Package org.drip.measure.chisquare

Class R1NonCentral

java.lang.Object

org.drip.measure.continuous.R1Distribution

org.drip.measure.chisquare.R1NonCentral

Direct Known Subclasses:

R1NonCentralCumulantInvariant

public class R1NonCentral
extends R1Distribution

R1NonCentral implements the Distribution Table 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 Standard Instance of R1NonCentral
• R1NonCentral Constructor
• Retrieve the R¹ Non-Central Parameters
• Retrieve the Gamma Estimator
• Retrieve the Digamma Estimator
• Retrieve the Lower Incomplete Gamma Estimator
• Retrieve the Modified Bessel First Kind 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 Variance of the Distribution
• Retrieve the Skewness of the Distribution
• Retrieve the Excess Kurtosis of the Distribution
• Construct the Moment Generating Function
• Compute the Cumulant
• Compute the Leading Non-central Moments
• Compute the Leading Central Moments
• Compute the Non-central Moment

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
R1NonCentral(R1NonCentralParameters parameters, R1ToR1 gammaEstimator, R1ToR1 digammaEstimator, R2ToR1 lowerIncompleteGammaEstimator, ModifiedBesselFirstKindEstimator modifiedBesselFirstKindEstimator)	R1NonCentral Constructor

Method Summary

Modifier and Type	Method	Description
double	cumulant(int n)	Compute the Cumulant
double	cumulative(double t)	Compute the cumulative under the distribution to the given value
double	density(double x)	Compute the Density under the Distribution at the given Variate
R1ToR1	digammaEstimator()	Retrieve the Digamma Estimator
double	excessKurtosis()	Retrieve the Excess Kurtosis of the Distribution
R1ToR1	gammaEstimator()	Retrieve the Gamma Estimator
double[]	leadingCentralMoments()	Compute the Leading Central Moments
double[]	leadingRawMoments()	Compute the Leading Non-central Moments
R2ToR1	lowerIncompleteGammaEstimator()	Retrieve the Lower Incomplete Gamma Estimator
double	mean()	Retrieve the Mean of the Distribution
ModifiedBesselFirstKindEstimator	modifiedBesselFirstKindEstimator()	Retrieve the Modified Bessel First Kind Estimator
R1ToR1	momentGeneratingFunction()	Construct the Moment Generating Function
double	nonCentralMoment(int n)	Compute the Non-central Moment
R1NonCentralParameters	parameters()	Retrieve the R1 Non-Central Parameters
double	skewness()	Retrieve the Skewness of the Distribution
static R1NonCentral	Standard(double degreesOfFreedom, double nonCentralityParameter, R1ToR1 gammaEstimator, R1ToR1 digammaEstimator, R2ToR1 lowerIncompleteGammaEstimator, ModifiedBesselFirstKindEstimator modifiedBesselFirstKindEstimator)	Construct the Standard Instance of R1NonCentral
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, differentialEntropy, expectedShortfall, fisherInformation, histogram, incremental, invCumulative, 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

R1NonCentral

public R1NonCentral(R1NonCentralParameters parameters,
R1ToR1 gammaEstimator,
R1ToR1 digammaEstimator,
R2ToR1 lowerIncompleteGammaEstimator,
ModifiedBesselFirstKindEstimator modifiedBesselFirstKindEstimator)
             throws java.lang.Exception

R1NonCentral Constructor

Parameters:

parameters - R¹ Non-central Parameters

gammaEstimator - Gamma Estimator

digammaEstimator - Digamma Estimator

lowerIncompleteGammaEstimator - Lower Incomplete Gamma Estimator

modifiedBesselFirstKindEstimator - Modified Bessel First Kind Estimator

Throws:

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

Method Details

Standard

public static final R1NonCentral Standard(double degreesOfFreedom,
double nonCentralityParameter,
R1ToR1 gammaEstimator,
R1ToR1 digammaEstimator,
R2ToR1 lowerIncompleteGammaEstimator,
ModifiedBesselFirstKindEstimator modifiedBesselFirstKindEstimator)

Construct the Standard Instance of R1NonCentral

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 Standard Instance of R1NonCentral

parameters

public R1NonCentralParameters parameters()

Retrieve the R¹ Non-Central Parameters

Returns:

The R¹ Non-Central 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

modifiedBesselFirstKindEstimator

public ModifiedBesselFirstKindEstimator modifiedBesselFirstKindEstimator()

Retrieve the Modified Bessel First Kind Estimator

Returns:

Modified Bessel First Kind 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 x)
               throws java.lang.Exception

Compute the Density under the Distribution at the given Variate

Specified by:

density in class R1Distribution

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 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

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 Skewness cannot be estimated

momentGeneratingFunction

public R1ToR1 momentGeneratingFunction()

Construct the Moment Generating Function

Overrides:

momentGeneratingFunction in class R1Distribution

Returns:

The Moment Generating Function

cumulant

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

Compute the Cumulant

Parameters:

n - Cumulant Index

Returns:

The Cumulant

Throws:

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

leadingRawMoments

public double[] leadingRawMoments()

Compute the Leading Non-central Moments

Returns:

Leading Non-central Moments

leadingCentralMoments

public double[] leadingCentralMoments()

Compute the Leading Central Moments

Returns:

Leading Central Moments

nonCentralMoment

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

Compute the Non-central Moment

Overrides:

nonCentralMoment in class R1Distribution

Parameters:

n - Moment Index

Returns:

The Non-central Moment

Throws:

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