Package org.drip.measure.exponential

Class R1RateDistribution

java.lang.Object

org.drip.measure.continuous.R1Univariate

org.drip.measure.exponential.R1RateDistribution

Direct Known Subclasses:

RealizedMinimaR1RateDistribution

public class R1RateDistribution
extends R1Univariate

R1RateDistribution implements the Rate Parameterization of the R¹ Exponential Distribution. The References are:

• Devroye, L. (1986): Non-Uniform Random Variate Generation Springer-Verlag New York
• Exponential Distribution (2019): Exponential Distribution https://en.wikipedia.org/wiki/Exponential_distribution
• Norton, M., V. Khokhlov, and S. Uryasev (2019): Calculating CVaR and bPOE for Common Probability Distributions with Application to Portfolio Optimization and Density Estimation Annals of Operations Research 299 (1-2) 1281-1315
• Ross, S. M. (2009): Introduction to Probability and Statistics for Engineers and Scientists 4^th Edition Associated Press New York, NY
• Schmidt, D. F., and D. Makalic (2009): Universal Models for the Exponential Distribution IEEE Transactions on Information Theory 55 (7) 3087-3090

• Module = Computational Core Module
• Library = Numerical Analysis Library
• Project = R^d Continuous/Discrete Probability Measures
• Package = R¹ Exponential Distribution Implementation/Properties

Author:

Lakshmi Krishnamurthy

Constructor Summary

Constructors

Constructor	Description
R1RateDistribution(double lambda)	R1RateDistribution Constructor

Method Summary

Modifier and Type	Method	Description
double	bPOE(double x)	Retrieve the Buffered Probability of Existence
double	centralMoment(int n)	Retrieve the nth Central Moment
double	cumulative(double t)	Compute the cumulative under the distribution to the given value
double	cvar(double p)	Retrieve the Quantile CVaR (Conditional Value-at-Risk) of the Distribution
double	density(double t)	Compute the Density under the Distribution at the given Variate
double	excessKurtosis()	Retrieve the Excess Kurtosis of the Distribution
double	fisherInformation()	Retrieve the Fisher Information of the Distribution
double	iqr()	Retrieve the Inter-quantile Range (IQR) of the Distribution
double	kullbackLeiblerDivergence(R1Univariate r1UnivariateOther)	Compute the Kullback-Leibler Divergence against the other R1 Distribution
double	lambda()	Retrieve the Lambda
double	mean()	Retrieve the Mean of the Distribution
double	median()	Retrieve the Median of the Distribution
double	mode()	Retrieve the Mode of the Distribution
R1ToR1	momentGeneratingFunction()	Construct the Moment Generating Function
double	nonCentralMoment(int n)	Retrieve the nth Non-central Moment
double	quantile(double p)	Retrieve the Quantile Variate of the Distribution
double	rate()	Retrieve the Rate Parameter
double	scale()	Retrieve the Scale Parameter
static R1RateDistribution	ScaleStandard(double beta)	Construct a Standard Scale Parameterized Instance of R1 Exponential Distribution
double	skewness()	Retrieve the Skewness of the Distribution
double[]	support()	Lay out the Support of the PDF Range
double	variance()	Retrieve the Variance of the Distribution

Methods inherited from class org.drip.measure.continuous.R1Univariate

differentialEntropy, expectedShortfall, histogram, incremental, invCumulative, populationCentralMeasures, probabilityGeneratingFunction, random, randomArray, supported, tukeyAnomaly, tukeyCriterion

Methods inherited from class java.lang.Object

equals, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

Constructor Details

R1RateDistribution

public R1RateDistribution(double lambda)
                   throws java.lang.Exception

R1RateDistribution Constructor

Parameters:

lambda - Rate Parameter

Throws:

java.lang.Exception - Thrown if lambda is invalid

Method Details

ScaleStandard

public static final R1RateDistribution ScaleStandard(double beta)

Construct a Standard Scale Parameterized Instance of R¹ Exponential Distribution

Parameters:

beta - The Scale Parameter Beta

Returns:

Scale Parameterized Instance of R¹ Exponential Distribution

lambda

public double lambda()

Retrieve the Lambda

Returns:

Lambda

rate

public double rate()

Retrieve the Rate Parameter

Returns:

Rate Parameter

scale

public double scale()

Retrieve the Scale Parameter

Returns:

Scale Parameter

support

public double[] support()

Description copied from class: R1Univariate

Lay out the Support of the PDF Range

Specified by:

support in class R1Univariate

Returns:

Support of the PDF Range

density

public double density(double t)
               throws java.lang.Exception

Description copied from class: R1Univariate

Compute the Density under the Distribution at the given Variate

Specified by:

density in class R1Univariate

Parameters:

t - Variate at which the Density needs to be computed

Returns:

The Density

Throws:

java.lang.Exception - Thrown if the input is invalid

cumulative

public double cumulative(double t)
                  throws java.lang.Exception

Description copied from class: R1Univariate

Compute the cumulative under the distribution to the given value

Specified by:

cumulative in class R1Univariate

Parameters:

t - Variate to which the cumulative is to be computed

Returns:

The cumulative

Throws:

java.lang.Exception - Thrown if the inputs are invalid

mean

public double mean()
            throws java.lang.Exception

Description copied from class: R1Univariate

Retrieve the Mean of the Distribution

Specified by:

mean in class R1Univariate

Returns:

The Mean of the Distribution

Throws:

java.lang.Exception - Thrown if the Mean cannot be estimated

median

public double median()
              throws java.lang.Exception

Description copied from class: R1Univariate

Retrieve the Median of the Distribution

Overrides:

median in class R1Univariate

Returns:

The Median of the Distribution

Throws:

java.lang.Exception - Thrown if the Median cannot be estimated

mode

public double mode()
            throws java.lang.Exception

Description copied from class: R1Univariate

Retrieve the Mode of the Distribution

Overrides:

mode in class R1Univariate

Returns:

The Mode of the Distribution

Throws:

java.lang.Exception - Thrown if the Mode cannot be estimated

quantile

public double quantile(double p)
                throws java.lang.Exception

Description copied from class: R1Univariate

Retrieve the Quantile Variate of the Distribution

Overrides:

quantile in class R1Univariate

Parameters:

p - The Quantile Fraction

Returns:

The Quantile Variate of the Distribution

Throws:

java.lang.Exception - Thrown if the Quantile Variate cannot be estimated

variance

public double variance()
                throws java.lang.Exception

Description copied from class: R1Univariate

Retrieve the Variance of the Distribution

Specified by:

variance in class R1Univariate

Returns:

The Variance of the Distribution

Throws:

java.lang.Exception - Thrown if the Variance cannot be estimated

skewness

public double skewness()
                throws java.lang.Exception

Description copied from class: R1Univariate

Retrieve the Skewness of the Distribution

Overrides:

skewness in class R1Univariate

Returns:

The Skewness of the Distribution

Throws:

java.lang.Exception - Thrown if the Skewness cannot be estimated

excessKurtosis

public double excessKurtosis()
                      throws java.lang.Exception

Description copied from class: R1Univariate

Retrieve the Excess Kurtosis of the Distribution

Overrides:

excessKurtosis in class R1Univariate

Returns:

The Excess Kurtosis of the Distribution

Throws:

java.lang.Exception - Thrown if the Skewness cannot be estimated

momentGeneratingFunction

public R1ToR1 momentGeneratingFunction()

Description copied from class: R1Univariate

Construct the Moment Generating Function

Overrides:

momentGeneratingFunction in class R1Univariate

Returns:

The Moment Generating Function

fisherInformation

public double fisherInformation()
                         throws java.lang.Exception

Description copied from class: R1Univariate

Retrieve the Fisher Information of the Distribution

Overrides:

fisherInformation in class R1Univariate

Returns:

The Fisher Information of the Distribution

Throws:

java.lang.Exception - Thrown if the Fisher Information cannot be estimated

kullbackLeiblerDivergence

public double kullbackLeiblerDivergence(R1Univariate r1UnivariateOther)
                                 throws java.lang.Exception

Description copied from class: R1Univariate

Compute the Kullback-Leibler Divergence against the other R¹ Distribution

Overrides:

kullbackLeiblerDivergence in class R1Univariate

Parameters:

r1UnivariateOther - Other R¹ Distribution

Returns:

Kullback-Leibler Divergence against the other R¹ Distribution

Throws:

java.lang.Exception - Thrown if the Kullback-Leibler Divergence cannot be estimated

cvar

public double cvar(double p)
            throws java.lang.Exception

Description copied from class: R1Univariate

Retrieve the Quantile CVaR (Conditional Value-at-Risk) of the Distribution

Overrides:

cvar in class R1Univariate

Parameters:

p - The Quantile

Returns:

The Quantile CVaR of the Distribution

Throws:

java.lang.Exception - Thrown if the Quantile CVaR cannot be estimated

bPOE

public double bPOE(double x)
            throws java.lang.Exception

Description copied from class: R1Univariate

Retrieve the Buffered Probability of Existence

Overrides:

bPOE in class R1Univariate

Parameters:

x - The Variate

Returns:

The Buffered Probability of Existence

Throws:

java.lang.Exception - Thrown if the Buffered Probability of Existence cannot be estimated

nonCentralMoment

public double nonCentralMoment(int n)
                        throws java.lang.Exception

Description copied from class: R1Univariate

Retrieve the n^th Non-central Moment

Overrides:

nonCentralMoment in class R1Univariate

Parameters:

n - Moment Number

Returns:

The n^th Non-central Moment

Throws:

java.lang.Exception - Thrown if the n^th Non-central Moment cannot be estimated

centralMoment

public double centralMoment(int n)
                     throws java.lang.Exception

Description copied from class: R1Univariate

Retrieve the n^th Central Moment

Overrides:

centralMoment in class R1Univariate

Parameters:

n - Moment Number

Returns:

The n^th Central Moment

Throws:

java.lang.Exception - Thrown if the n^th Central Moment cannot be estimated

iqr

public double iqr()
           throws java.lang.Exception

Description copied from class: R1Univariate

Retrieve the Inter-quantile Range (IQR) of the Distribution

Overrides:

iqr in class R1Univariate

Returns:

The Inter-quantile Range of the Distribution

Throws:

java.lang.Exception - Thrown if the Inter-quantile Range cannot be estimated