umontreal.iro.lecuyer.probdist
Class PowerDist

java.lang.Object
  umontreal.iro.lecuyer.probdist.ContinuousDistribution
      umontreal.iro.lecuyer.probdist.PowerDist

All Implemented Interfaces:

Distribution

public class PowerDist
extends ContinuousDistribution

Extends the class ContinuousDistribution for the power distribution with shape parameter c > 0, over the interval [a, b], where a < b. It has density

f (x) = c(x - a)^c-1/(b - a)^c

for a < x < b, and 0 elsewhere. It has distribution function

F(x) = (x - a)^c/(b - a)^c for a <= x <= b,

with F(x) = 0 for x <= a and F(x) = 1 for x >= b.

Field Summary

Fields inherited from class umontreal.iro.lecuyer.probdist.ContinuousDistribution

decPrec

Constructor Summary

PowerDist(double c)
Constructs a PowerDist object with parameters a = 0, b = 1 and c = c.

PowerDist(double b, double c)
Constructs a PowerDist object with parameters a = 0, b = b and c = c.

PowerDist(double a, double b, double c)
Constructs a PowerDist object with parameters a = a, b = b and c = c.

Method Summary

double	barF(double x) Returns the complementary distribution function.
static double	barF(double a, double b, double c, double x) Computes the complementary distribution function.
double	cdf(double x) Returns the distribution function F(x).
static double	cdf(double a, double b, double c, double x) Computes the distribution function.
double	density(double x) Returns f (x), the density evaluated at x.
static double	density(double a, double b, double c, double x) Computes the density function.
double	getA() Returns the parameter a.
double	getB() Returns the parameter b.
double	getC() Returns the parameter c.
static PowerDist	getInstanceFromMLE(double[] x, int n, double a, double b) Creates a new instance of a power distribution with parameters a and b, with c estimated using the maximum likelihood method based on the n observations x[i], i = 0,…, n - 1.
double	getMean() Returns the mean.
static double	getMean(double a, double b, double c) Returns the mean a + (b - a)c/(c + 1) of the power distribution with parameters a, b and c.
static double[]	getMLE(double[] x, int n, double a, double b) Estimates the parameter c of the power distribution from the n observations x[i], i = 0, 1,…, n - 1, using the maximum likelihood method and assuming that a and b are known.
double[]	getParams() Return a table containing the parameters of the current distribution.
double	getStandardDeviation() Returns the standard deviation.
static double	getStandardDeviation(double a, double b, double c) Computes and returns the standard deviation of the power distribution with parameters a, b and c.
double	getVariance() Returns the variance.
static double	getVariance(double a, double b, double c) Computes and returns the variance (b - a)2c/[(c + 1)2(c + 2)] of the power distribution with parameters a, b and c.
double	inverseF(double u) Returns the inverse distribution function x = F-1(u).
static double	inverseF(double a, double b, double c, double u) Computes the inverse of the distribution function.
void	setParams(double a, double b, double c) Sets the parameters a, b and c for this object.
String	toString()

Methods inherited from class umontreal.iro.lecuyer.probdist.ContinuousDistribution

getXinf, getXsup, inverseBisection, inverseBrent, setXinf, setXsup

Methods inherited from class java.lang.Object

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

Constructor Detail

PowerDist

public PowerDist(double a,
                 double b,
                 double c)

Constructs a PowerDist object with parameters a = a, b = b and c = c.

PowerDist

public PowerDist(double b,
                 double c)

Constructs a PowerDist object with parameters a = 0, b = b and c = c.

PowerDist

public PowerDist(double c)

Constructs a PowerDist object with parameters a = 0, b = 1 and c = c.

Method Detail

density

public double density(double x)

Description copied from class: ContinuousDistribution

Returns f (x), the density evaluated at x.

Specified by:

density in class ContinuousDistribution

Parameters:

x - value at which the density is evaluated

Returns:

density function evaluated at x

cdf

public double cdf(double x)

Description copied from interface: Distribution

Returns the distribution function F(x).

Parameters:

x - value at which the distribution function is evaluated

Returns:

distribution function evaluated at x

barF

public double barF(double x)

Description copied from class: ContinuousDistribution

Returns the complementary distribution function. The default implementation computes bar(F)(x) = 1 - F(x).

Specified by:

barF in interface Distribution

Overrides:

barF in class ContinuousDistribution

Parameters:

x - value at which the complementary distribution function is evaluated

Returns:

complementary distribution function evaluated at x

inverseF

public double inverseF(double u)

Description copied from class: ContinuousDistribution

Returns the inverse distribution function x = F^-1(u). Restrictions: u∈[0, 1].

Specified by:

inverseF in interface Distribution

Overrides:

inverseF in class ContinuousDistribution

Parameters:

u - value at which the inverse distribution function is evaluated

Returns:

the inverse distribution function evaluated at u

getMean

public double getMean()

Description copied from class: ContinuousDistribution

Returns the mean.

Specified by:

getMean in interface Distribution

Overrides:

getMean in class ContinuousDistribution

Returns:

the mean

getVariance

public double getVariance()

Description copied from class: ContinuousDistribution

Returns the variance.

Specified by:

getVariance in interface Distribution

Overrides:

getVariance in class ContinuousDistribution

Returns:

the variance

getStandardDeviation

public double getStandardDeviation()

Description copied from class: ContinuousDistribution

Returns the standard deviation.

Specified by:

getStandardDeviation in interface Distribution

Overrides:

getStandardDeviation in class ContinuousDistribution

Returns:

the standard deviation

density

public static double density(double a,
                             double b,
                             double c,
                             double x)

Computes the density function.

Parameters:

a - left limit of interval

b - right limit of interval

c - shape parameter

x - the value at which the density is evaluated

Returns:

returns the density function

cdf

public static double cdf(double a,
                         double b,
                         double c,
                         double x)

Computes the distribution function.

Parameters:

a - left limit of interval

b - right limit of interval

c - shape parameter

x - the value at which the distribution is evaluated

Returns:

returns the distribution function

barF

public static double barF(double a,
                          double b,
                          double c,
                          double x)

Computes the complementary distribution function.

Parameters:

a - left limit of interval

b - right limit of interval

c - shape parameter

x - the value at which the complementary distribution is evaluated

Returns:

returns the complementary distribution function

inverseF

public static double inverseF(double a,
                              double b,
                              double c,
                              double u)

Computes the inverse of the distribution function.

Parameters:

a - left limit of interval

b - right limit of interval

c - shape parameter

u - the value at which the inverse distribution is evaluated

Returns:

returns the inverse of the distribution function

getMLE

public static double[] getMLE(double[] x,
                              int n,
                              double a,
                              double b)

Estimates the parameter c of the power distribution from the n observations x[i], i = 0, 1,…, n - 1, using the maximum likelihood method and assuming that a and b are known. The estimate is returned in a one-element array: [c].

Parameters:

x - the list of observations to use to evaluate parameters

n - the number of observations to use to evaluate parameters

a - left limit of interval

b - right limit of interval

Returns:

returns the shape parameter [hat(c)]

getInstanceFromMLE

public static PowerDist getInstanceFromMLE(double[] x,
                                           int n,
                                           double a,
                                           double b)

Creates a new instance of a power distribution with parameters a and b, with c estimated using the maximum likelihood method based on the n observations x[i], i = 0,…, n - 1.

Parameters:

x - the list of observations to use to evaluate parameters

n - the number of observations to use to evaluate parameters

a - left limit of interval

b - right limit of interval

getMean

public static double getMean(double a,
                             double b,
                             double c)

Returns the mean a + (b - a)c/(c + 1) of the power distribution with parameters a, b and c.

Parameters:

a - left limit of interval

b - right limit of interval

c - shape parameter

Returns:

returns the mean

getVariance

public static double getVariance(double a,
                                 double b,
                                 double c)

Computes and returns the variance (b - a)²c/[(c + 1)²(c + 2)] of the power distribution with parameters a, b and c.

Parameters:

a - left limit of interval

b - right limit of interval

c - shape parameter

Returns:

returns the variance

getStandardDeviation

public static double getStandardDeviation(double a,
                                          double b,
                                          double c)

Computes and returns the standard deviation of the power distribution with parameters a, b and c.

Returns:

the standard deviation of the power distribution

getA

public double getA()

Returns the parameter a.

Returns:

the left limit of interval a

getB

public double getB()

Returns the parameter b.

Returns:

the right limit of interval b

getC

public double getC()

Returns the parameter c.

Returns:

the shape parameter c

setParams

public void setParams(double a,
                      double b,
                      double c)

Sets the parameters a, b and c for this object.

Parameters:

a - left limit of interval

b - right limit of interval

c - shape parameter

getParams

public double[] getParams()

Return a table containing the parameters of the current distribution. This table is put in regular order: [a, b, c].

Returns:

[a, b,c]

toString

public String toString()

Overrides:

toString in class Object