498 lines
16 KiB
Go
498 lines
16 KiB
Go
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// Copyright (c) 2026 Petr Balvín <opensource@petrbalvin.org> (https://petrbalvin.org)
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// SPDX-License-Identifier: MIT
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package stats
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import (
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"math"
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"sourcedock.dev/petrbalvin/tensor/internal/core"
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"strings"
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"testing"
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)
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// TestNoncentralChiSquareClosed pins the noncentral χ² on the exact
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// closed form its df = 1 corner carries: χ²(1, λ) is the square of a
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// N(√λ, 1) draw, so the CDF is Φ(√x−√λ) − Φ(−√x−√λ), and on the λ = 0
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// reduction to the central law.
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func TestNoncentralChiSquareClosed(t *testing.T) {
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for _, lambda := range []float64{0.5, 1, 4, 9, 25} {
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for _, x := range []float64{0.5, 1, 2, 4, 9, 16} {
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got, err := NoncentralChiSquareCDF(x, 1, lambda)
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if err != nil {
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t.Fatalf("NoncentralChiSquareCDF(%g, 1, %g): %v", x, lambda, err)
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}
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root := math.Sqrt(lambda)
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want := NormalCDF(math.Sqrt(x)-root) - NormalCDF(-math.Sqrt(x)-root)
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if math.Abs(got-want) > 1e-13 {
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t.Fatalf("NoncentralChiSquareCDF(%g, 1, %g) = %.16g, want %.16g", x, lambda, got, want)
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}
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}
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}
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for _, df := range []int{1, 2, 5, 10} {
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for _, x := range []float64{0.5, 2, 7} {
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got, err := NoncentralChiSquareCDF(x, df, 0)
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if err != nil {
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t.Fatalf("NoncentralChiSquareCDF(%g, %d, 0): %v", x, df, err)
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}
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want, err := ChiSquareCDF(x, df)
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if err != nil || math.Abs(got-want) > 1e-14 {
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t.Fatalf("λ = 0 reduction at df %d: %v vs %v (%v)", df, got, want, err)
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}
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}
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}
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if v, _ := NoncentralChiSquareCDF(-1, 3, 2); v != 0 {
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t.Fatalf("CDF below the support = %v, want 0", v)
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}
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if v, _ := NoncentralChiSquareDensity(-1, 3, 2); v != 0 {
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t.Fatalf("density below the support = %v, want 0", v)
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}
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}
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// TestNoncentralChiSquareDensityIntegral integrates the density against
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// the CDF. The walk runs after the substitution x = s², which leaves
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// 2s·f(s²) smooth at the origin for every df (the density itself
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// behaves like x^{df/2−1} there, too flat a start for Simpson's error
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// estimate on the lower degrees). Simpson must then reproduce the CDF
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// to better than 1e-10 relative, the double route the closed forms
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// cannot cover.
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func TestNoncentralChiSquareDensityIntegral(t *testing.T) {
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type grid struct {
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df int
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lambda float64
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x float64
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}
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for _, g := range []grid{
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{2, 1, 6}, {3, 1, 6}, {3, 4, 10}, {5, 3, 12}, {10, 25, 60},
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} {
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n := 200000
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s := math.Sqrt(g.x)
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sh := s / float64(n)
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f := func(sv float64) float64 {
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if sv == 0 {
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return 0
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}
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xv := sv * sv
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d, err := NoncentralChiSquareDensity(xv, g.df, g.lambda)
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if err != nil {
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t.Fatalf("NoncentralChiSquareDensity: %v", err)
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}
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return 2 * sv * d
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}
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sum := f(0) + f(s)
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for i := 1; i < n; i++ {
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w := 4.0
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if i%2 == 0 {
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w = 2
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}
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sum += w * f(float64(i)*sh)
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}
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integral := sum * sh / 3
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cdf, err := NoncentralChiSquareCDF(g.x, g.df, g.lambda)
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if err != nil {
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t.Fatalf("NoncentralChiSquareCDF: %v", err)
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}
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if rel := math.Abs(integral-cdf) / cdf; rel > 1e-10 {
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t.Fatalf("df = %d, λ = %g, x = %g: density integral %.15g vs CDF %.15g (rel %g)",
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g.df, g.lambda, g.x, integral, cdf, rel)
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}
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}
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}
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// noncentralTOracle evaluates E[Φ(t√(V/ν) − δ)] for V ~ χ²(ν) by
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// Simpson after the substitution V = u², which leaves the integrand
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// smooth for every ν; the u = 0 limit is finite only for ν = 1.
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func noncentralTOracle(t float64, df int, delta float64) float64 {
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uhi := 40.0
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n := 200000
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h := uhi / float64(n)
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logGammaB := func(a float64) float64 { l, _ := math.Lgamma(a); return l }
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f := func(u float64) float64 {
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if u == 0 {
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if df == 1 {
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return math.Sqrt(2/math.Pi) * NormalCDF(-delta)
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}
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return 0
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}
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v := u * u
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fv := 2 * u * math.Exp((float64(df)/2-1)*math.Log(v)-v/2-(float64(df)/2)*math.Ln2-logGammaB(float64(df)/2))
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return NormalCDF(t*math.Sqrt(v/float64(df))-delta) * fv
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}
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sum := f(0) + f(uhi)
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for i := 1; i < n; i++ {
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w := 4.0
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if i%2 == 0 {
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w = 2
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}
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sum += w * f(float64(i)*h)
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}
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return sum * h / 3
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}
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// TestNoncentralTAgainstQuadrature holds the Lenth series against a
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// direct quadrature of E[Φ(t√(V/ν) − δ)], an independent route that
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// shares no code with the series, at a grid spanning both signs of t
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// and δ and degrees of freedom from 1 to 10. The large-δ cases are the
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// underflow round: a noncentrality whose Poisson weight seed e^{−δ²/2}
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// is below the double floor used to silence the whole series, and each
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// of them once answered a silent 0. Tolerance 1e-9, an order above the
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// quadrature's own accuracy.
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func TestNoncentralTAgainstQuadrature(t *testing.T) {
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for _, df := range []int{1, 2, 5, 10} {
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for _, delta := range []float64{-3, -0.5, 0.5, 2} {
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for _, tv := range []float64{-2, -0.5, 0.4, 1, 3} {
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got, err := NoncentralTCDF(tv, df, delta)
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if err != nil {
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t.Fatalf("NoncentralTCDF(%g, %d, %g): %v", tv, df, delta, err)
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}
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want := noncentralTOracle(tv, df, delta)
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if math.Abs(got-want) > 1e-9 {
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t.Fatalf("NoncentralTCDF(%g, %d, %g) = %.15g, want quadrature %.15g",
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tv, df, delta, got, want)
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}
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}
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}
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}
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for _, c := range []struct {
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tv, delta float64
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df int
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}{
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{45, 45, 5},
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{50, 40, 3},
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{-45, -45, 5},
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{100, 45, 5},
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} {
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got, err := NoncentralTCDF(c.tv, c.df, c.delta)
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if err != nil {
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t.Fatalf("NoncentralTCDF(%g, %d, %g): %v", c.tv, c.df, c.delta, err)
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}
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want := noncentralTOracle(c.tv, c.df, c.delta)
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if math.Abs(got-want) > 1e-9 {
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t.Fatalf("NoncentralTCDF(%g, %d, %g) = %.15g, want quadrature %.15g",
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c.tv, c.df, c.delta, got, want)
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}
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}
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}
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// TestNoncentralUnderflowSurvival pins the deep-noncentrality window
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// where the mixture weights' raw seed underflows: the χ² CDF at its
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// own mean answers a half, the far tail answers a genuinely computed
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// negligible value rather than a silent zero, the F CDF saturates at
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// 1 past the overflow of ν₁x, and the χ² density at the origin keeps
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// the finite df = 2 limit. Past the term budget the refusal is
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// explicit.
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func TestNoncentralUnderflowSurvival(t *testing.T) {
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atMean, err := NoncentralChiSquareCDF(2005, 5, 2000)
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if err != nil {
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t.Fatal(err)
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}
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if atMean < 0.48 || atMean > 0.52 {
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t.Fatalf("NoncentralChiSquareCDF at the mean 2005 = %g, want near a half", atMean)
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}
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tail, err := NoncentralChiSquareCDF(1005, 5, 2000)
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if err != nil {
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t.Fatal(err)
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}
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if !(tail >= 0 && tail < 1e-30) {
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t.Fatalf("NoncentralChiSquareCDF(1005, 5, 2000) = %g, want a negligible non-negative tail", tail)
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}
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for _, x := range []float64{math.MaxFloat64, math.Inf(1)} {
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f, err := NoncentralFCDF(x, 4, 10, 3)
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if err != nil {
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t.Fatal(err)
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}
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if math.Abs(f-1) > 1e-15 {
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t.Fatalf("NoncentralFCDF(%g, 4, 10, 3) = %g, want 1 to rounding", x, f)
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}
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}
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d, err := NoncentralChiSquareDensity(0, 2, 3)
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if err != nil {
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t.Fatal(err)
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}
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if want := 0.5 * math.Exp(-1.5); d != want {
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t.Fatalf("NoncentralChiSquareDensity(0, 2, 3) = %g, want the limit %g", d, want)
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}
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if _, err := NoncentralChiSquareCDF(10, 5, 4e5); err == nil || !strings.Contains(err.Error(), "budget") {
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t.Fatalf("lambda 4e5: error = %v, want the budget refusal", err)
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}
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if _, err := NoncentralTCDF(10, 5, 500); err == nil || !strings.Contains(err.Error(), "budget") {
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t.Fatalf("delta 500: error = %v, want the budget refusal", err)
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}
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}
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// TestNoncentralTIdentityReductions pins the exact corners: δ = 0 is
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// the central Student t, t = 0 is Φ(−δ), and the two reflection
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// identities of the law hold to rounding.
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func TestNoncentralTIdentityReductions(t *testing.T) {
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for _, df := range []int{1, 3, 8} {
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for _, tv := range []float64{-4, -1, 0.3, 2} {
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got, err := NoncentralTCDF(tv, df, 0)
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if err != nil {
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t.Fatalf("NoncentralTCDF(%g, %d, 0): %v", tv, df, err)
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}
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want, err := StudentTCDF(tv, df)
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if err != nil || got != want {
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t.Fatalf("δ = 0 reduction at (%g, %d): %v vs %v (%v)", tv, df, got, want, err)
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}
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}
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}
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for _, delta := range []float64{-3, -0.5, 1, 4} {
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got, _ := NoncentralTCDF(0, 5, delta)
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if want := NormalCDF(-delta); math.Abs(got-want) > 1e-15 {
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t.Fatalf("NoncentralTCDF(0, 5, %g) = %.16g, want Φ(−δ) = %.16g", delta, got, want)
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}
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}
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for _, delta := range []float64{-2, 1.5} {
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for _, tv := range []float64{-1, 0.7, 2} {
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pos, _ := NoncentralTCDF(tv, 4, delta)
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reflected, _ := NoncentralTCDF(-tv, 4, -delta)
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if math.Abs(pos+reflected-1) > 1e-14 {
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t.Fatalf("reflection broken at t = %g, δ = %g: %.17g", tv, delta, pos+reflected)
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}
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mirrored, _ := NoncentralTCDF(-tv, 4, delta)
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flipped, _ := NoncentralTCDF(tv, 4, -delta)
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if math.Abs(flipped-(1-mirrored)) > 1e-14 {
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t.Fatalf("sign symmetry broken at t = %g, δ = %g: %.17g vs %.17g",
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tv, delta, flipped, 1-mirrored)
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}
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}
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}
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}
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// TestNoncentralFIdentityReductions pins the noncentral F on its λ = 0
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// central reduction and on the df₁ = 1 identity with the noncentral t:
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// F(1, ν, λ) is the squared t(ν, √λ), so P(F ≤ y) is the t CDF across
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// ±√y. The CDF must also fall as the noncentrality grows.
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func TestNoncentralFIdentityReductions(t *testing.T) {
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for _, df2 := range []int{1, 4, 12} {
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for _, x := range []float64{0.3, 1, 2.5} {
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got, err := NoncentralFCDF(x, 1, df2, 0)
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if err != nil {
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t.Fatalf("NoncentralFCDF(%g, 1, %d, 0): %v", x, df2, err)
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}
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want, err := BetaIncomplete(x/(x+float64(df2)), 0.5, float64(df2)/2)
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if err != nil || math.Abs(got-want) > 1e-14 {
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t.Fatalf("central reduction at x = %g, df₂ = %d: %v vs %v (%v)", x, df2, got, want, err)
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}
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}
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}
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for _, lambda := range []float64{1, 4} {
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for _, df2 := range []int{2, 6} {
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for _, y := range []float64{0.5, 2, 6} {
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got, err := NoncentralFCDF(y, 1, df2, lambda)
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if err != nil {
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t.Fatalf("NoncentralFCDF(%g, 1, %d, %g): %v", y, df2, lambda, err)
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}
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root := math.Sqrt(lambda)
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hi, _ := NoncentralTCDF(math.Sqrt(y), df2, root)
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lo, _ := NoncentralTCDF(-math.Sqrt(y), df2, root)
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if math.Abs(got-(hi-lo)) > 1e-13 {
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t.Fatalf("t² identity at y = %g, ν = %d, λ = %g: %.15g vs %.15g",
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y, df2, lambda, got, hi-lo)
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}
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}
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}
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}
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central, _ := NoncentralFCDF(2, 3, 8, 0)
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shifted, _ := NoncentralFCDF(2, 3, 8, 5)
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if shifted >= central {
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t.Fatalf("a larger λ lowered the CDF from %g to %g", central, shifted)
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}
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}
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// TestNoncentralQuantileRoundTrips inverts each noncentral CDF and
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|||
|
|
// checks the CDF at the quantile returns q.
|
|||
|
|
func TestNoncentralQuantileRoundTrips(t *testing.T) {
|
|||
|
|
for _, q := range []float64{0.01, 0.25, 0.5, 0.9, 0.99} {
|
|||
|
|
x, err := NoncentralChiSquareQuantile(q, 5, 3)
|
|||
|
|
if err != nil {
|
|||
|
|
t.Fatalf("NoncentralChiSquareQuantile(%g): %v", q, err)
|
|||
|
|
}
|
|||
|
|
back, _ := NoncentralChiSquareCDF(x, 5, 3)
|
|||
|
|
if math.Abs(back-q) > 1e-10 {
|
|||
|
|
t.Fatalf("χ² round trip q = %g: CDF(quantile) = %v", q, back)
|
|||
|
|
}
|
|||
|
|
tq, err := NoncentralTQuantile(q, 5, 2)
|
|||
|
|
if err != nil {
|
|||
|
|
t.Fatalf("NoncentralTQuantile(%g): %v", q, err)
|
|||
|
|
}
|
|||
|
|
tback, _ := NoncentralTCDF(tq, 5, 2)
|
|||
|
|
if math.Abs(tback-q) > 1e-10 {
|
|||
|
|
t.Fatalf("t round trip q = %g: CDF(quantile) = %v", q, tback)
|
|||
|
|
}
|
|||
|
|
fq, err := NoncentralFQuantile(q, 4, 10, 2)
|
|||
|
|
if err != nil {
|
|||
|
|
t.Fatalf("NoncentralFQuantile(%g): %v", q, err)
|
|||
|
|
}
|
|||
|
|
fback, _ := NoncentralFCDF(fq, 4, 10, 2)
|
|||
|
|
if math.Abs(fback-q) > 1e-10 {
|
|||
|
|
t.Fatalf("F round trip q = %g: CDF(quantile) = %v", q, fback)
|
|||
|
|
}
|
|||
|
|
}
|
|||
|
|
// The t quantile leans towards δ.
|
|||
|
|
neg, _ := NoncentralTQuantile(0.5, 5, -2)
|
|||
|
|
if neg >= 0 {
|
|||
|
|
t.Fatalf("median of t(5, −2) = %g, want negative", neg)
|
|||
|
|
}
|
|||
|
|
}
|
|||
|
|
|
|||
|
|
// TestNoncentralFMonteCarlo rebuilds the noncentral F from the
|
|||
|
|
// package's own samplers: a χ²(df₁+2J) numerator with J drawn from the
|
|||
|
|
// Poisson, over an independent central χ²(df₂) denominator, checked
|
|||
|
|
// against the analytic CDF the mixture code computes. The sampler and
|
|||
|
|
// the mixture share no code. Statistical tolerance 0.01, far above the
|
|||
|
|
// 2σ of 50 000 draws.
|
|||
|
|
func TestNoncentralFMonteCarlo(t *testing.T) {
|
|||
|
|
g := core.NewGenerator(11)
|
|||
|
|
const n = 50000
|
|||
|
|
x := 2.0
|
|||
|
|
got, err := NoncentralFCDF(x, 4, 10, 3)
|
|||
|
|
if err != nil {
|
|||
|
|
t.Fatalf("NoncentralFCDF: %v", err)
|
|||
|
|
}
|
|||
|
|
j, err := PoissonDraws(g, n, 1.5)
|
|||
|
|
if err != nil {
|
|||
|
|
t.Fatalf("PoissonDraws: %v", err)
|
|||
|
|
}
|
|||
|
|
count := 0.0
|
|||
|
|
for i := range n {
|
|||
|
|
df := min(
|
|||
|
|
// The Poisson(1.5) tail never reaches 30; the fold is a
|
|||
|
|
// contract guard, not a working branch.
|
|||
|
|
4+2*int(j.FloatAt(i)), 64)
|
|||
|
|
num, err := ChiSquareDraws(g, 1, df)
|
|||
|
|
if err != nil {
|
|||
|
|
t.Fatalf("ChiSquareDraws: %v", err)
|
|||
|
|
}
|
|||
|
|
den, err := ChiSquareDraws(g, 1, 10)
|
|||
|
|
if err != nil {
|
|||
|
|
t.Fatalf("ChiSquareDraws: %v", err)
|
|||
|
|
}
|
|||
|
|
f := num.FloatAt(0) * 10 / (4 * den.FloatAt(0))
|
|||
|
|
if f <= x {
|
|||
|
|
count++
|
|||
|
|
}
|
|||
|
|
}
|
|||
|
|
if math.Abs(count/n-got) > 0.01 {
|
|||
|
|
t.Fatalf("sampler CDF = %.4f, analytic %.4f", count/n, got)
|
|||
|
|
}
|
|||
|
|
}
|
|||
|
|
|
|||
|
|
// TestNoncentralChiSquareMonteCarlo rebuilds the noncentral χ² from
|
|||
|
|
// PoissonDraws and ChiSquareDraws, the sampler route the mixture CDF
|
|||
|
|
// has no code in common with, at a tolerance the 200 000 draws can
|
|||
|
|
// carry.
|
|||
|
|
func TestNoncentralChiSquareMonteCarlo(t *testing.T) {
|
|||
|
|
g := core.NewGenerator(13)
|
|||
|
|
const n = 200000
|
|||
|
|
x := 6.0
|
|||
|
|
got, err := NoncentralChiSquareCDF(x, 3, 4)
|
|||
|
|
if err != nil {
|
|||
|
|
t.Fatalf("NoncentralChiSquareCDF: %v", err)
|
|||
|
|
}
|
|||
|
|
j, err := PoissonDraws(g, n, 2)
|
|||
|
|
if err != nil {
|
|||
|
|
t.Fatalf("PoissonDraws: %v", err)
|
|||
|
|
}
|
|||
|
|
// One shared χ²(3) stream rescaled per draw would not follow
|
|||
|
|
// χ²(3+2J), so the check walks the mixture identity the other way:
|
|||
|
|
// P(χ²(3+2J) ≤ x) averaged over the drawn J equals the CDF.
|
|||
|
|
total := 0.0
|
|||
|
|
for i := range n {
|
|||
|
|
df := min(3+2*int(j.FloatAt(i)), 400)
|
|||
|
|
p, err := ChiSquareCDF(x, df)
|
|||
|
|
if err != nil {
|
|||
|
|
t.Fatalf("ChiSquareCDF: %v", err)
|
|||
|
|
}
|
|||
|
|
total += p
|
|||
|
|
}
|
|||
|
|
if math.Abs(total/n-got) > 0.01 {
|
|||
|
|
t.Fatalf("sampler-route CDF = %v, want ≈ %v", total/n, got)
|
|||
|
|
}
|
|||
|
|
}
|
|||
|
|
|
|||
|
|
// TestNoncentralErrors pins the parameter contracts of the noncentral
|
|||
|
|
// family.
|
|||
|
|
func TestNoncentralErrors(t *testing.T) {
|
|||
|
|
if _, err := NoncentralChiSquareCDF(1, 0, 1); err == nil {
|
|||
|
|
t.Fatal("df = 0: want an error")
|
|||
|
|
}
|
|||
|
|
if _, err := NoncentralChiSquareCDF(1, 3, -1); err == nil {
|
|||
|
|
t.Fatal("negative λ: want an error")
|
|||
|
|
}
|
|||
|
|
if _, err := NoncentralChiSquareCDF(1, 3, math.Inf(1)); err == nil {
|
|||
|
|
t.Fatal("λ = +Inf: want an error")
|
|||
|
|
}
|
|||
|
|
if _, err := NoncentralChiSquareCDF(math.NaN(), 3, 1); err == nil {
|
|||
|
|
t.Fatal("NaN x: want an error")
|
|||
|
|
}
|
|||
|
|
if _, err := NoncentralChiSquareDensity(1, 0, 1); err == nil {
|
|||
|
|
t.Fatal("density df = 0: want an error")
|
|||
|
|
}
|
|||
|
|
if _, err := NoncentralChiSquareQuantile(0.5, 0, 1); err == nil {
|
|||
|
|
t.Fatal("quantile df = 0: want an error")
|
|||
|
|
}
|
|||
|
|
if _, err := NoncentralChiSquareQuantile(1.5, 3, 1); err == nil {
|
|||
|
|
t.Fatal("q outside [0, 1]: want an error")
|
|||
|
|
}
|
|||
|
|
if _, err := NoncentralChiSquareQuantile(0.5, 0, 1); err == nil {
|
|||
|
|
t.Fatal("quantile df = 0: want an error")
|
|||
|
|
}
|
|||
|
|
if _, err := NoncentralChiSquareQuantile(0.5, 3, math.NaN()); err == nil {
|
|||
|
|
t.Fatal("quantile NaN λ: want an error")
|
|||
|
|
}
|
|||
|
|
if v, err := NoncentralChiSquareCDF(math.Inf(1), 3, 2); err != nil || v != 1 {
|
|||
|
|
t.Fatalf("CDF at +Inf = %v, %v, want 1", v, err)
|
|||
|
|
}
|
|||
|
|
if v, err := NoncentralChiSquareDensity(0, 1, 2); err != nil || !math.IsInf(v, 1) {
|
|||
|
|
t.Fatalf("density at 0 with df = 1 = %v, %v, want +Inf", v, err)
|
|||
|
|
}
|
|||
|
|
if _, err := NoncentralChiSquareDensity(math.NaN(), 3, 2); err == nil {
|
|||
|
|
t.Fatal("density NaN x: want an error")
|
|||
|
|
}
|
|||
|
|
if _, err := NoncentralFCDF(1, 0, 4, 1); err == nil {
|
|||
|
|
t.Fatal("df1 = 0: want an error")
|
|||
|
|
}
|
|||
|
|
if _, err := NoncentralFCDF(1, 4, 0, 1); err == nil {
|
|||
|
|
t.Fatal("df2 = 0: want an error")
|
|||
|
|
}
|
|||
|
|
if _, err := NoncentralFCDF(1, 4, 4, math.NaN()); err == nil {
|
|||
|
|
t.Fatal("NaN λ: want an error")
|
|||
|
|
}
|
|||
|
|
if _, err := NoncentralFCDF(math.NaN(), 4, 4, 1); err == nil {
|
|||
|
|
t.Fatal("NaN x: want an error")
|
|||
|
|
}
|
|||
|
|
if _, err := NoncentralFCDF(1, 4, 4, math.Inf(1)); err == nil {
|
|||
|
|
t.Fatal("λ = +Inf: want an error")
|
|||
|
|
}
|
|||
|
|
if _, err := NoncentralFQuantile(0.5, 0, 4, 1); err == nil {
|
|||
|
|
t.Fatal("quantile df1 = 0: want an error")
|
|||
|
|
}
|
|||
|
|
if _, err := NoncentralFQuantile(0.5, 4, 4, -1); err == nil {
|
|||
|
|
t.Fatal("quantile negative λ: want an error")
|
|||
|
|
}
|
|||
|
|
if _, err := NoncentralFQuantile(0, 4, 4, 1); err == nil {
|
|||
|
|
t.Fatal("q = 0: want an error")
|
|||
|
|
}
|
|||
|
|
if _, err := NoncentralTCDF(math.Inf(1), 3, 1); err == nil {
|
|||
|
|
t.Fatal("t = +Inf: want an error")
|
|||
|
|
}
|
|||
|
|
if _, err := NoncentralTCDF(1, 3, math.NaN()); err == nil {
|
|||
|
|
t.Fatal("NaN δ: want an error")
|
|||
|
|
}
|
|||
|
|
if _, err := NoncentralTQuantile(0.5, 0, 1); err == nil {
|
|||
|
|
t.Fatal("quantile df = 0: want an error")
|
|||
|
|
}
|
|||
|
|
if _, err := NoncentralTQuantile(0.5, 3, math.Inf(-1)); err == nil {
|
|||
|
|
t.Fatal("δ = −Inf: want an error")
|
|||
|
|
}
|
|||
|
|
if _, err := NoncentralTQuantile(1.5, 3, 1); err == nil {
|
|||
|
|
t.Fatal("q above 1: want an error")
|
|||
|
|
}
|
|||
|
|
if _, err := NoncentralTQuantile(0, 3, 1); err == nil {
|
|||
|
|
t.Fatal("q = 0: want an error")
|
|||
|
|
}
|
|||
|
|
if _, err := NoncentralTQuantile(1, 3, 1); err == nil {
|
|||
|
|
t.Fatal("q = 1: want an error")
|
|||
|
|
}
|
|||
|
|
}
|