Package org.drip.measure.chisquare

Class R1NonCentralCumulantInvariant

java.lang.Object

org.drip.measure.continuous.R1Distribution

org.drip.measure.chisquare.R1NonCentral

org.drip.measure.chisquare.R1NonCentralCumulantInvariant

public class R1NonCentralCumulantInvariant
extends R1NonCentral

R1NonCentralCumulantInvariant implements the Cumulant Invariant Transformation for the R¹ Non-central Chi-Square Distribution. The References are:

• Johnson, N. L., S. Kotz, and N. Balakrishnan (1995): Continuous Univariate Distributions 2^nd Edition John Wiley and Sons
• Muirhead, R. (2005): Aspects of Multivariate Statistical Theory 2^nd Edition Wiley
• Non-central Chi-Squared Distribution (2019): Chi-Squared Function https://en.wikipedia.org/wiki/Noncentral_chi-squared_distribution
• Sankaran, M. (1963): Approximations to the Non-Central Chi-Square Distribution Biometrika 50 (1-2) 199-204
• Young, D. S. (2010): tolerance: An R Package for Estimating Tolerance Intervals Journal of Statistical Software 36 (5) 1-39

It provides the following Functionality:

• Construct the Second Cumulant Invariant Instance of R1NonCentralCumulantInvariant
• Construct the Third Cumulant Invariant Instance of R1NonCentralCumulantInvariant
• Construct the Fourth Cumulant Invariant Instance of R1NonCentralCumulantInvariant
• R1NonCentralCumulantInvariant Constructor
• Retrieve the Sankaran (1963) "B"
• Compute the Density under the Distribution at the given Variate
• Transform x into the Wilson-Hilferty Variate
• Transform the Wilson-Hilferty Variate into x

Module	Computational Core Module
Library	Numerical Analysis Library
Project	Rd Continuous/Discrete Probability Measures
Package	Chi-Square Distribution Implementation/Properties

Author:

Lakshmi Krishnamurthy

Constructor Summary

Constructors

Constructor	Description
R1NonCentralCumulantInvariant(R1NonCentralParameters r1NonCentralParameters, R1ToR1 gammaEstimator, R1ToR1 digammaEstimator, R2ToR1 lowerIncompleteGammaEstimator, ModifiedBesselFirstKindEstimator modifiedBesselFirstKindEstimator, double sankaranB)	R1NonCentralCumulantInvariant Constructor

Method Summary

Modifier and Type	Method	Description
double	cumulative(double x)	Transform x into the Wilson-Hilferty Variate
double	density(double x)	Compute the Density under the Distribution at the given Variate
static R1NonCentralCumulantInvariant	FourthCumulant(int degreesOfFreedom, double nonCentralityParameter, R1ToR1 gammaEstimator, R1ToR1 digammaEstimator, R2ToR1 lowerIncompleteGammaEstimator, ModifiedBesselFirstKindEstimator modifiedBesselFirstKindEstimator)	Construct the Fourth Cumulant Invariant Instance of R1NonCentralCumulantInvariant
double	invCumulative(double y)	Transform the Wilson-Hilferty Variate into x
double	sankaranB()	Retrieve the Sankaran (1963) "B"
static R1NonCentralCumulantInvariant	SecondCumulant(int degreesOfFreedom, double nonCentralityParameter, R1ToR1 gammaEstimator, R1ToR1 digammaEstimator, R2ToR1 lowerIncompleteGammaEstimator, ModifiedBesselFirstKindEstimator modifiedBesselFirstKindEstimator)	Construct the Second Cumulant Invariant Instance of R1NonCentralCumulantInvariant
static R1NonCentralCumulantInvariant	ThirdCumulant(int degreesOfFreedom, double nonCentralityParameter, R1ToR1 gammaEstimator, R1ToR1 digammaEstimator, R2ToR1 lowerIncompleteGammaEstimator, ModifiedBesselFirstKindEstimator modifiedBesselFirstKindEstimator)	Construct the Third Cumulant Invariant Instance of R1NonCentralCumulantInvariant

Methods inherited from class org.drip.measure.chisquare.R1NonCentral

cumulant, digammaEstimator, excessKurtosis, gammaEstimator, leadingCentralMoments, leadingRawMoments, lowerIncompleteGammaEstimator, mean, modifiedBesselFirstKindEstimator, momentGeneratingFunction, nonCentralMoment, parameters, skewness, Standard, support, variance

Methods inherited from class org.drip.measure.continuous.R1Distribution

bPOE, centralMoment, cvar, differentialEntropy, expectedShortfall, fisherInformation, histogram, incremental, iqr, kullbackLeiblerDivergence, median, mode, 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

R1NonCentralCumulantInvariant

public R1NonCentralCumulantInvariant(R1NonCentralParameters r1NonCentralParameters,
R1ToR1 gammaEstimator,
R1ToR1 digammaEstimator,
R2ToR1 lowerIncompleteGammaEstimator,
ModifiedBesselFirstKindEstimator modifiedBesselFirstKindEstimator,
double sankaranB)
                              throws java.lang.Exception

R1NonCentralCumulantInvariant Constructor

Parameters:

r1NonCentralParameters - R¹ Non-central Parameters

gammaEstimator - Gamma Estimator

digammaEstimator - Digamma Estimator

lowerIncompleteGammaEstimator - Lower Incomplete Gamma Estimator

modifiedBesselFirstKindEstimator - Modified Bessel First Kind Estimator

sankaranB - Sankaran (1963) "B"

Throws:

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

Method Details

SecondCumulant

public static final R1NonCentralCumulantInvariant SecondCumulant(int degreesOfFreedom,
double nonCentralityParameter,
R1ToR1 gammaEstimator,
R1ToR1 digammaEstimator,
R2ToR1 lowerIncompleteGammaEstimator,
ModifiedBesselFirstKindEstimator modifiedBesselFirstKindEstimator)

Construct the Second Cumulant Invariant Instance of R1NonCentralCumulantInvariant

Parameters:

degreesOfFreedom - Degrees of Freedom

nonCentralityParameter - Non-centrality Parameter

gammaEstimator - Gamma Estimator

digammaEstimator - Digamma Estimator

lowerIncompleteGammaEstimator - Lower Incomplete Gamma Estimator

modifiedBesselFirstKindEstimator - Modified Bessel First Kind Estimator

Returns:

The Second Cumulant Invariant Instance of R1NonCentralCumulantInvariant

ThirdCumulant

public static final R1NonCentralCumulantInvariant ThirdCumulant(int degreesOfFreedom,
double nonCentralityParameter,
R1ToR1 gammaEstimator,
R1ToR1 digammaEstimator,
R2ToR1 lowerIncompleteGammaEstimator,
ModifiedBesselFirstKindEstimator modifiedBesselFirstKindEstimator)

Construct the Third Cumulant Invariant Instance of R1NonCentralCumulantInvariant

Parameters:

degreesOfFreedom - Degrees of Freedom

nonCentralityParameter - Non-centrality Parameter

gammaEstimator - Gamma Estimator

digammaEstimator - Digamma Estimator

lowerIncompleteGammaEstimator - Lower Incomplete Gamma Estimator

modifiedBesselFirstKindEstimator - Modified Bessel First Kind Estimator

Returns:

The Third Cumulant Invariant Instance of R1NonCentralCumulantInvariant

FourthCumulant

public static final R1NonCentralCumulantInvariant FourthCumulant(int degreesOfFreedom,
double nonCentralityParameter,
R1ToR1 gammaEstimator,
R1ToR1 digammaEstimator,
R2ToR1 lowerIncompleteGammaEstimator,
ModifiedBesselFirstKindEstimator modifiedBesselFirstKindEstimator)

Construct the Fourth Cumulant Invariant Instance of R1NonCentralCumulantInvariant

Parameters:

degreesOfFreedom - Degrees of Freedom

nonCentralityParameter - Non-centrality Parameter

gammaEstimator - Gamma Estimator

digammaEstimator - Digamma Estimator

lowerIncompleteGammaEstimator - Lower Incomplete Gamma Estimator

modifiedBesselFirstKindEstimator - Modified Bessel First Kind Estimator

Returns:

The Fourth Cumulant Invariant Instance of R1NonCentralCumulantInvariant

sankaranB

public double sankaranB()

Retrieve the Sankaran (1963) "B"

Returns:

Sankaran (1963) "B"

density

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

Compute the Density under the Distribution at the given Variate

Overrides:

density in class R1NonCentral

Parameters:

x - 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 x)
                  throws java.lang.Exception

Transform x into the Wilson-Hilferty Variate

Overrides:

cumulative in class R1NonCentral

Parameters:

x - X

Returns:

The Wilson-Hilferty Variate

Throws:

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

invCumulative

public double invCumulative(double y)
                     throws java.lang.Exception

Transform the Wilson-Hilferty Variate into x

Overrides:

invCumulative in class R1Distribution

Parameters:

y - The Wilson-Hilferty Variate

Returns:

The Wilson-Hilferty Variate transformed back to x

Throws:

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