/// <summary>
/// Returns the value "x" such that q == GammaQ(a, x);
/// </summary>
/// <param name="a">Requires a > 0</param>
/// <param name="q">Requires 0 ≤ q ≤ 1</param>
public static double GammaQInv(double a, double q)
{
if ((!(a > 0) || double.IsInfinity(a)) ||
(!(q >= 0 && q <= 1)))
{
Policies.ReportDomainError("GammaQInv(a: {0}, q: {1}): Requires finite a > 0; q in [0,1]", a, q);
return(double.NaN);
}
if (q == 0)
{
return(double.PositiveInfinity);
}
if (q == 1)
{
return(0);
}
// Q(1,x) = e^-x
if (a == 1)
{
return(-Math.Log(q));
}
// Q(1/2, x) = erfc(sqrt(x))
if (a == 0.5)
{
return(Squared(Math2.ErfcInv(q)));
}
double guess = _IgammaInv.GammaPInvGuess(a, 1 - q, q, out bool has10Digits);
var(lower, upper) = _IgammaInv.GammaQInvLimits(a, q);
// there can be some numerical uncertainties in the limits,
// particularly for denormalized values, so adjust the limits
if (upper == 0 && lower == 0)
{
return(0.0);
}
if (guess <= DoubleLimits.MinNormalValue)
{
return(guess);
}
if (guess <= lower)
{
lower = DoubleLimits.MinNormalValue;
}
if (guess > upper)
{
upper = guess;
}
return(_IgammaInv.SolveGivenAQ(a, q, guess, lower, upper));
}
static double InverseNegativeBinomialCornishFisher(double n, double sf, double sfc, double p, double q)
{
// mean:
double m = n * (sfc) / sf;
double t = Math.Sqrt(n * (sfc));
// standard deviation:
double sigma = t / sf;
// skewness
double sk = (1 + sfc) / t;
// kurtosis:
double k = (6 - sf * (5 + sfc)) / (n * (sfc));
// Get the inverse of a std normal distribution:
double x = Math2.ErfcInv(p > q ? 2 * q : 2 * p) * Constants.Sqrt2;
// Set the sign:
if (p < 0.5)
{
x = -x;
}
double x2 = x * x;
// w is correction term due to skewness
double w = x + sk * (x2 - 1) / 6;
//
// Add on correction due to kurtosis.
//
if (n >= 10)
{
w += k * x * (x2 - 3) / 24 + sk * sk * x * (2 * x2 - 5) / -36;
}
w = m + sigma * w;
if (w < DoubleLimits.MinNormalValue)
{
return(DoubleLimits.MinNormalValue);
}
return(w);
}
static double InversePoissonCornishFisher(double lambda, double p, double q)
{
// mean:
double m = lambda;
// standard deviation:
double sigma = Math.Sqrt(lambda);
// skewness
double sk = 1 / sigma;
// kurtosis:
// double k = 1/lambda;
// Get the inverse of a std normal distribution:
double x = Math2.ErfcInv(p > q ? 2 * q : 2 * p) * Constants.Sqrt2;
// Set the sign:
if (p < 0.5)
{
x = -x;
}
double x2 = x * x;
// w is correction term due to skewness
// double w = x + sk * (x2 - 1) / 6;
double w = x + (x2 - 1) / (6 * sigma);
//if(lambda >= 10)
// w += k * x * (x2 - 3) / 24 + sk * sk * x * (2 * x2 - 5) / -36;
if (lambda >= 10)
{
w += x / 72 * (1 - x2) / lambda;
}
w = m + sigma * w;
return(w);
}
