fix(stats): hold the Student-t tails past t-squared overflow

Assisted-by: GLM 5.3 Flash

stats/noncentral_test.go

@@ -219,6 +219,36 @@ func TestNoncentralUnderflowSurvival(t *testing.T) {
 	}
 }
 
+// TestNoncentralTCDFHugeFiniteT pins the far corner of the signed axis: a
+// finite t whose square overflows drives the beta argument to Inf/Inf, a
+// NaN the incomplete beta refused under its own name. The CDF there is 1
+// below rounding for t on the δ side and 0 above it, the same limits the
+// central law answers.
+func TestNoncentralTCDFHugeFiniteT(t *testing.T) {
+	for _, c := range []struct {
+		tv    float64
+		df    int
+		delta float64
+		want  float64
+	}{
+		{1e200, 3, 2, 1},
+		{1e155, 1, 0.5, 1},
+		{-1e200, 5, 1, 0},
+		{-1e155, 2, -3, 0},
+	} {
+		got, err := NoncentralTCDF(c.tv, c.df, c.delta)
+		if err != nil {
+			t.Fatalf("NoncentralTCDF(%g, %d, %g): %v", c.tv, c.df, c.delta, err)
+		}
+		if math.IsNaN(got) || got < 0 || got > 1 {
+			t.Fatalf("NoncentralTCDF(%g, %d, %g) = %g, want a probability", c.tv, c.df, c.delta, got)
+		}
+		if math.Abs(got-c.want) > 1e-15 {
+			t.Fatalf("NoncentralTCDF(%g, %d, %g) = %.17g, want %g", c.tv, c.df, c.delta, got, c.want)
+		}
+	}
+}
+
 // TestNoncentralTIdentityReductions pins the exact corners: δ = 0 is
 // the central Student t, t = 0 is Φ(−δ), and the two reflection
 // identities of the law hold to rounding.