commons-math-legacy/src/test/java/org/apache/commons/math4/legacy/stat/inference/InferenceTestUtilsTest.java

/*
 * Licensed to the Apache Software Foundation (ASF) under one or more
 * contributor license agreements.  See the NOTICE file distributed with
 * this work for additional information regarding copyright ownership.
 * The ASF licenses this file to You under the Apache License, Version 2.0
 * (the "License"); you may not use this file except in compliance with
 * the License.  You may obtain a copy of the License at
 *
 *      http://www.apache.org/licenses/LICENSE-2.0
 *
 * Unless required by applicable law or agreed to in writing, software
 * distributed under the License is distributed on an "AS IS" BASIS,
 * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
 * See the License for the specific language governing permissions and
 * limitations under the License.
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package org.apache.commons.math4.legacy.stat.inference;

import java.util.ArrayList;
import java.util.List;

import org.apache.commons.statistics.distribution.NormalDistribution;
import org.apache.commons.math4.legacy.exception.DimensionMismatchException;
import org.apache.commons.math4.legacy.exception.NotPositiveException;
import org.apache.commons.math4.legacy.exception.NotStrictlyPositiveException;
import org.apache.commons.math4.legacy.exception.NullArgumentException;
import org.apache.commons.math4.legacy.exception.NumberIsTooSmallException;
import org.apache.commons.math4.legacy.exception.OutOfRangeException;
import org.apache.commons.math4.legacy.stat.descriptive.SummaryStatistics;
import org.apache.commons.math4.legacy.core.jdkmath.AccurateMath;
import org.junit.Assert;
import org.junit.Test;


/**
 * Test cases for the TestUtils class.
 *
 */
public class InferenceTestUtilsTest {
    @Test
    public void testChiSquare() {

        // Target values computed using R version 1.8.1
        // Some assembly required ;-)
        //      Use sum((obs - exp)^2/exp) for the chi-square statistic and
        //      1 - pchisq(sum((obs - exp)^2/exp), length(obs) - 1) for the p-value

        long[] observed = {10, 9, 11};
        double[] expected = {10, 10, 10};
        Assert.assertEquals("chi-square statistic", 0.2,  InferenceTestUtils.chiSquare(expected, observed), 10E-12);
        Assert.assertEquals("chi-square p-value", 0.904837418036, InferenceTestUtils.chiSquareTest(expected, observed), 1E-10);

        long[] observed1 = { 500, 623, 72, 70, 31 };
        double[] expected1 = { 485, 541, 82, 61, 37 };
        Assert.assertEquals( "chi-square test statistic", 9.023307936427388, InferenceTestUtils.chiSquare(expected1, observed1), 1E-10);
        Assert.assertEquals("chi-square p-value", 0.06051952647453607, InferenceTestUtils.chiSquareTest(expected1, observed1), 1E-9);
        Assert.assertTrue("chi-square test reject", InferenceTestUtils.chiSquareTest(expected1, observed1, 0.07));
        Assert.assertFalse("chi-square test accept", InferenceTestUtils.chiSquareTest(expected1, observed1, 0.05));

        try {
            InferenceTestUtils.chiSquareTest(expected1, observed1, 95);
            Assert.fail("alpha out of range, OutOfRangeException expected");
        } catch (OutOfRangeException ex) {
            // expected
        }

        long[] tooShortObs = { 0 };
        double[] tooShortEx = { 1 };
        try {
            InferenceTestUtils.chiSquare(tooShortEx, tooShortObs);
            Assert.fail("arguments too short, DimensionMismatchException expected");
        } catch (DimensionMismatchException ex) {
            // expected
        }

        // unmatched arrays
        long[] unMatchedObs = { 0, 1, 2, 3 };
        double[] unMatchedEx = { 1, 1, 2 };
        try {
            InferenceTestUtils.chiSquare(unMatchedEx, unMatchedObs);
            Assert.fail("arrays have different lengths, DimensionMismatchException expected");
        } catch (DimensionMismatchException ex) {
            // expected
        }

        // 0 expected count
        expected[0] = 0;
        try {
            InferenceTestUtils.chiSquareTest(expected, observed, .01);
            Assert.fail("bad expected count, NotStrictlyPositiveException expected");
        } catch (NotStrictlyPositiveException ex) {
            // expected
        }

        // negative observed count
        expected[0] = 1;
        observed[0] = -1;
        try {
            InferenceTestUtils.chiSquareTest(expected, observed, .01);
            Assert.fail("bad expected count, NotPositiveException expected");
        } catch (NotPositiveException ex) {
            // expected
        }

    }

    @Test
    public void testChiSquareIndependence() {

        // Target values computed using R version 1.8.1

        long[][] counts = { {40, 22, 43}, {91, 21, 28}, {60, 10, 22}};
        Assert.assertEquals( "chi-square test statistic", 22.709027688, InferenceTestUtils.chiSquare(counts), 1E-9);
        Assert.assertEquals("chi-square p-value", 0.000144751460134, InferenceTestUtils.chiSquareTest(counts), 1E-9);
        Assert.assertTrue("chi-square test reject", InferenceTestUtils.chiSquareTest(counts, 0.0002));
        Assert.assertFalse("chi-square test accept", InferenceTestUtils.chiSquareTest(counts, 0.0001));

        long[][] counts2 = {{10, 15}, {30, 40}, {60, 90} };
        Assert.assertEquals( "chi-square test statistic", 0.168965517241, InferenceTestUtils.chiSquare(counts2), 1E-9);
        Assert.assertEquals("chi-square p-value",0.918987499852, InferenceTestUtils.chiSquareTest(counts2), 1E-9);
        Assert.assertFalse("chi-square test accept", InferenceTestUtils.chiSquareTest(counts2, 0.1));

        // ragged input array
        long[][] counts3 = { {40, 22, 43}, {91, 21, 28}, {60, 10}};
        try {
            InferenceTestUtils.chiSquare(counts3);
            Assert.fail("Expecting DimensionMismatchException");
        } catch (DimensionMismatchException ex) {
            // expected
        }

        // insufficient data
        long[][] counts4 = {{40, 22, 43}};
        try {
            InferenceTestUtils.chiSquare(counts4);
            Assert.fail("Expecting DimensionMismatchException");
        } catch (DimensionMismatchException ex) {
            // expected
        }
        long[][] counts5 = {{40}, {40}, {30}, {10}};
        try {
            InferenceTestUtils.chiSquare(counts5);
            Assert.fail("Expecting DimensionMismatchException");
        } catch (DimensionMismatchException ex) {
            // expected
        }

        // negative counts
        long[][] counts6 = {{10, -2}, {30, 40}, {60, 90} };
        try {
            InferenceTestUtils.chiSquare(counts6);
            Assert.fail("Expecting NotPositiveException");
        } catch (NotPositiveException ex) {
            // expected
        }

        // bad alpha
        try {
            InferenceTestUtils.chiSquareTest(counts, 0);
            Assert.fail("Expecting OutOfRangeException");
        } catch (OutOfRangeException ex) {
            // expected
        }
    }

    @Test
    public void testChiSquareLargeTestStatistic() {
        double[] exp = new double[] {
                3389119.5, 649136.6, 285745.4, 25357364.76, 11291189.78, 543628.0,
                232921.0, 437665.75
        };

        long[] obs = new long[] {
                2372383, 584222, 257170, 17750155, 7903832, 489265, 209628, 393899
        };
        org.apache.commons.math4.legacy.stat.inference.ChiSquareTest csti =
            new org.apache.commons.math4.legacy.stat.inference.ChiSquareTest();
        double cst = csti.chiSquareTest(exp, obs);
        Assert.assertEquals("chi-square p-value", 0.0, cst, 1E-3);
        Assert.assertEquals( "chi-square test statistic",
                114875.90421929007, InferenceTestUtils.chiSquare(exp, obs), 1E-9);
    }

    /** Contingency table containing zeros - PR # 32531 */
    @Test
    public void testChiSquareZeroCount() {
        // Target values computed using R version 1.8.1
        long[][] counts = { {40, 0, 4}, {91, 1, 2}, {60, 2, 0}};
        Assert.assertEquals( "chi-square test statistic", 9.67444662263,
                InferenceTestUtils.chiSquare(counts), 1E-9);
        Assert.assertEquals("chi-square p-value", 0.0462835770603,
                InferenceTestUtils.chiSquareTest(counts), 1E-9);
    }

    private double[] tooShortObs = { 1.0 };
    private double[] emptyObs = {};
    private SummaryStatistics emptyStats = new SummaryStatistics();

    @Test
    public void testOneSampleT() {
        double[] observed =
            {93.0, 103.0, 95.0, 101.0, 91.0, 105.0, 96.0, 94.0, 101.0,  88.0, 98.0, 94.0, 101.0, 92.0, 95.0 };
        double mu = 100.0;
        SummaryStatistics sampleStats = null;
        sampleStats = new SummaryStatistics();
        for (int i = 0; i < observed.length; i++) {
            sampleStats.addValue(observed[i]);
        }

        // Target comparison values computed using R version 1.8.1 (Linux version)
        Assert.assertEquals("t statistic",  -2.81976445346,
                InferenceTestUtils.t(mu, observed), 10E-10);
        Assert.assertEquals("t statistic",  -2.81976445346,
                InferenceTestUtils.t(mu, sampleStats), 10E-10);
        Assert.assertEquals("p value", 0.0136390585873,
                InferenceTestUtils.tTest(mu, observed), 10E-10);
        Assert.assertEquals("p value", 0.0136390585873,
                InferenceTestUtils.tTest(mu, sampleStats), 10E-10);

        try {
            InferenceTestUtils.t(mu, (double[]) null);
            Assert.fail("arguments too short, NullArgumentException expected");
        } catch (NullArgumentException ex) {
            // expected
        }

        try {
            InferenceTestUtils.t(mu, (SummaryStatistics) null);
            Assert.fail("arguments too short, NullArgumentException expected");
        } catch (NullArgumentException ex) {
            // expected
        }

        try {
            InferenceTestUtils.t(mu, emptyObs);
            Assert.fail("arguments too short, NumberIsTooSmallException expected");
        } catch (NumberIsTooSmallException ex) {
            // expected
        }

        try {
            InferenceTestUtils.t(mu, emptyStats);
            Assert.fail("arguments too short, NumberIsTooSmallException expected");
        } catch (NumberIsTooSmallException ex) {
            // expected
        }

        try {
            InferenceTestUtils.t(mu, tooShortObs);
            Assert.fail("insufficient data to compute t statistic, NumberIsTooSmallException expected");
        } catch (NumberIsTooSmallException ex) {
            // expected
        }
        try {
            InferenceTestUtils.tTest(mu, tooShortObs);
            Assert.fail("insufficient data to perform t test, NumberIsTooSmallException expected");
        } catch (NumberIsTooSmallException ex) {
            // expected
        }

        try {
            InferenceTestUtils.t(mu, (SummaryStatistics) null);
            Assert.fail("insufficient data to compute t statistic, NullArgumentException expected");
        } catch (NullArgumentException ex) {
            // expected
        }
        try {
            InferenceTestUtils.tTest(mu, (SummaryStatistics) null);
            Assert.fail("insufficient data to perform t test, NullArgumentException expected");
        } catch (NullArgumentException ex) {
            // expected
        }
    }

    @Test
    public void testOneSampleTTest() {
        double[] oneSidedP =
            {2d, 0d, 6d, 6d, 3d, 3d, 2d, 3d, -6d, 6d, 6d, 6d, 3d, 0d, 1d, 1d, 0d, 2d, 3d, 3d };
        SummaryStatistics oneSidedPStats = new SummaryStatistics();
        for (int i = 0; i < oneSidedP.length; i++) {
            oneSidedPStats.addValue(oneSidedP[i]);
        }
        // Target comparison values computed using R version 1.8.1 (Linux version)
        Assert.assertEquals("one sample t stat", 3.86485535541,
                InferenceTestUtils.t(0d, oneSidedP), 10E-10);
        Assert.assertEquals("one sample t stat", 3.86485535541,
                InferenceTestUtils.t(0d, oneSidedPStats),1E-10);
        Assert.assertEquals("one sample p value", 0.000521637019637,
                InferenceTestUtils.tTest(0d, oneSidedP) / 2d, 10E-10);
        Assert.assertEquals("one sample p value", 0.000521637019637,
                InferenceTestUtils.tTest(0d, oneSidedPStats) / 2d, 10E-5);
        Assert.assertTrue("one sample t-test reject", InferenceTestUtils.tTest(0d, oneSidedP, 0.01));
        Assert.assertTrue("one sample t-test reject", InferenceTestUtils.tTest(0d, oneSidedPStats, 0.01));
        Assert.assertFalse("one sample t-test accept", InferenceTestUtils.tTest(0d, oneSidedP, 0.0001));
        Assert.assertFalse("one sample t-test accept", InferenceTestUtils.tTest(0d, oneSidedPStats, 0.0001));

        try {
            InferenceTestUtils.tTest(0d, oneSidedP, 95);
            Assert.fail("alpha out of range, OutOfRangeException expected");
        } catch (OutOfRangeException ex) {
            // expected
        }

        try {
            InferenceTestUtils.tTest(0d, oneSidedPStats, 95);
            Assert.fail("alpha out of range, OutOfRangeException expected");
        } catch (OutOfRangeException ex) {
            // expected
        }

    }

    @Test
    public void testTwoSampleTHeterscedastic() {
        double[] sample1 = { 7d, -4d, 18d, 17d, -3d, -5d, 1d, 10d, 11d, -2d };
        double[] sample2 = { -1d, 12d, -1d, -3d, 3d, -5d, 5d, 2d, -11d, -1d, -3d };
        SummaryStatistics sampleStats1 = new SummaryStatistics();
        for (int i = 0; i < sample1.length; i++) {
            sampleStats1.addValue(sample1[i]);
        }
        SummaryStatistics sampleStats2 = new SummaryStatistics();
        for (int i = 0; i < sample2.length; i++) {
            sampleStats2.addValue(sample2[i]);
        }

        // Target comparison values computed using R version 1.8.1 (Linux version)
        Assert.assertEquals("two sample heteroscedastic t stat", 1.60371728768,
                InferenceTestUtils.t(sample1, sample2), 1E-10);
        Assert.assertEquals("two sample heteroscedastic t stat", 1.60371728768,
                InferenceTestUtils.t(sampleStats1, sampleStats2), 1E-10);
        Assert.assertEquals("two sample heteroscedastic p value", 0.128839369622,
                InferenceTestUtils.tTest(sample1, sample2), 1E-10);
        Assert.assertEquals("two sample heteroscedastic p value", 0.128839369622,
                InferenceTestUtils.tTest(sampleStats1, sampleStats2), 1E-10);
        Assert.assertTrue("two sample heteroscedastic t-test reject",
                InferenceTestUtils.tTest(sample1, sample2, 0.2));
        Assert.assertTrue("two sample heteroscedastic t-test reject",
                InferenceTestUtils.tTest(sampleStats1, sampleStats2, 0.2));
        Assert.assertFalse("two sample heteroscedastic t-test accept", InferenceTestUtils.tTest(sample1, sample2, 0.1));
        Assert.assertFalse("two sample heteroscedastic t-test accept", InferenceTestUtils.tTest(sampleStats1, sampleStats2, 0.1));

        try {
            InferenceTestUtils.tTest(sample1, sample2, .95);
            Assert.fail("alpha out of range, OutOfRangeException expected");
        } catch (OutOfRangeException ex) {
            // expected
        }

        try {
            InferenceTestUtils.tTest(sampleStats1, sampleStats2, .95);
            Assert.fail("alpha out of range, OutOfRangeException expected");
        } catch (OutOfRangeException ex) {
            // expected
        }

        try {
            InferenceTestUtils.tTest(sample1, tooShortObs, .01);
            Assert.fail("insufficient data, NumberIsTooSmallException expected");
        } catch (NumberIsTooSmallException ex) {
            // expected
        }

        try {
            InferenceTestUtils.tTest(sampleStats1, (SummaryStatistics) null, .01);
            Assert.fail("insufficient data, NullArgumentException expected");
        } catch (NullArgumentException ex) {
            // expected
        }

        try {
            InferenceTestUtils.tTest(sample1, tooShortObs);
            Assert.fail("insufficient data, NumberIsTooSmallException expected");
        } catch (NumberIsTooSmallException ex) {
            // expected
        }

        try {
            InferenceTestUtils.tTest(sampleStats1, (SummaryStatistics) null);
            Assert.fail("insufficient data, NullArgumentException expected");
        } catch (NullArgumentException ex) {
            // expected
        }

        try {
            InferenceTestUtils.t(sample1, tooShortObs);
            Assert.fail("insufficient data, NumberIsTooSmallException expected");
        } catch (NumberIsTooSmallException ex) {
            // expected
        }

        try {
            InferenceTestUtils.t(sampleStats1, (SummaryStatistics) null);
            Assert.fail("insufficient data, NullArgumentException expected");
        } catch (NullArgumentException ex) {
            // expected
        }
    }
    @Test
    public void testTwoSampleTHomoscedastic() {
        double[] sample1 ={2, 4, 6, 8, 10, 97};
        double[] sample2 = {4, 6, 8, 10, 16};
        SummaryStatistics sampleStats1 = new SummaryStatistics();
        for (int i = 0; i < sample1.length; i++) {
            sampleStats1.addValue(sample1[i]);
        }
        SummaryStatistics sampleStats2 = new SummaryStatistics();
        for (int i = 0; i < sample2.length; i++) {
            sampleStats2.addValue(sample2[i]);
        }

        // Target comparison values computed using R version 1.8.1 (Linux version)
        Assert.assertEquals("two sample homoscedastic t stat", 0.73096310086,
                InferenceTestUtils.homoscedasticT(sample1, sample2), 10E-11);
        Assert.assertEquals("two sample homoscedastic p value", 0.4833963785,
                InferenceTestUtils.homoscedasticTTest(sampleStats1, sampleStats2), 1E-10);
        Assert.assertTrue("two sample homoscedastic t-test reject",
                InferenceTestUtils.homoscedasticTTest(sample1, sample2, 0.49));
        Assert.assertFalse("two sample homoscedastic t-test accept", InferenceTestUtils.homoscedasticTTest(sample1, sample2, 0.48));
    }

    @Test
    public void testSmallSamples() {
        double[] sample1 = {1d, 3d};
        double[] sample2 = {4d, 5d};

        // Target values computed using R, version 1.8.1 (linux version)
        Assert.assertEquals(-2.2360679775, InferenceTestUtils.t(sample1, sample2),
                1E-10);
        Assert.assertEquals(0.198727388935, InferenceTestUtils.tTest(sample1, sample2),
                1E-10);
    }

    @Test
    public void testPaired() {
        double[] sample1 = {1d, 3d, 5d, 7d};
        double[] sample2 = {0d, 6d, 11d, 2d};
        double[] sample3 = {5d, 7d, 8d, 10d};

        // Target values computed using R, version 1.8.1 (linux version)
        Assert.assertEquals(-0.3133, InferenceTestUtils.pairedT(sample1, sample2), 1E-4);
        Assert.assertEquals(0.774544295819, InferenceTestUtils.pairedTTest(sample1, sample2), 1E-10);
        Assert.assertEquals(0.001208, InferenceTestUtils.pairedTTest(sample1, sample3), 1E-6);
        Assert.assertFalse(InferenceTestUtils.pairedTTest(sample1, sample3, .001));
        Assert.assertTrue(InferenceTestUtils.pairedTTest(sample1, sample3, .002));
    }

    private double[] classA =
      {93.0, 103.0, 95.0, 101.0};
    private double[] classB =
      {99.0, 92.0, 102.0, 100.0, 102.0};
    private double[] classC =
      {110.0, 115.0, 111.0, 117.0, 128.0};

    private List<double[]> classes = new ArrayList<>();
    private OneWayAnova oneWayAnova = new OneWayAnova();

    @Test
    public void testOneWayAnovaUtils() {
        classes.add(classA);
        classes.add(classB);
        classes.add(classC);
        Assert.assertEquals(oneWayAnova.anovaFValue(classes),
                InferenceTestUtils.oneWayAnovaFValue(classes), 10E-12);
        Assert.assertEquals(oneWayAnova.anovaPValue(classes),
                InferenceTestUtils.oneWayAnovaPValue(classes), 10E-12);
        Assert.assertEquals(oneWayAnova.anovaTest(classes, 0.01),
                InferenceTestUtils.oneWayAnovaTest(classes, 0.01));
    }
    @Test
    public void testGTestGoodnesOfFit() throws Exception {
        double[] exp = new double[]{
            0.54d, 0.40d, 0.05d, 0.01d
        };

        long[] obs = new long[]{
            70, 79, 3, 4
        };
        Assert.assertEquals("G test statistic",
                13.144799, InferenceTestUtils.g(exp, obs), 1E-5);
        double p_gtgf = InferenceTestUtils.gTest(exp, obs);
        Assert.assertEquals("g-Test p-value", 0.004333, p_gtgf, 1E-5);

        Assert.assertTrue(InferenceTestUtils.gTest(exp, obs, 0.05));
}

    @Test
    public void testGTestIndependence() throws Exception {
        long[] obs1 = new long[]{
            268, 199, 42
        };

        long[] obs2 = new long[]{
            807, 759, 184
        };

        double g = InferenceTestUtils.gDataSetsComparison(obs1, obs2);

        Assert.assertEquals("G test statistic",
                7.3008170, g, 1E-4);
        double p_gti = InferenceTestUtils.gTestDataSetsComparison(obs1, obs2);

        Assert.assertEquals("g-Test p-value", 0.0259805, p_gti, 1E-4);
        Assert.assertTrue(InferenceTestUtils.gTestDataSetsComparison(obs1, obs2, 0.05));
    }

    @Test
    public void testRootLogLikelihood() {
        // positive where k11 is bigger than expected.
        Assert.assertTrue(InferenceTestUtils.rootLogLikelihoodRatio(904, 21060, 1144, 283012) > 0.0);

        // negative because k11 is lower than expected
        Assert.assertTrue(InferenceTestUtils.rootLogLikelihoodRatio(36, 21928, 60280, 623876) < 0.0);

        Assert.assertEquals(AccurateMath.sqrt(2.772589), InferenceTestUtils.rootLogLikelihoodRatio(1, 0, 0, 1), 0.000001);
        Assert.assertEquals(-AccurateMath.sqrt(2.772589), InferenceTestUtils.rootLogLikelihoodRatio(0, 1, 1, 0), 0.000001);
        Assert.assertEquals(AccurateMath.sqrt(27.72589), InferenceTestUtils.rootLogLikelihoodRatio(10, 0, 0, 10), 0.00001);

        Assert.assertEquals(AccurateMath.sqrt(39.33052), InferenceTestUtils.rootLogLikelihoodRatio(5, 1995, 0, 100000), 0.00001);
        Assert.assertEquals(-AccurateMath.sqrt(39.33052), InferenceTestUtils.rootLogLikelihoodRatio(0, 100000, 5, 1995), 0.00001);

        Assert.assertEquals(AccurateMath.sqrt(4730.737), InferenceTestUtils.rootLogLikelihoodRatio(1000, 1995, 1000, 100000), 0.001);
        Assert.assertEquals(-AccurateMath.sqrt(4730.737), InferenceTestUtils.rootLogLikelihoodRatio(1000, 100000, 1000, 1995), 0.001);

        Assert.assertEquals(AccurateMath.sqrt(5734.343), InferenceTestUtils.rootLogLikelihoodRatio(1000, 1000, 1000, 100000), 0.001);
        Assert.assertEquals(AccurateMath.sqrt(5714.932), InferenceTestUtils.rootLogLikelihoodRatio(1000, 1000, 1000, 99000), 0.001);
    }

    @Test
    public void testKSOneSample() throws Exception {
       final NormalDistribution unitNormal = NormalDistribution.of(0d, 1d);
       final double[] sample = KolmogorovSmirnovTestTest.gaussian;
       final double tol = 1e-10;
       Assert.assertEquals(0.3172069207622391, InferenceTestUtils.kolmogorovSmirnovTest(unitNormal, sample), tol);
       Assert.assertEquals(0.0932947561266756, InferenceTestUtils.kolmogorovSmirnovStatistic(unitNormal, sample), tol);
    }

    @Test
    public void testKSTwoSample() throws Exception {
        final double tol = 1e-10;
        final double[] smallSample1 = {
            6, 7, 9, 13, 19, 21, 22, 23, 24
        };
        final double[] smallSample2 = {
            10, 11, 12, 16, 20, 27, 28, 32, 44, 54
        };
        Assert.assertEquals(0.105577085453247, InferenceTestUtils.kolmogorovSmirnovTest(smallSample1, smallSample2, false), tol);
        final double d = InferenceTestUtils.kolmogorovSmirnovStatistic(smallSample1, smallSample2);
        Assert.assertEquals(0.5, d, tol);
        Assert.assertEquals(0.105577085453247, InferenceTestUtils.exactP(d, smallSample1.length, smallSample2.length, false), tol);
    }
}