Package org.drip.measure.chisquare

Class R1WilsonHilferty

java.lang.Object

org.drip.measure.continuous.R1Distribution

org.drip.measure.chisquare.R1WilsonHilferty

Direct Known Subclasses:

R1CentralWilsonHilferty, R1NonCentralWilsonHaferty

public abstract class R1WilsonHilferty
extends R1Distribution

R1WilsonHilferty implements the Normal Proxy Version for the R¹ Chi-Square Distribution using the Wilson-Hilferty Transformation. The References are:

• Abramowitz, M., and I. A. Stegun (2007): Handbook of Mathematics Functions Dover Book on Mathematics
• Backstrom, T., and J. Fischer (2018): Fast Randomization for Distributed Low Bit-rate Coding of Speech and Audio IEEE/ACM Transactions on Audio, Speech, and Language Processing 26 (1) 19-30
• Chi-Squared Distribution (2019): Chi-Squared Function https://en.wikipedia.org/wiki/Chi-squared_distribution
• Johnson, N. L., S. Kotz, and N. Balakrishnan (1994): Continuous Univariate Distributions 2^nd Edition John Wiley and Sons
• National Institute of Standards and Technology (2019): Chi-Squared Distribution https://www.itl.nist.gov/div898/handbook/eda/section3/eda3666.htm

It provides the following Functionality:

• Retrieve the Degrees of Freedom
• Retrieve the R¹ Univariate Normal
• Transform x into the Wilson-Hilferty Variate
• Transform the Wilson-Hilferty Variate into x
• 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
• Compute the inverse cumulative under the distribution corresponding to the given value
• Retrieve the Mean of the Distribution
• Retrieve the Median 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
• Construct the Moment Generating Function
• Construct the Probability Generating Function

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

Author:

Lakshmi Krishnamurthy

Method Summary

Modifier and Type	Method	Description
double	cumulative(double t)	Compute the cumulative under the distribution to the given value
double	degreesOfFreedom()	Retrieve the Degrees of Freedom
double	density(double t)	Compute the Density under the Distribution at the given Variate
double	excessKurtosis()	Retrieve the Excess Kurtosis of the Distribution
double	invCumulative(double y)	Compute the inverse cumulative under the distribution corresponding to the given value
abstract double	inverseTransform(double wilsonHilferty)	Transform the Wilson-Hilferty Variate into x
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
R1ToR1	probabilityGeneratingFunction()	Construct the Probability Generating Function
R1UnivariateNormal	r1UnivariateNormal()	Retrieve the R1 Univariate Normal
double	skewness()	Retrieve the Skewness of the Distribution
double[]	support()	Lay out the Support of the PDF Range
abstract double	transform(double x)	Transform x into the Wilson-Hilferty Variate
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, iqr, kullbackLeiblerDivergence, nonCentralMoment, populationCentralMeasures, quantile, random, randomArray, supported, tukeyAnomaly, tukeyCriterion

Methods inherited from class java.lang.Object

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

Method Details

degreesOfFreedom

public double degreesOfFreedom()

Retrieve the Degrees of Freedom

Returns:

The Degrees of Freedom

r1UnivariateNormal

public R1UnivariateNormal r1UnivariateNormal()

Retrieve the R¹ Univariate Normal

Returns:

The R¹ Univariate Normal

transform

public abstract double transform(double x)

Transform x into the Wilson-Hilferty Variate

Parameters:

x - X

Returns:

The Wilson-Hilferty Variate

inverseTransform

public abstract double inverseTransform(double wilsonHilferty)

Transform the Wilson-Hilferty Variate into x

Parameters:

wilsonHilferty - The Wilson-Hilferty Variate

Returns:

The Wilson-Hilferty Variate transformed back to x

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

invCumulative

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

Compute the inverse cumulative under the distribution corresponding to the given value

Overrides:

invCumulative in class R1Distribution

Parameters:

y - Value corresponding to which the inverse cumulative is to be computed

Returns:

The inverse cumulative

Throws:

java.lang.Exception - Thrown if the Input is 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

median

public double median()
              throws java.lang.Exception

Retrieve the Median of the Distribution

Overrides:

median in class R1Distribution

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

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 Skewness cannot be estimated

momentGeneratingFunction

public R1ToR1 momentGeneratingFunction()

Construct the Moment Generating Function

Overrides:

momentGeneratingFunction in class R1Distribution

Returns:

The Moment Generating Function

probabilityGeneratingFunction

public R1ToR1 probabilityGeneratingFunction()

Construct the Probability Generating Function

Overrides:

probabilityGeneratingFunction in class R1Distribution

Returns:

The Probability Generating Function