Features:

	- Composite R^1 Exponential Distribution Shell (1, 2)
	- Minimum Among Exponential Distribution #1 (3, 4)
	- Minimum Among Exponential Distribution #2 (5, 6, 7)
	- Composite R^1 Exponential Distribution Constructor (8, 9, 10)
	- Minimum of Realized Exponential Distributions (11, 12)
	- Composite R^1 Rate Distribution Shell (15, 16)
	- Realized R^1 Rate Distribution Standard (17, 18, 19)
	- Realized R^1 Rate Distribution Constructor (20, 21, 22)
	- R^1 Exponential Rate Distribution Array (23, 24, 25)
	- Probability of Index as Minimum (26, 27, 28)
	- Composite R^1 Order Statistics Shell (29, 30)
	- Exponential IID Joint Moment #1 (31, 32, 33)
	- Exponential IID Joint Moment #2 (34, 35, 36)
	- Exponential IID Joint Moment #3 (37, 38, 39)
	- Exponential IID Joint Moment #4 (40, 41, 42)
	- Two IID Exponential Sum Shell (43, 44)
	- Smaller R^1 Exponential Rate Distribution (45, 46)
	- Larger R^1 Exponential Rate Distribution (47, 48)
	- Two IID Exponential Sum Constructor (49, 50, 51)
	- Two IID Exponential Sum Support (52)
	- Two IID Exponential Sum Density (53, 54, 55)
	- Two IID Sum Quadrature Count (56, 57)
	- Two IID Exponential Sum Mean (58, 59, 60)


Bug Fixes/Re-organization:

	- Composite R^1 To Minima Realization (13, 14)


Samples:


IdeaDRIP:

ReleaseNotes/12_22_2022.txt

@@ -0,0 +1,37 @@
+
+Features:
+
+ - Composite R^1 Exponential Distribution Shell (1, 2)
+ - Minimum Among Exponential Distribution #1 (3, 4)
+ - Minimum Among Exponential Distribution #2 (5, 6, 7)
+ - Composite R^1 Exponential Distribution Constructor (8, 9, 10)
+ - Minimum of Realized Exponential Distributions (11, 12)
+ - Composite R^1 Rate Distribution Shell (15, 16)
+ - Realized R^1 Rate Distribution Standard (17, 18, 19)
+ - Realized R^1 Rate Distribution Constructor (20, 21, 22)
+ - R^1 Exponential Rate Distribution Array (23, 24, 25)
+ - Probability of Index as Minimum (26, 27, 28)
+ - Composite R^1 Order Statistics Shell (29, 30)
+ - Exponential IID Joint Moment #1 (31, 32, 33)
+ - Exponential IID Joint Moment #2 (34, 35, 36)
+ - Exponential IID Joint Moment #3 (37, 38, 39)
+ - Exponential IID Joint Moment #4 (40, 41, 42)
+ - Two IID Exponential Sum Shell (43, 44)
+ - Smaller R^1 Exponential Rate Distribution (45, 46)
+ - Larger R^1 Exponential Rate Distribution (47, 48)
+ - Two IID Exponential Sum Constructor (49, 50, 51)
+ - Two IID Exponential Sum Support (52)
+ - Two IID Exponential Sum Density (53, 54, 55)
+ - Two IID Sum Quadrature Count (56, 57)
+ - Two IID Exponential Sum Mean (58, 59, 60)
+
+
+Bug Fixes/Re-organization:
+
+ - Composite R^1 To Minima Realization (13, 14)
+
+
+Samples:
+
+
+IdeaDRIP:

src/main/java/org/drip/measure/exponential/CompositeR1OrderStatistics.java

@@ -0,0 +1,175 @@
+
+package org.drip.measure.exponential;
+
+/*
+ * -*- 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>CompositeR1OrderStatistics</i> generates the Order Statistics for a Composite Set of R<sup>1</sup>
+ * Exponential Distributions. 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 CompositeR1OrderStatistics
+{
+
+ /**
+ * Compute the Joint Moment of a Set of i.i.d. Distributions
+ * 
+ * @param r1RateDistribution R<sup>1</sup> Exponential Distribution
+ * @param variateCount Variate Count
+ * @param index1 First Index
+ * @param index2 Second Index
+ * 
+ * @return Joint Moment of a Set of i.i.d. Distributions
+ * 
+ * @throws Exception Thrown if the Inputs are Invalid
+ */
+
+ public static final double IIDJointMoment (
+ final R1RateDistribution r1RateDistribution,
+ final int variateCount,
+ final int index1,
+ final int index2)
+ throws Exception
+ {
+ if (null == r1RateDistribution ||
+ 0 > index1 || index1 >= variateCount ||
+ 0 > index2 || index2 >= variateCount || index1 == index2
+ )
+ {
+ throw new Exception (
+ "CompositeR1OrderStatistics::IIDJointMoment => Invalid Inputs"
+ );
+ }
+
+ double expectationMaxIndex = 0.;
+ double expectationMinIndex = 0.;
+ double expectationMinIndexSquared = 0.;
+ int maxIndex = index1 > index2 ? index1 : index2;
+ int minIndex = index1 < index2 ? index1 : index2;
+
+ double inverseRate = 1. / r1RateDistribution.rate();
+
+ for (int k = 0;
+ k < minIndex;
+ ++k)
+ {
+ double expectation = inverseRate / (variateCount - k);
+ expectationMinIndexSquared += expectation * expectation;
+ expectationMinIndex += expectation;
+ }
+
+ for (int k = 0;
+ k < maxIndex;
+ ++k)
+ {
+ expectationMaxIndex += inverseRate / (variateCount - k);
+ }
+
+ return expectationMinIndex * expectationMaxIndex + expectationMinIndexSquared +
+ expectationMinIndex * expectationMinIndex;
+ }
+}

src/main/java/org/drip/measure/exponential/R1RateDistribution.java

@@ -163,9 +163,9 @@ public R1RateDistribution (
  }
 
  /**
- * Retrieve the Rate Parameter Lambda
+ * Retrieve the Lambda
  * 
- * @return Rate Parameter Lambda
+ * @return Lambda
  */
 
  public double lambda()
@@ -174,9 +174,9 @@ public double lambda()
  }
 
  /**
- * Retrieve the Rate Parameter Lambda
+ * Retrieve the Rate Parameter
  * 
- * @return Rate Parameter Lambda
+ * @return Rate Parameter
  */
 
  public double rate()