/// <summary>
/// Computes the inverse of the cumulative distribution function (InvCDF) for the distribution
/// at the given probability. This is also known as the quantile or percent point function.
/// </summary>
/// <param name="p">The location at which to compute the inverse cumulative density.</param>
/// <param name="location">The location (μ) of the distribution.</param>
/// <param name="scale">The scale (σ) of the distribution. Range: σ > 0.</param>
/// <param name="freedom">The degrees of freedom (ν) for the distribution. Range: ν > 0.</param>
/// <returns>the inverse cumulative density at <paramref name="p"/>.</returns>
/// <seealso cref="InverseCumulativeDistribution"/>
/// <remarks>WARNING: currently not an explicit implementation, hence slow and unreliable.</remarks>
public static double InvCDF(double location, double scale, double freedom, double p)
{
if (scale <= 0.0 || freedom <= 0.0)
{
throw new ArgumentException(Resources.InvalidDistributionParameters);
}
// TODO JVG we can probably do a better job for Cauchy special case
if (double.IsPositiveInfinity(freedom))
{
return(Normal.InvCDF(location, scale, p));
}
if (p == 0.5d)
{
return(location);
}
// TODO PERF: We must implement this explicitly instead of solving for CDF^-1
return(Brent.FindRoot(x =>
{
var k = (x - location) / scale;
var h = freedom / (freedom + (k * k));
var ib = 0.5 * SpecialFunctions.BetaRegularized(freedom / 2.0, 0.5, h);
return x <= location ? ib - p : 1.0 - ib - p;
}, -800, 800, accuracy: 1e-12));
}
