Class CumulativeStudentDistribution

Namespace

NanoQL.Numerics

Assembly

NanoQL.dll

The Student t-distribution's cumulative distribution function for n degrees of freedom (see mathworld.wolfram.com): F(x) = ½ + ½·sgn(x)·[I(1; n/2, ½) − I(n/(n + x²); n/2, ½)], I(z; a, b) the regularized incomplete beta function.

public sealed class CumulativeStudentDistribution

Inheritance

object

CumulativeStudentDistribution

Inherited Members

object.GetType()

object.ToString()

object.Equals(object)

object.Equals(object, object)

object.ReferenceEquals(object, object)

object.GetHashCode()

Remarks

Immutable. QuantLib's formula loses digits in two places, which nanoql computes otherwise (Divergences/StudentDistributionTests.cs). Where n/(n + x²) exceeds ½, near the centre and everywhere as n grows, its incomplete beta takes its complement at 1 − n/(n + x²), losing the argument's digits (6.4e-10 relative at n 1e6 and 4.6e-9 at 1e8, at x 1.5); there nanoql takes I(x²/(n + x²); ½, n/2), the same value by the beta function's reflection, its argument computed directly. And below the centre, ½ − ½·(1 − I) cancels as F(x) falls, to 0 where F(x) is below about 1e-16 (at n 30, x −30); there nanoql takes ½·I(n/(n + x²); n/2, ½), or its reflection's complement, directly. Both routes take Boost's regularised incomplete beta (ibeta, ported for the quantile), as QuantLib's own loses digits in its log-gamma prefactor as n grows (7.6e-10 at n 1e6 with the complementary argument alone): within 8e-14 relative of a high-precision evaluation for n from 1 to 1e9. At n 1 below the centre Boost's complement is its arcsine form on 1 − z, which rounds 1 − z, so there nanoql takes ½·(1 − I(z; ½, ½)), F lying in [¼, ½]: within 2.9e-14 of the Cauchy distribution's ½ + atan(x)/π, where QuantLib's value is 0.5 at x −1e-8, 6.4e-9 off. QuantLib's formula above the centre where n/(n + x²) is at most ½, bit for bit.

Constructors

CumulativeStudentDistribution(int)

Initializes a new instance of the CumulativeStudentDistribution class.

public CumulativeStudentDistribution(int n)

Parameters

n int

The degrees of freedom, positive.

Exceptions

ArgumentOutOfRangeException

n isn't positive.

Methods

Evaluate(double)

The cumulative distribution function.

public double Evaluate(double x)

Parameters

x double

The point.

Returns

double

F(x).

Exceptions

NanoQLException

The incomplete beta's continued fraction doesn't converge.