com.imsl.stat.distributions

Class ParetoPD

java.lang.Object

com.imsl.stat.distributions.ProbabilityDistribution

com.imsl.stat.distributions.ParetoPD

All Implemented Interfaces:

PDFGradientInterface, PDFHessianInterface, Serializable, Cloneable

public class ParetoPD
extends ProbabilityDistribution
implements Serializable, Cloneable, PDFHessianInterface

The Pareto probability distribution.

See Also:

Example, Serialized Form

Constructor Summary

Constructors

Constructor and Description
ParetoPD() Constructor for the Pareto probability distribution.

Method Summary

Modifier and Type	Method and Description
double[]	getMethodOfMomentsEstimates(double[] x) Returns the method-of-moments estimates given the sample data.
double[]	getParameterLowerBounds() Returns the lower bounds of the parameters.
double[]	getParameterUpperBounds() Returns the upper bounds of the parameters.
double[]	getPDFGradient(double x, double... params) Returns the analytic gradient of the pdf.
double[][]	getPDFHessian(double x, double... params) Returns the analytic Hessian of the pdf.
double	pdf(double x, double... params) Returns the value of the Pareto probability density function.

Methods inherited from class com.imsl.stat.distributions.ProbabilityDistribution

getNumberOfParameters, getPDFGradientApproximation, getPDFHessianApproximation, getRangeOfX, setRangeOfX

Methods inherited from class java.lang.Object

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

Constructor Detail

ParetoPD

public ParetoPD()

Constructor for the Pareto probability distribution.

Method Detail

getParameterLowerBounds

public double[] getParameterLowerBounds()

Returns the lower bounds of the parameters.

Specified by:

getParameterLowerBounds in class ProbabilityDistribution

Returns:

a double array of length 2 containing the lower bounds for the scale parameter (\(x_m\gt0\)) and the shape parameter (\(k\gt0\))

getParameterUpperBounds

public double[] getParameterUpperBounds()

Returns the upper bounds of the parameters.

Specified by:

getParameterUpperBounds in class ProbabilityDistribution

Returns:

a double array of length 2 containing the upper bounds for the scale parameter (\(x_m\gt0\)) and the shape parameter (\(k\gt0\))

pdf

public double pdf(double x,
                  double... params)

Returns the value of the Pareto probability density function.

The probability density function of the Pareto distribution is

$$f\left(x;x_m,k\right)=\left\{\begin{array}{ll} \frac{kx_m^k}{x^{k+1}} & x\ge x_m, \\[3pt] 0 & x\lt x_m.\end{array}\right.$$

where \(x_m\gt0\) is the scale parameter and \(k\gt0\) is the shape parameter.

Specified by:

pdf in class ProbabilityDistribution

Parameters:

x - a double, the value (quantile) at which to evaluate the pdf

params - a double array containing the scale and shape parameters. The parameters can also be given in the form pdf(x,a,b), where a=\(x_m\) and b=\(k\) are scalars.

Returns:

a double, the probability density function at x and the given parameter values

getPDFGradient

public double[] getPDFGradient(double x,
                               double... params)

Returns the analytic gradient of the pdf.

Specified by:

getPDFGradient in interface PDFGradientInterface

Parameters:

x - a double, the value at which to evaluate the gradient

params - a double array specifying the scale parameter (\(x_m\gt0\)) and the shape parameter (\(k\gt0\))

Returns:

a double array containing the first partial derivative of the pdf with respect to the parameters

getPDFHessian

public double[][] getPDFHessian(double x,
                                double... params)

Returns the analytic Hessian of the pdf.

Specified by:

getPDFHessian in interface PDFHessianInterface

Parameters:

x - a double, the value at which to evaluate the Hessian

params - a double array specifying the scale parameter (\(x_m\gt0\)) and the shape parameter (\(k\gt0\))

Returns:

a double array containing the second partial derivatives of the pdf with respect to the parameters

getMethodOfMomentsEstimates

public double[] getMethodOfMomentsEstimates(double[] x)

Returns the method-of-moments estimates given the sample data.

Parameters:

x - a double array containing the data

Returns:

a double array containing method-of-moments estimates for the parameters of the Pareto distribution