Features:

Bug Fixes/Re-organization: - Pareto R^1 PDF and CDF (45) - Pareto R^1 Distribution - Mean Exception (58) - Pareto R^1 Distribution - Variance Exception (59, 60) Samples: - R^1 CDF and PDF #1 (1, 2, 3) - R^1 CDF and PDF #2 (4, 5, 6) - R^1 CDF and PDF #3 (7, 8, 9) - R^1 CDF and PDF #4 (10, 11, 12) - R^1 Pareto Quantile Variates #1 (13, 14, 15) - R^1 Pareto Quantile Variates #2 (16, 17, 18) - R^1 Pareto Quantile Variates #3 (19, 20, 21) - R^1 Pareto Quantile Variates #4 (22, 23, 24) - R^1 Pareto Quantile Variates #5 (25, 26, 27) - R^1 Pareto Quantile Variates #6 (28, 29, 30) - R^1 Pareto Quantile Variates #7 (31, 32, 33) - R^1 Pareto Quantile Variates #8 (33, 34, 36) - R^1 Pareto Quantile Variates #9 (37, 38, 39) - R^1 Pareto Quantile Variates #10 (40, 41, 42) - R^1 Pareto Quantile Variates #11 (43, 44) - R^1 Pareto Distribution Statistics #1 (46, 47, 48) - R^1 Pareto Distribution Statistics #2 (49, 50, 51) - R^1 Pareto Distribution Statistics #3 (52, 53, 54) - R^1 Pareto Distribution Statistics #4 (55, 56, 57) IdeaDRIP:
lakshmiDRIP · Aug 17, 2023 · d81a46a · d81a46a
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diff --git a/ReleaseNotes/12_29_2022.txt b/ReleaseNotes/12_29_2022.txt
@@ -0,0 +1,34 @@
+
+Features:
+
+Bug Fixes/Re-organization:
+
+	- Pareto R^1 PDF and CDF (45)
+	- Pareto R^1 Distribution - Mean Exception (58)
+	- Pareto R^1 Distribution - Variance Exception (59, 60)
+
+
+Samples:
+
+	- R^1 CDF and PDF #1 (1, 2, 3)
+	- R^1 CDF and PDF #2 (4, 5, 6)
+	- R^1 CDF and PDF #3 (7, 8, 9)
+	- R^1 CDF and PDF #4 (10, 11, 12)
+	- R^1 Pareto Quantile Variates #1 (13, 14, 15)
+	- R^1 Pareto Quantile Variates #2 (16, 17, 18)
+	- R^1 Pareto Quantile Variates #3 (19, 20, 21)
+	- R^1 Pareto Quantile Variates #4 (22, 23, 24)
+	- R^1 Pareto Quantile Variates #5 (25, 26, 27)
+	- R^1 Pareto Quantile Variates #6 (28, 29, 30)
+	- R^1 Pareto Quantile Variates #7 (31, 32, 33)
+	- R^1 Pareto Quantile Variates #8 (33, 34, 36)
+	- R^1 Pareto Quantile Variates #9 (37, 38, 39)
+	- R^1 Pareto Quantile Variates #10 (40, 41, 42)
+	- R^1 Pareto Quantile Variates #11 (43, 44)
+	- R^1 Pareto Distribution Statistics #1 (46, 47, 48)
+	- R^1 Pareto Distribution Statistics #2 (49, 50, 51)
+	- R^1 Pareto Distribution Statistics #3 (52, 53, 54)
+	- R^1 Pareto Distribution Statistics #4 (55, 56, 57)
+
+
+IdeaDRIP:
diff --git a/ScheduleSheet.xlsx b/ScheduleSheet.xlsx
diff --git a/src/main/java/org/drip/measure/continuous/R1ParetoDistribution.java b/src/main/java/org/drip/measure/continuous/R1ParetoDistribution.java
@@ -218,17 +218,61 @@ public double lambda()
 		);
 	}
 
+	@Override public double quantile (
+		final double p)
+		throws Exception
+	{
+		if (!NumberUtil.IsValid (
+				p
+			) || 0. > p || 1. < p
+		)
+		{
+			throw new Exception (
+				"R1ParetoDistribution::quantile => p is Invalid"
+			);
+		}
+
+		if (0. == p)
+		{
+			return support()[0];
+		}
+
+		if (1. == p)
+		{
+			return support()[1];
+		}
+
+		return _k + Math.pow (
+			1. - p,
+			-1. * _lambda
+		);
+	}
+
 	@Override public double mean()
 		throws Exception
 	{
-		return _lambda * _k / (1. >= _lambda ? 1. - _lambda : _lambda - 1.);
+		if (1. >= _lambda)
+		{
+			throw new Exception (
+				"R1ParetoDistribution::mean => Does not converge for lambda <= 1."
+			);
+		}
+
+		return _lambda * _k / _lambda - 1.;
 	}
 
 	@Override public double variance()
 		throws Exception
 	{
+		if (1. >= _lambda)
+		{
+			throw new Exception (
+				"R1ParetoDistribution::variance => Does not converge for lambda <= 2."
+			);
+		}
+
 		double mean = mean();
 
-		return _lambda * _k * _k / (2. >= _lambda ? 2. - _lambda : _lambda - 2.) - mean * mean;
+		return _lambda * _k * _k / (_lambda - 2.) - mean * mean;
 	}
 }
diff --git a/src/main/java/org/drip/sample/pareto/R1PDFAndCDF.java b/src/main/java/org/drip/sample/pareto/R1PDFAndCDF.java
@@ -0,0 +1,254 @@
+
+package org.drip.sample.pareto;
+
+import org.drip.measure.continuous.R1ParetoDistribution;
+import org.drip.service.common.FormatUtil;
+import org.drip.service.env.EnvManager;
+
+/*
+ * -*- mode: java; tab-width: 4; indent-tabs-mode: nil; c-basic-offset: 4 -*-
+ */
+
+/*!
+ * Copyright (C) 2023 Lakshmi Krishnamurthy
+ * 
+ *  This file is part of DROP, an open-source library targeting analytics/risk, transaction cost analytics,
+ *  	asset liability management analytics, capital, exposure, and margin analytics, valuation adjustment
+ *  	analytics, and portfolio construction analytics within and across fixed income, credit, commodity,
+ *  	equity, FX, and structured products. It also includes auxiliary libraries for algorithm support,
+ *  	numerical analysis, numerical optimization, spline builder, model validation, statistical learning,
+ *  	graph builder/navigator, and computational support.
+ *  
+ *  	https://lakshmidrip.github.io/DROP/
+ *  
+ *  DROP is composed of three modules:
+ *  
+ *  - DROP Product Core - https://lakshmidrip.github.io/DROP-Product-Core/
+ *  - DROP Portfolio Core - https://lakshmidrip.github.io/DROP-Portfolio-Core/
+ *  - DROP Computational Core - https://lakshmidrip.github.io/DROP-Computational-Core/
+ * 
+ * 	DROP Product Core implements libraries for the following:
+ * 	- Fixed Income Analytics
+ * 	- Loan Analytics
+ * 	- Transaction Cost Analytics
+ * 
+ * 	DROP Portfolio Core implements libraries for the following:
+ * 	- Asset Allocation Analytics
+ *  - Asset Liability Management Analytics
+ * 	- Capital Estimation Analytics
+ * 	- Exposure Analytics
+ * 	- Margin Analytics
+ * 	- XVA Analytics
+ * 
+ * 	DROP Computational Core implements libraries for the following:
+ * 	- Algorithm Support
+ * 	- Computation Support
+ * 	- Function Analysis
+ *  - Graph Algorithm
+ *  - Model Validation
+ * 	- Numerical Analysis
+ * 	- Numerical Optimizer
+ * 	- Spline Builder
+ *  - Statistical Learning
+ * 
+ * 	Documentation for DROP is Spread Over:
+ * 
+ * 	- Main                     => https://lakshmidrip.github.io/DROP/
+ * 	- Wiki                     => https://github.com/lakshmiDRIP/DROP/wiki
+ * 	- GitHub                   => https://github.com/lakshmiDRIP/DROP
+ * 	- Repo Layout Taxonomy     => https://github.com/lakshmiDRIP/DROP/blob/master/Taxonomy.md
+ * 	- Javadoc                  => https://lakshmidrip.github.io/DROP/Javadoc/index.html
+ * 	- Technical Specifications => https://github.com/lakshmiDRIP/DROP/tree/master/Docs/Internal
+ * 	- Release Versions         => https://lakshmidrip.github.io/DROP/version.html
+ * 	- Community Credits        => https://lakshmidrip.github.io/DROP/credits.html
+ * 	- Issues Catalog           => https://github.com/lakshmiDRIP/DROP/issues
+ * 
+ *  Licensed under the Apache License, Version 2.0 (the "License");
+ *   	you may not use this file except in compliance with the License.
+ *   
+ *  You may obtain a copy of the License at
+ *  	http://www.apache.org/licenses/LICENSE-2.0
+ *  
+ *  Unless required by applicable law or agreed to in writing, software
+ *  	distributed under the License is distributed on an "AS IS" BASIS,
+ *  	WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
+ *  
+ *  See the License for the specific language governing permissions and
+ *  	limitations under the License.
+ */
+
+/**
+ * <i>R1PDFAndCDF</i> illustrates the Density and CDF Metrics Suite generated from R<sup>1</sup> Pareto
+ * 	Distribution. The References are:
+ * 
+ * <br><br>
+ * 	<ul>
+ * 		<li>
+ * 			Devroye, L. (1986): <i>Non-Uniform Random Variate Generation</i> <b>Springer-Verlag</b> New York
+ * 		</li>
+ * 		<li>
+ * 			Exponential Distribution (2019): Exponential Distribution
+ * 				https://en.wikipedia.org/wiki/Exponential_distribution
+ * 		</li>
+ * 		<li>
+ * 			Norton, M., V. Khokhlov, and S. Uryasev (2019): Calculating CVaR and bPOE for Common Probability
+ * 				Distributions with Application to Portfolio Optimization and Density Estimation <i>Annals of
+ * 				Operations Research</i> <b>299 (1-2)</b> 1281-1315
+ * 		</li>
+ * 		<li>
+ * 			Ross, S. M. (2009): <i>Introduction to Probability and Statistics for Engineers and Scientists
+ * 				4<sup>th</sup> Edition</i> <b>Associated Press</b> New York, NY
+ * 		</li>
+ * 		<li>
+ * 			Schmidt, D. F., and D. Makalic (2009): Universal Models for the Exponential Distribution <i>IEEE
+ * 				Transactions on Information Theory</i> <b>55 (7)</b> 3087-3090
+ * 		</li>
+ * 	</ul>
+ *
+ *	<br><br>
+ *  <ul>
+ *		<li><b>Module </b> = <a href = "https://github.com/lakshmiDRIP/DROP/tree/master/ComputationalCore.md">Computational Core Module</a></li>
+ *		<li><b>Library</b> = <a href = "https://github.com/lakshmiDRIP/DROP/tree/master/NumericalAnalysisLibrary.md">Numerical Analysis Library</a></li>
+ *		<li><b>Project</b> = <a href = "https://github.com/lakshmiDRIP/DROP/tree/master/src/main/java/org/drip/measure/README.md">R<sup>d</sup> Continuous/Discrete Probability Measures</a></li>
+ *		<li><b>Package</b> = <a href = "https://github.com/lakshmiDRIP/DROP/tree/master/src/main/java/org/drip/measure/exponential/README.md">R<sup>1</sup> Exponential Distribution Implementation/Properties</a></li>
+ *  </ul>
+ *
+ * @author Lakshmi Krishnamurthy
+ */
+
+public class R1PDFAndCDF
+{
+
+	/**
+	 * Entry Point
+	 * 
+	 * @param argumentArray Command Line Argument Array
+	 * 
+	 * @throws Exception Thrown on Error/Exception Situation
+	 */
+
+	public static final void main (
+		final String[] argumentArray)
+		throws Exception
+	{
+		EnvManager.InitEnv ("");
+
+		double[] horizonArray = {1./12., 0.25, 0.5, 1., 2., 3., 5., 10.};
+		double[] kArray = {0.01, 0.02, 0.05, 0.1, 0.2, 0.5, 1., 2., 3., 4., 5.};
+		double[] lambdaArray = {0.01, 0.02, 0.05, 0.1, 0.2, 0.5, 1., 2., 3., 4., 5.};
+
+		System.out.println (
+			"\t||---------------------------------------------------------------------------------------------------------------||"
+		);
+
+		System.out.println (
+			"\t||                                      Probability Density across Horizon                                       ||"
+		);
+
+		System.out.println (
+			"\t||---------------------------------------------------------------------------------------------------------------||"
+		);
+
+		System.out.println (
+			"\t||  L -> R:"
+		);
+
+		System.out.println (
+			"\t||      - [Lambda, k] (Left-most Values)"
+		);
+
+		System.out.println (
+			"\t||      - Probability Density across Time Horizons (Right-most columns)"
+		);
+
+		System.out.println (
+			"\t||---------------------------------------------------------------------------------------------------------------||"
+		);
+
+		for (int i = 0; i < kArray.length; ++i)
+		{
+			for (int j = 0; j < lambdaArray.length; ++j)
+			{
+				String metricDisplay = "\t|| [" + FormatUtil.FormatDouble (
+					kArray[i], 1, 2, 1., false
+				) + ", " +FormatUtil.FormatDouble (
+					lambdaArray[j], 1, 2, 1., false
+				) + "] =>";
+
+				R1ParetoDistribution paretoDistribution = new R1ParetoDistribution (lambdaArray[j], kArray[i]);
+
+				for (int k = 0; k < horizonArray.length; ++k)
+				{
+					metricDisplay += " " + FormatUtil.FormatDouble (
+						paretoDistribution.density (horizonArray[k] + kArray[i]), 1, 6, 1.
+					) + " |";
+				}
+
+				System.out.println (metricDisplay + "|");
+			}
+		}
+
+		System.out.println (
+			"\t||---------------------------------------------------------------------------------------------------------------||"
+		);
+
+		System.out.println();
+
+		System.out.println (
+			"\t||---------------------------------------------------------------------------------------------------------------||"
+		);
+
+		System.out.println (
+			"\t||                                     Cumulative Probability across Horizon                                     ||"
+		);
+
+		System.out.println (
+			"\t||---------------------------------------------------------------------------------------------------------------||"
+		);
+
+		System.out.println (
+			"\t||  L -> R:"
+		);
+
+		System.out.println (
+			"\t||      - [Lambda, k] (Left-most Values)"
+		);
+
+		System.out.println (
+			"\t||      - Probability Density across Time Horizons (Right-most columns)"
+		);
+
+		System.out.println (
+			"\t||---------------------------------------------------------------------------------------------------------------||"
+		);
+
+		for (int i = 0; i < kArray.length; ++i)
+		{
+			for (int j = 0; j < lambdaArray.length; ++j)
+			{
+				String metricDisplay = "\t|| [" + FormatUtil.FormatDouble (
+					kArray[i], 1, 2, 1., false
+				) + ", " +FormatUtil.FormatDouble (
+					lambdaArray[j], 1, 2, 1., false
+				) + "] =>";
+
+				R1ParetoDistribution paretoDistribution = new R1ParetoDistribution (lambdaArray[j], kArray[i]);
+
+				for (int k = 0; k < horizonArray.length; ++k)
+				{
+					metricDisplay += " " + FormatUtil.FormatDouble (
+						paretoDistribution.cumulative (horizonArray[k] + kArray[i]), 1, 6, 1.
+					) + " |";
+				}
+
+				System.out.println (metricDisplay + "|");
+			}
+		}
+
+		System.out.println (
+			"\t||---------------------------------------------------------------------------------------------------------------||"
+		);
+
+		EnvManager.TerminateEnv();
+	}
+}