internal static void ApproximateExcessDistributionParametersV4(IList <double> sortedData, out double a, out double c, out double u)
{
// The upper tail is defined here by an ECDF interpolating linearly from (u,0) to (x_i, i/n) for data x_1, x_2, ..., x_n all greater than u.
// This is the model from which we compute the upper tail parameters, using method of moments.
// This midpoint version works slightly better than the plain ECDF
double MidpointMSE(IList <double> tailData, double scaleParam, double shapeParam)
{
int n = tailData.Count;
double sum = 0;
for (int i = 0; i < n - 1; i++)
{
double GHat = TailCDF(0.5 * (tailData[i] + tailData[i + 1]) - tailData[0], scaleParam, shapeParam);
double residual = (2.0 * i + 1) / (2.0 * n) - GHat;
sum += residual * residual;
}
return(sum / (n - 1));
}
double GetScore(double uval, out double scaleParam, out double shapeParam)
{
var tailData = GetTailData(sortedData, uval);
EstimateParamsMOM(tailData, out double scaleEst, out double shapeEst);
scaleParam = scaleEst;
shapeParam = shapeEst;
double score = MidpointMSE(tailData, scaleEst, shapeEst);
return(score);
}
// Try several choices of u evenly spaced over (x_0, x_n-3), and keep the best fit
var uValues = Interpolation.Linspace(sortedData[0], sortedData[sortedData.Count - 5], sortedData.Count / 4);
double bestU = 0;
double bestA = 0;
double bestC = 0;
double bestScore = double.PositiveInfinity;
for (int i = 0; i < uValues.Length; i++)
{
double score = GetScore(uValues[i], out double scaleEst, out double shapeEst);
if (score < bestScore)
{
bestScore = score;
bestU = uValues[i];
bestA = scaleEst;
bestC = shapeEst;
}
}
// --- Refine the best so far by bisection search ---
double delta = uValues[1] - uValues[0];
for (int i = 0; i < 10; i++)
{
delta *= 0.5;
double forwardU = Math.Min(bestU + delta, sortedData[sortedData.Count - 3]); // Don't go so high that we don't have data to work with
double forwardScore = GetScore(forwardU, out double forwardScale, out double forwardShape);
double backwardScore = GetScore(bestU - delta, out double backwardScale, out double backwardShape);
if (forwardScore < bestScore)
{
bestScore = forwardScore;
bestU = forwardU;
bestA = forwardScale;
bestC = forwardShape;
}
if (backwardScore < bestScore)
{
bestScore = backwardScore;
bestU -= delta;
bestA = backwardScale;
bestC = backwardShape;
}
}
u = bestU;
a = bestA;
c = bestC;
}