print(f"U={U}, m={m}, n={n}, p-value={p_value}")
+*/
+TEST_CASE("pval_cvm_2samp_exact test", "[pval_cvm_2samp_exact][xsf_tests]") {
+    using test_case = std::tuple<double, int, int, double, double>;
+    auto [s, m, n, pval_expected, rtol] = GENERATE(
+        test_case{8862.0, 14, 8, 0.2679738562091503, 1e-10},
+        test_case{3491.0000000000005, 14, 5, 0.34657722738218094, 1e-10},
+        test_case{12559.0, 5, 26, 0.11812654860485784, 1e-10},
+        test_case{8901.0, 23, 5, 0.9907610907610908, 1e-10} //, test_case{119376.0, 20, 21, 0.5716351061359124, 1e-10}
+    );
+
+    const int64_t lcm = std::lcm(m, n);
+    const int64_t K = (m + n) * lcm * lcm + 1;
+
+    std::vector<int64_t> buf1((m + 1) * K, 0);
+    std::vector<int64_t> buf2((m + 1) * K, 0);
+
+    using mdspan_2d = std::mdspan<int64_t, std::dextents<size_t, 2>>;
+    mdspan_2d gs(buf1.data(), static_cast<size_t>(m + 1), static_cast<size_t>(K));
+    mdspan_2d next_gs(buf2.data(), static_cast<size_t>(m + 1), static_cast<size_t>(K));
+
+    xsf::cvm_2samp_freq_table(m, n, gs, next_gs);
+    auto pval = xsf::pval_cvm_2samp_exact(s, m, n, gs);
+    const double rel_error = xsf::extended_relative_error(pval, pval_expected);
+    CAPTURE(s, m, n, K, pval, pval_expected, rel_error);
+    REQUIRE(rel_error <= rtol);
+}