/// <summary>
/// Computes the cumulative distribution (CDF) of the distribution at x, i.e. P(X ≤ x).
/// </summary>
/// <param name="x">The location at which to compute the cumulative distribution function.</param>
/// <param name="alpha">The stability (α) of the distribution. Range: 2 ≥ α > 0.</param>
/// <param name="beta">The skewness (β) of the distribution. Range: 1 ≥ β ≥ -1.</param>
/// <param name="scale">The scale (c) of the distribution. Range: c > 0.</param>
/// <param name="location">The location (μ) of the distribution.</param>
/// <returns>the cumulative distribution at location <paramref name="x"/>.</returns>
/// <seealso cref="CumulativeDistribution"/>
public static double CDF(double alpha, double beta, double scale, double location, double x)
{
if (alpha <= 0.0 || alpha > 2.0 || beta < -1.0 || beta > 1.0 || scale <= 0.0)
{
throw new ArgumentException(Resources.InvalidDistributionParameters);
}
if (alpha == 2d)
{
return(Normal.CDF(location, Constants.Sqrt2 * scale, x));
}
if (alpha == 1d && beta == 0d)
{
return(Cauchy.CDF(location, scale, x));
}
if (alpha == 0.5d && beta == 1d)
{
return(SpecialFunctions.Erfc(Math.Sqrt(scale / (2 * (x - location)))));
}
throw new NotSupportedException();
}
/// <summary>
/// Computes the log probability density of the distribution (lnPDF) at x, i.e. ln(∂P(X ≤ x)/∂x).
/// </summary>
/// <param name="alpha">The stability (α) of the distribution. Range: 2 ≥ α > 0.</param>
/// <param name="beta">The skewness (β) of the distribution. Range: 1 ≥ β ≥ -1.</param>
/// <param name="scale">The scale (c) of the distribution. Range: c > 0.</param>
/// <param name="location">The location (μ) of the distribution.</param>
/// <param name="x">The location at which to compute the density.</param>
/// <returns>the log density at <paramref name="x"/>.</returns>
/// <seealso cref="DensityLn"/>
public static double PDFLn(double alpha, double beta, double scale, double location, double x)
{
if (alpha <= 0.0 || alpha > 2.0 || beta < -1.0 || beta > 1.0 || scale <= 0.0)
{
throw new ArgumentException(Resources.InvalidDistributionParameters);
}
if (alpha == 2d)
{
return(Normal.PDFLn(location, Constants.Sqrt2 * scale, x));
}
if (alpha == 1d && beta == 0d)
{
return(Cauchy.PDFLn(location, scale, x));
}
if (alpha == 0.5d && beta == 1d && x >= location)
{
return(Math.Log(scale / Constants.Pi2) / 2 - scale / (2 * (x - location)) - 1.5 * Math.Log(x - location));
}
throw new NotSupportedException();
}