// <copyright file="TruncatedPareto.cs" company="Math.NET">
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// Math.NET Numerics, part of the Math.NET Project
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// http://numerics.mathdotnet.com
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// http://github.com/mathnet/mathnet-numerics
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//
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// Copyright (c) 2009-2019 Math.NET
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//
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// Permission is hereby granted, free of charge, to any person
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// obtaining a copy of this software and associated documentation
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// files (the "Software"), to deal in the Software without
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// restriction, including without limitation the rights to use,
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// copy, modify, merge, publish, distribute, sublicense, and/or sell
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// copies of the Software, and to permit persons to whom the
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// Software is furnished to do so, subject to the following
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// conditions:
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//
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// The above copyright notice and this permission notice shall be
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// included in all copies or substantial portions of the Software.
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//
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// THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND,
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// EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES
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// OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND
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// NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT
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// HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY,
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// WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING
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// FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR
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// OTHER DEALINGS IN THE SOFTWARE.
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// </copyright>
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using IStation.Numerics.Random;
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using System;
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using System.Collections.Generic;
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namespace IStation.Numerics.Distributions
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{
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public class TruncatedPareto : IContinuousDistribution
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{
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System.Random _random;
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/// <summary>
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/// Initializes a new instance of the TruncatedPareto class.
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/// </summary>
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/// <param name="scale">The scale (xm) of the distribution. Range: xm > 0.</param>
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/// <param name="shape">The shape (α) of the distribution. Range: α > 0.</param>
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/// <param name="truncation">The truncation (T) of the distribution. Range: T > xm.</param>
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/// <param name="randomSource">The random number generator which is used to draw random samples.</param>
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/// <exception cref="ArgumentException">If <paramref name="scale"/> or <paramref name="shape"/> are non-positive or if T ≤ xm.</exception>
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public TruncatedPareto(double scale, double shape, double truncation, System.Random randomSource = null)
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{
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if (!IsValidParameterSet(scale, shape, truncation))
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{
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throw new ArgumentException("Invalid parametrization for the distribution.");
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}
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_random = randomSource ?? SystemRandomSource.Default;
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Scale = scale;
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Shape = shape;
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Truncation = truncation;
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}
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/// <summary>
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/// A string representation of the distribution.
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/// </summary>
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/// <returns>a string representation of the distribution.</returns>
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public override string ToString()
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{
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return $"Truncated Pareto(Scale = {Scale}, Shape = {Shape}, Truncation = {Truncation})";
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}
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/// <summary>
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/// Tests whether the provided values are valid parameters for this distribution.
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/// </summary>
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/// <param name="scale">The scale (xm) of the distribution. Range: xm > 0.</param>
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/// <param name="shape">The shape (α) of the distribution. Range: α > 0.</param>
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/// <param name="truncation">The truncation (T) of the distribution. Range: T > xm.</param>
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public static bool IsValidParameterSet(double scale, double shape, double truncation)
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{
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var allFinite = scale.IsFinite() && shape.IsFinite() && truncation.IsFinite();
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return allFinite && scale > 0.0 && shape > 0.0 && truncation > scale;
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}
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/// <summary>
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/// Gets the random number generator which is used to draw random samples.
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/// </summary>
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public System.Random RandomSource
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{
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get => _random;
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set => _random = value ?? SystemRandomSource.Default;
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}
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/// <summary>
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/// Gets the scale (xm) of the distribution. Range: xm > 0.
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/// </summary>
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public double Scale { get; }
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/// <summary>
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/// Gets the shape (α) of the distribution. Range: α > 0.
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/// </summary>
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public double Shape { get; }
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/// <summary>
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/// Gets the truncation (T) of the distribution. Range: T > 0.
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/// </summary>
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public double Truncation { get; }
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/// <summary>
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/// Gets the n-th raw moment of the distribution.
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/// </summary>
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/// <param name="n">The order (n) of the moment. Range: n ≥ 1.</param>
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/// <returns>the n-th moment of the distribution.</returns>
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public double GetMoment(int n)
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{
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double moment;
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if (Shape.AlmostEqual(n))
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{
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moment = ((Shape * Math.Pow(Scale, n)) / (1 - Math.Pow(Scale / Truncation, Shape))) * Math.Log(Truncation / Scale);
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}
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else
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{
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moment = ((Shape * Math.Pow(Scale, n)) / (Shape - n)) * ((1 - Math.Pow((Scale / Truncation), (Shape - n))) / (1 - Math.Pow(Scale / Truncation, Shape)));
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}
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return moment;
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}
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/// <summary>
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/// Gets the mean of the truncated Pareto distribution.
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/// </summary>
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public double Mean => GetMoment(1);
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/// <summary>
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/// Gets the variance of the truncated Pareto distribution.
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/// </summary>
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public double Variance => GetMoment(2) - Math.Pow(GetMoment(1), 2);
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/// <summary>
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/// Gets the standard deviation of the truncated Pareto distribution.
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/// </summary>
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public double StdDev => Math.Sqrt(Variance);
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/// <summary>
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/// Gets the mode of the truncated Pareto distribution (not supported).
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/// </summary>
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public double Mode => throw new NotSupportedException();
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/// <summary>
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/// Gets the minimum of the truncated Pareto distribution.
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/// </summary>
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public double Minimum => Scale;
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/// <summary>
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/// Gets the maximum of the truncated Pareto distribution.
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/// </summary>
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public double Maximum => Truncation;
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/// <summary>
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/// Gets the entropy of the truncated Pareto distribution (not supported).
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/// </summary>
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public double Entropy => throw new NotSupportedException();
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/// <summary>
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/// Gets the skewness of the truncated Pareto distribution.
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/// </summary>
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public double Skewness
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{
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get
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{
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var mean = Mean;
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var variance = Variance;
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var std = StdDev;
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return (GetMoment(3) - 3.0 * mean * variance - mean * mean * mean) / (std * std * std);
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}
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}
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/// <summary>
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/// Gets the median of the truncated Pareto distribution.
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/// </summary>
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public double Median => Scale * Math.Pow(1.0 - (1.0 / 2.0) * (1.0 - Math.Pow(Scale / Truncation, Shape)), -(1.0 / Shape));
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/// <summary>
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/// Generates a sample from the truncated Pareto distribution.
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/// </summary>
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/// <returns>a sample from the distribution.</returns>
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public double Sample()
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{
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return SampleUnchecked(_random, Scale, Shape, Truncation);
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}
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/// <summary>
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/// Fills an array with samples generated from the distribution.
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/// </summary>
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/// <param name="values">The array to fill with the samples.</param>
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public void Samples(double[] values)
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{
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SamplesUnchecked(_random, values, Scale, Shape, Truncation);
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}
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/// <summary>
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/// Generates a sequence of samples from the truncated Pareto distribution.
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/// </summary>
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/// <returns>a sequence of samples from the distribution.</returns>
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public IEnumerable<double> Samples()
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{
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return SamplesUnchecked(_random, Scale, Shape, Truncation);
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}
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/// <summary>
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/// Generates a sample from the truncated Pareto distribution.
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/// </summary>
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/// <param name="rnd">The random number generator to use.</param>
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/// <param name="scale">The scale (xm) of the distribution. Range: xm > 0.</param>
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/// <param name="shape">The shape (α) of the distribution. Range: α > 0.</param>
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/// <param name="truncation">The truncation (T) of the distribution. Range: T > xm.</param>
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/// <returns>a sample from the distribution.</returns>
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public static double Sample(System.Random rnd, double scale, double shape, double truncation)
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{
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if (!IsValidParameterSet(scale, shape, truncation))
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{
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throw new ArgumentException("Invalid parametrization for the distribution.");
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}
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return SampleUnchecked(rnd, scale, shape, truncation);
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}
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/// <summary>
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/// Fills an array with samples generated from the distribution.
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/// </summary>
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/// <param name="rnd">The random number generator to use.</param>
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/// <param name="values">The array to fill with the samples.</param>
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/// <param name="scale">The scale (xm) of the distribution. Range: xm > 0.</param>
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/// <param name="shape">The shape (α) of the distribution. Range: α > 0.</param>
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/// <param name="truncation">The truncation (T) of the distribution. Range: T > xm.</param>
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public static void Samples(System.Random rnd, double[] values, double scale, double shape, double truncation)
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{
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if (!IsValidParameterSet(scale, shape, truncation))
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{
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throw new ArgumentException("Invalid parametrization for the distribution.");
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}
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SamplesUnchecked(rnd, values, scale, shape, truncation);
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}
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/// <summary>
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/// Generates a sequence of samples from the truncated Pareto distribution.
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/// </summary>
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/// <param name="rnd">The random number generator to use.</param>
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/// <param name="scale">The scale (xm) of the distribution. Range: xm > 0.</param>
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/// <param name="shape">The shape (α) of the distribution. Range: α > 0.</param>
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/// <param name="truncation">The truncation (T) of the distribution. Range: T > xm.</param>
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/// <returns>a sequence of samples from the distribution.</returns>
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public static IEnumerable<double> Samples(System.Random rnd, double scale, double shape, double truncation)
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{
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if (!IsValidParameterSet(scale, shape, truncation))
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{
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throw new ArgumentException("Invalid parametrization for the distribution.");
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}
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return SamplesUnchecked(rnd, scale, shape, truncation);
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}
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internal static double SampleUnchecked(System.Random rnd, double scale, double shape, double truncation)
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{
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double uniform = rnd.NextDouble();
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return InvCDFUncheckedImpl(scale, shape, truncation, uniform);
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}
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internal static void SamplesUnchecked(System.Random rnd, double[] values, double scale, double shape, double truncation)
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{
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if (values.Length == 0)
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{
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return;
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}
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double[] uniforms = rnd.NextDoubles(values.Length);
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for (var j = 0; j < values.Length; ++j)
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{
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values[j] = InvCDFUncheckedImpl(scale, shape, truncation, uniforms[j]);
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}
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}
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internal static IEnumerable<double> SamplesUnchecked(System.Random rnd, double scale, double shape, double truncation)
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{
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while (true)
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{
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yield return SampleUnchecked(rnd, scale, shape, truncation);
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}
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}
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/// <summary>
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/// Computes the probability density of the distribution (PDF) at x, i.e. ∂P(X ≤ x)/∂x.
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/// </summary>
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/// <param name="x">The location at which to compute the density.</param>
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/// <returns>the density at <paramref name="x"/>.</returns>
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/// <seealso cref="PDF"/>
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public double Density(double x)
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{
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return DensityImpl(Scale, Shape, Truncation, x);
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}
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/// <summary>
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/// Computes the log probability density of the distribution (lnPDF) at x, i.e. ln(∂P(X ≤ x)/∂x).
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/// </summary>
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/// <param name="x">The location at which to compute the log density.</param>
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/// <returns>the log density at <paramref name="x"/>.</returns>
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/// <seealso cref="PDFLn"/>
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public double DensityLn(double x)
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{
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return DensityLnImpl(Scale, Shape, Truncation, x);
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}
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/// <summary>
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/// Computes the cumulative distribution (CDF) of the distribution at x, i.e. P(X ≤ x).
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/// </summary>
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/// <param name="x">The location at which to compute the cumulative distribution function.</param>
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/// <returns>the cumulative distribution at location <paramref name="x"/>.</returns>
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/// <seealso cref="CDF"/>
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public double CumulativeDistribution(double x)
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{
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return CumulativeDistributionImpl(Scale, Shape, Truncation, x);
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}
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/// <summary>
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/// Computes the inverse cumulative distribution (CDF) of the distribution at p, i.e. solving for P(X ≤ x) = p.
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/// </summary>
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/// <param name="p">The location at which to compute the inverse cumulative distribution function.</param>
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/// <returns>the inverse cumulative distribution at location <paramref name="p"/>.</returns>
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public double InvCDF(double p)
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{
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return InvCDFUncheckedImpl(Scale, Shape, Truncation, p);
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}
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/// <summary>
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/// Computes the inverse cumulative distribution (CDF) of the distribution at p, i.e. solving for P(X ≤ x) = p.
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/// </summary>
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/// <param name="scale">The scale (xm) of the distribution. Range: xm > 0.</param>
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/// <param name="shape">The shape (α) of the distribution. Range: α > 0.</param>
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/// <param name="truncation">The truncation (T) of the distribution. Range: T > xm.</param>
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/// <param name="p">The location at which to compute the inverse cumulative distribution function.</param>
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/// <returns>the inverse cumulative distribution at location <paramref name="p"/>.</returns>
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/// <seealso cref="CumulativeDistribution"/>
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public static double ICDF(double scale, double shape, double truncation, double p)
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{
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if (!IsValidParameterSet(scale, shape, truncation))
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{
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throw new ArgumentException("Invalid parametrization for the distribution.");
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}
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return InvCDFUncheckedImpl(scale, shape, truncation, p);
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}
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/// <summary>
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/// Computes the probability density of the distribution (PDF) at x, i.e. ∂P(X ≤ x)/∂x.
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/// </summary>
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/// <param name="scale">The scale (xm) of the distribution. Range: xm > 0.</param>
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/// <param name="shape">The shape (α) of the distribution. Range: α > 0.</param>
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/// <param name="truncation">The truncation (T) of the distribution. Range: T > xm.</param>
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/// <param name="x">The location at which to compute the density.</param>
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/// <returns>the density at <paramref name="x"/>.</returns>
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/// <seealso cref="Density"/>
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public static double PDF(double scale, double shape, double truncation, double x)
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{
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if (!IsValidParameterSet(scale, shape, truncation))
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{
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throw new ArgumentException("Invalid parametrization for the distribution.");
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}
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return DensityImpl(scale, shape, truncation, x);
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}
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/// <summary>
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/// Computes the log probability density of the distribution (lnPDF) at x, i.e. ln(∂P(X ≤ x)/∂x).
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/// </summary>
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/// <param name="scale">The scale (xm) of the distribution. Range: xm > 0.</param>
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/// <param name="shape">The shape (α) of the distribution. Range: α > 0.</param>
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/// <param name="truncation">The truncation (T) of the distribution. Range: T > xm.</param>
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/// <param name="x">The location at which to compute the log density.</param>
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/// <returns>the log density at <paramref name="x"/>.</returns>
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/// <seealso cref="DensityLn"/>
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public static double PDFLn(double scale, double shape, double truncation, double x)
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{
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if (!IsValidParameterSet(scale, shape, truncation))
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{
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throw new ArgumentException("Invalid parametrization for the distribution.");
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}
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return DensityLnImpl(scale, shape, truncation, x);
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}
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/// <summary>
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/// Computes the cumulative distribution (CDF) of the distribution at x, i.e. P(X ≤ x).
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/// </summary>
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/// <param name="scale">The scale (xm) of the distribution. Range: xm > 0.</param>
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/// <param name="shape">The shape (α) of the distribution. Range: α > 0.</param>
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/// <param name="truncation">The truncation (T) of the distribution. Range: T > xm.</param>
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/// <param name="x">The location at which to compute the cumulative distribution function.</param>
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/// <returns>the cumulative distribution at location <paramref name="x"/>.</returns>
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/// <seealso cref="CumulativeDistribution"/>
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public static double CDF(double scale, double shape, double truncation, double x)
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{
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if (!IsValidParameterSet(scale, shape, truncation))
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{
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throw new ArgumentException("Invalid parametrization for the distribution.");
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}
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return CumulativeDistributionImpl(scale, shape, truncation, x);
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}
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internal static double DensityImpl(double scale, double shape, double truncation, double x)
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{
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if (x < scale || x > truncation)
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return 0;
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else
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return (shape * Math.Pow(scale, shape) * Math.Pow(x, -shape - 1)) / (1 - Math.Pow(scale / truncation, shape));
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}
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internal static double DensityLnImpl(double scale, double shape, double truncation, double x)
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{
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return Math.Log(DensityImpl(scale, shape, truncation, x));
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}
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internal static double CumulativeDistributionImpl(double scale, double shape, double truncation, double x)
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{
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if (x <= scale)
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return 0;
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else if (x >= truncation)
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return 1;
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else
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return (1 - Math.Pow(scale, shape) * Math.Pow(x, -shape)) / (1 - Math.Pow(scale / truncation, shape));
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}
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internal static double InvCDFUncheckedImpl(double scale, double shape, double truncation, double p)
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{
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var numerator = p * Math.Pow(truncation, shape) - p * Math.Pow(scale, shape) - Math.Pow(truncation, shape);
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var denominator = Math.Pow(truncation, shape) * Math.Pow(scale, shape);
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return Math.Pow(-numerator / denominator, -(1 / shape));
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}
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}
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}
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