Package org.drip.measure.gamma

Class R1ShapeScaleDistribution

java.lang.Object

org.drip.measure.continuous.R1Distribution

org.drip.measure.gamma.R1ShapeScaleDistribution

Direct Known Subclasses:

ErlangDistribution, R1GammaToExponential, R1GammaToMaxwellBoltzmannSquared, R1ShapeScaleDiscrete

public class R1ShapeScaleDistribution
extends R1Distribution

R1ShapeScaleDistribution implements the Shape and Scale Parameterization of the R¹ Gamma Distribution. The References are:

• Devroye, L. (1986): Non-Uniform Random Variate Generation Springer-Verlag New York
• Gamma Distribution (2019): Gamma Distribution https://en.wikipedia.org/wiki/Chi-squared_distribution
• Louzada, F., P. L. Ramos, and E. Ramos (2019): A Note on Bias of Closed-Form Estimators for the Gamma Distribution Derived From Likelihood Equations The American Statistician 73 (2) 195-199
• Minka, T. (2002): Estimating a Gamma distribution https://tminka.github.io/papers/minka-gamma.pdf
• Ye, Z. S., and N. Chen (2017): Closed-Form Estimators for the Gamma Distribution Derived from Likelihood Equations The American Statistician 71 (2) 177-181

It provides the following Functionality:

• Construct a Gamma Distribution from Shape and Rate Parameters
• Shape Summation Based R1ShapeScaleDistribution
• Construct the Standard R1ShapeScaleDistribution Instance
• R1ShapeScaleDistribution Constructor
• Retrieve the Shape-Scale Parameters
• Retrieve the Gamma Estimator
• Retrieve the Digamma Estimator
• Retrieve the Lower Incomplete Gamma Estimator
• Lay out the Support of the PDF Range
• Compute the Density under the Distribution at the given Variate
• Compute the cumulative under the distribution to the given value
• Retrieve the Mean of the Distribution
• Retrieve the Mode of the Distribution
• Retrieve the Variance of the Distribution
• Retrieve the Skewness of the Distribution
• Retrieve the Excess Kurtosis of the Distribution
• Retrieve the Differential Entropy of the Distribution
• Retrieve the Moment Generating Function of the Distribution
• Retrieve the Central Limit Theorem Equivalent Normal Distribution Proxy
• Compute the Logarithmic Expectation
• Compute the Banneheke-Ekayanake Approximation for the Median when k gte 1
• Compute the Ramanujan-Choi Approximation for the Median
• Compute the Chen-Rubin Median Lower Bound
• Compute the Chen-Rubin Median Upper Bound
• Generate a Scaled Gamma Distribution
• Retrieve the Array of Natural Parameters
• Retrieve the Array of Natural Statistics
• Generate the Exponential Family Representation
• Compute the Laplacian
• Generate a Random Variable using the Ahrens-Dieter (1982) Scheme
• Generate a Random Variable using the Marsaglia (1977) Scheme

Module	Computational Core Module
Library	Numerical Analysis Library
Project	Rd Continuous/Discrete Probability Measures
Package	R1 Gamma Distribution Implementation/Properties

Author:

Lakshmi Krishnamurthy

Constructor Summary

Constructors

Constructor	Description
R1ShapeScaleDistribution(ShapeScaleParameters shapeScaleParameters, R1ToR1 gammaEstimator, R1ToR1 digammaEstimator, R2ToR1 lowerIncompleteGammaEstimator)	R1ShapeScaleDistribution Constructor

Method Summary

Modifier and Type	Method	Description
double	bannehekeEkayanakeMedianApproximation()	Compute the Banneheke-Ekayanake Approximation for the Median when k gte 1
double	chenRubinMedianLowerBound()	Compute the Chen-Rubin Median Lower Bound
double	chenRubinMedianUpperBound()	Compute the Chen-Rubin Median Upper Bound
R1UnivariateNormal	cltProxy()	Retrieve the Central Limit Theorem Equivalent Normal Distribution Proxy
double	cumulative(double t)	Compute the cumulative under the distribution to the given value
double	density(double t)	Compute the Density under the Distribution at the given Variate
double	differentialEntropy()	Retrieve the Differential Entropy of the Distribution
R1ToR1	digammaEstimator()	Retrieve the Digamma Estimator
double	excessKurtosis()	Retrieve the Excess Kurtosis of the Distribution
ExponentialFamilyRepresentation	exponentialFamilyRepresentation(double x)	Generate the Exponential Family Representation
static R1ShapeScaleDistribution	FromShapeAndRate(double shapeParameter, double rateParameter, R1ToR1 gammaEstimator, R1ToR1 digammaEstimator, R2ToR1 lowerIncompleteGammaEstimator)	Construct a Gamma Distribution from Shape and Rate Parameters
static R1ShapeScaleDistribution	FromShapeAndSummation(double[] shapeParameterArray, double scaleParameter, R1ToR1 gammaEstimator, R1ToR1 digammaEstimator, R2ToR1 lowerIncompleteGammaEstimator)	Shape Summation Based R1ShapeScaleDistribution
R1ToR1	gammaEstimator()	Retrieve the Gamma Estimator
double	laplacian(double s)	Compute the Laplacian
double	logarithmicExpectation()	Compute the Logarithmic Expectation
R2ToR1	lowerIncompleteGammaEstimator()	Retrieve the Lower Incomplete Gamma Estimator
double	mean()	Retrieve the Mean of the Distribution
double	mode()	Retrieve the Mode of the Distribution
R1ToR1	momentGeneratingFunction()	Retrieve the Moment Generating Function of the Distribution
double[]	naturalParameters()	Retrieve the Array of Natural Parameters
double[]	naturalStatistics(double x)	Retrieve the Array of Natural Statistics
double	ramanujanChoiMedianApproximation()	Compute the Ramanujan-Choi Approximation for the Median
double	randomAhrensDieter1982()	Generate a Random Variable using the Ahrens-Dieter (1982) Scheme
double	randomMarsaglia1977()	Generate a Random Variable using the Marsaglia (1977) Scheme
R1ShapeScaleDistribution	scale(double scaleFactor)	Generate a Scaled Gamma Distribution
ShapeScaleParameters	shapeScaleParameters()	Retrieve the Shape-Scale Parameters
double	skewness()	Retrieve the Skewness of the Distribution
static R1ShapeScaleDistribution	Standard(double shapeParameter, double scaleParameter, R1ToR1 gammaEstimator, R1ToR1 digammaEstimator, R2ToR1 lowerIncompleteGammaEstimator)	Construct the Standard R1ShapeScaleDistribution Instance
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.R1Distribution

bPOE, centralMoment, cvar, expectedShortfall, fisherInformation, histogram, incremental, invCumulative, iqr, kullbackLeiblerDivergence, median, nonCentralMoment, populationCentralMeasures, probabilityGeneratingFunction, quantile, random, randomArray, supported, tukeyAnomaly, tukeyCriterion

Methods inherited from class java.lang.Object

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

Constructor Details

R1ShapeScaleDistribution

public R1ShapeScaleDistribution(ShapeScaleParameters shapeScaleParameters,
R1ToR1 gammaEstimator,
R1ToR1 digammaEstimator,
R2ToR1 lowerIncompleteGammaEstimator)
                         throws java.lang.Exception

R1ShapeScaleDistribution Constructor

Parameters:

shapeScaleParameters - Shape-Scale Parameters

gammaEstimator - Gamma Estimator

digammaEstimator - Digamma Estimator

lowerIncompleteGammaEstimator - Lower Incomplete Gamma Estimator

Throws:

java.lang.Exception - Thrown if the Inputs are Invalid

Method Details

FromShapeAndRate

public static final R1ShapeScaleDistribution FromShapeAndRate(double shapeParameter,
double rateParameter,
R1ToR1 gammaEstimator,
R1ToR1 digammaEstimator,
R2ToR1 lowerIncompleteGammaEstimator)

Construct a Gamma Distribution from Shape and Rate Parameters

Parameters:

shapeParameter - Shape Parameter

rateParameter - Rate Parameter

gammaEstimator - Gamma Estimator

digammaEstimator - Digamma Estimator

lowerIncompleteGammaEstimator - Lower Incomplete Gamma Estimator

Returns:

Gamma Distribution from Shape Alpha and Rate Beta Parameters

FromShapeAndSummation

public static final R1ShapeScaleDistribution FromShapeAndSummation(double[] shapeParameterArray,
double scaleParameter,
R1ToR1 gammaEstimator,
R1ToR1 digammaEstimator,
R2ToR1 lowerIncompleteGammaEstimator)

Shape Summation Based R1ShapeScaleDistribution

Parameters:

shapeParameterArray - Shape Parameter Array

scaleParameter - Scale Parameter

gammaEstimator - Gamma Estimator

digammaEstimator - Digamma Estimator

lowerIncompleteGammaEstimator - Lower Incomplete Gamma Estimator

Returns:

Shape Summation Based R1ShapeScaleDistribution

Standard

public static final R1ShapeScaleDistribution Standard(double shapeParameter,
double scaleParameter,
R1ToR1 gammaEstimator,
R1ToR1 digammaEstimator,
R2ToR1 lowerIncompleteGammaEstimator)

Construct the Standard R1ShapeScaleDistribution Instance

Parameters:

shapeParameter - Shape Parameter

scaleParameter - Scale Parameter

gammaEstimator - Gamma Estimator

digammaEstimator - Digamma Estimator

lowerIncompleteGammaEstimator - Lower Incomplete Gamma Estimator

Returns:

The R1ShapeScaleDistribution Instance

shapeScaleParameters

public ShapeScaleParameters shapeScaleParameters()

Retrieve the Shape-Scale Parameters

Returns:

The Shape-Scale Parameters

gammaEstimator

public R1ToR1 gammaEstimator()

Retrieve the Gamma Estimator

Returns:

Gamma Estimator

digammaEstimator

public R1ToR1 digammaEstimator()

Retrieve the Digamma Estimator

Returns:

Digamma Estimator

lowerIncompleteGammaEstimator

public R2ToR1 lowerIncompleteGammaEstimator()

Retrieve the Lower Incomplete Gamma Estimator

Returns:

Lower Incomplete Gamma Estimator

support

public double[] support()

Lay out the Support of the PDF Range

Specified by:

support in class R1Distribution

Returns:

Support of the PDF Range

density

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

Compute the Density under the Distribution at the given Variate

Specified by:

density in class R1Distribution

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

Compute the cumulative under the distribution to the given value

Specified by:

cumulative in class R1Distribution

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

Retrieve the Mean of the Distribution

Specified by:

mean in class R1Distribution

Returns:

The Mean of the Distribution

Throws:

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

mode

public double mode()
            throws java.lang.Exception

Retrieve the Mode of the Distribution

Overrides:

mode in class R1Distribution

Returns:

The Mode of the Distribution

Throws:

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

variance

public double variance()
                throws java.lang.Exception

Retrieve the Variance of the Distribution

Specified by:

variance in class R1Distribution

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

Retrieve the Skewness of the Distribution

Overrides:

skewness in class R1Distribution

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

Retrieve the Excess Kurtosis of the Distribution

Overrides:

excessKurtosis in class R1Distribution

Returns:

The Excess Kurtosis of the Distribution

Throws:

java.lang.Exception - Thrown if the Excess Kurtosis cannot be estimated

differentialEntropy

public double differentialEntropy()
                           throws java.lang.Exception

Retrieve the Differential Entropy of the Distribution

Overrides:

differentialEntropy in class R1Distribution

Returns:

The Differential Entropy of the Distribution

Throws:

java.lang.Exception - Thrown if the Differential Entropy cannot be estimated

momentGeneratingFunction

public R1ToR1 momentGeneratingFunction()

Retrieve the Moment Generating Function of the Distribution

Overrides:

momentGeneratingFunction in class R1Distribution

Returns:

The Moment Generating Function of the Distribution

Throws:

java.lang.Exception - Thrown if the Moment Generating Function cannot be estimated

cltProxy

public R1UnivariateNormal cltProxy()

Retrieve the Central Limit Theorem Equivalent Normal Distribution Proxy

Returns:

The Central Limit Theorem Equivalent Normal Distribution Proxy

logarithmicExpectation

public double logarithmicExpectation()
                              throws java.lang.Exception

Compute the Logarithmic Expectation

Returns:

The Logarithmic Expectation

Throws:

java.lang.Exception - Thrown if the Logarithmic Expectation cannot be computed

bannehekeEkayanakeMedianApproximation

public double bannehekeEkayanakeMedianApproximation()
                                             throws java.lang.Exception

Compute the Banneheke-Ekayanake Approximation for the Median when k gte 1

Returns:

The Banneheke-Ekayanake Approximation for the Median

Throws:

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

ramanujanChoiMedianApproximation

public double ramanujanChoiMedianApproximation()

Compute the Ramanujan-Choi Approximation for the Median

Returns:

The Ramanujan-Choi Approximation for the Median

chenRubinMedianLowerBound

public double chenRubinMedianLowerBound()

Compute the Chen-Rubin Median Lower Bound

Returns:

The Chen-Rubin Median Lower Bound

chenRubinMedianUpperBound

public double chenRubinMedianUpperBound()

Compute the Chen-Rubin Median Upper Bound

Returns:

The Chen-Rubin Median Upper Bound

scale

public R1ShapeScaleDistribution scale(double scaleFactor)

Generate a Scaled Gamma Distribution

Parameters:

scaleFactor - The Gamma Distribution Scale Factor

Returns:

Scaled Gamma Distribution

naturalParameters

public double[] naturalParameters()

Retrieve the Array of Natural Parameters

Returns:

Array of Natural Parameters

naturalStatistics

public double[] naturalStatistics(double x)

Retrieve the Array of Natural Statistics

Parameters:

x - X

Returns:

Array of Natural Statistics

exponentialFamilyRepresentation

public ExponentialFamilyRepresentation exponentialFamilyRepresentation(double x)

Generate the Exponential Family Representation

Parameters:

x - X

Returns:

Exponential Family Representation

laplacian

public double laplacian(double s)
                 throws java.lang.Exception

Compute the Laplacian

Parameters:

s - S

Returns:

The Laplacian

Throws:

java.lang.Exception - Thrown if the Inputs are Invalid

randomAhrensDieter1982

public double randomAhrensDieter1982()
                              throws java.lang.Exception

Generate a Random Variable using the Ahrens-Dieter (1982) Scheme

Returns:

Random Variable using the Ahrens-Dieter (1982) Scheme

Throws:

java.lang.Exception - Thrown if the Random Instance cannot be estimated

randomMarsaglia1977

public double randomMarsaglia1977()
                           throws java.lang.Exception

Generate a Random Variable using the Marsaglia (1977) Scheme

Returns:

Random Variable using the Marsaglia (1977) Scheme

Throws:

java.lang.Exception - Thrown if the Random Instance cannot be estimated