public void StudentTest2()
{
// make sure Student t is consistent with its definition
// we are going to take a sample that we expect to be t-distributed
Sample tSample = new Sample();
// begin with an underlying normal distribution
Distribution xDistribution = new NormalDistribution();
// compute a bunch of t satistics from the distribution
for (int i = 0; i < 100000; i++) {
// take a small sample from the underlying distribution
// (as the sample gets large, the t distribution becomes normal)
Random rng = new Random(314159+i);
double p = xDistribution.InverseLeftProbability(rng.NextDouble());
double q = 0.0;
for (int j = 0; j < 5; j++) {
double x = xDistribution.InverseLeftProbability(rng.NextDouble());
q += x * x;
}
q = q / 5;
double t = p / Math.Sqrt(q);
tSample.Add(t);
}
Distribution tDistribution = new StudentDistribution(5);
TestResult result = tSample.KolmogorovSmirnovTest(tDistribution);
Console.WriteLine(result.LeftProbability);
}
public void CauchyStudentAgreement()
{
StudentDistribution S = new StudentDistribution(1);
CauchyDistribution C = new CauchyDistribution();
// don't compare moments directly, because NaN != NaN
foreach (double P in probabilities) {
double xS = S.InverseLeftProbability(P);
double xC = C.InverseLeftProbability(P);
Console.WriteLine("{0} {1} {2}", P, xS, xC);
Assert.IsTrue(TestUtilities.IsNearlyEqual(xS, xC));
Assert.IsTrue(TestUtilities.IsNearlyEqual(S.ProbabilityDensity(xS), C.ProbabilityDensity(xC)));
}
}
public void StudentTest()
{
// make sure Student t is consistent with its definition
// we are going to take a sample that we expect to be t-distributed
Sample tSample = new Sample();
// begin with an underlying normal distribution
Distribution xDistribution = new NormalDistribution(1.0, 2.0);
// compute a bunch of t satistics from the distribution
for (int i = 0; i < 200000; i++) {
// take a small sample from the underlying distribution
// (as the sample gets large, the t distribution becomes normal)
Random rng = new Random(i);
Sample xSample = new Sample();
for (int j = 0; j < 5; j++) {
double x = xDistribution.InverseLeftProbability(rng.NextDouble());
xSample.Add(x);
}
// compute t for the sample
double t = (xSample.Mean - xDistribution.Mean) / (xSample.PopulationStandardDeviation.Value / Math.Sqrt(xSample.Count));
tSample.Add(t);
//Console.WriteLine(t);
}
// t's should be t-distrubuted; use a KS test to check this
Distribution tDistribution = new StudentDistribution(4);
TestResult result = tSample.KolmogorovSmirnovTest(tDistribution);
Console.WriteLine(result.LeftProbability);
//Assert.IsTrue(result.LeftProbability < 0.95);
// t's should be demonstrably not normally distributed
Console.WriteLine(tSample.KolmogorovSmirnovTest(new NormalDistribution()).LeftProbability);
//Assert.IsTrue(tSample.KolmogorovSmirnovTest(new NormalDistribution()).LeftProbability > 0.95);
}
public void TTestDistribution()
{
// start with a normally distributed population
Distribution xDistribution = new NormalDistribution(2.0, 3.0);
Random rng = new Random(1);
// draw 100 samples from it and compute the t statistic for each
Sample tSample = new Sample();
for (int i = 0; i < 100; i++) {
// each sample has 9 values
Sample xSample = new Sample();
for (int j = 0; j < 9; j++) {
xSample.Add(xDistribution.GetRandomValue(rng));
}
//Sample xSample = CreateSample(xDistribution, 10, i);
TestResult tResult = xSample.StudentTTest(2.0);
double t = tResult.Statistic;
Console.WriteLine("t = {0}", t);
tSample.Add(t);
}
// sanity check our sample of t's
Assert.IsTrue(tSample.Count == 100);
// check that the t statistics are distributed as expected
Distribution tDistribution = new StudentDistribution(9);
// check on the mean
Console.WriteLine("m = {0} vs. {1}", tSample.PopulationMean, tDistribution.Mean);
Assert.IsTrue(tSample.PopulationMean.ConfidenceInterval(0.95).ClosedContains(tDistribution.Mean), String.Format("{0} vs. {1}", tSample.PopulationMean, tDistribution.Mean));
// check on the standard deviation
Console.WriteLine("s = {0} vs. {1}", tSample.PopulationStandardDeviation, tDistribution.StandardDeviation);
Assert.IsTrue(tSample.PopulationStandardDeviation.ConfidenceInterval(0.95).ClosedContains(tDistribution.StandardDeviation));
// do a KS test
TestResult ksResult = tSample.KolmogorovSmirnovTest(tDistribution);
Assert.IsTrue(ksResult.LeftProbability < 0.95);
Console.WriteLine("D = {0}", ksResult.Statistic);
// check that we can distinguish the t distribution from a normal distribution?
}
