umontreal.iro.lecuyer.probdist
Class GammaDist

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

All Implemented Interfaces:

Distribution

public class GammaDist
extends ContinuousDistribution

Extends the class ContinuousDistribution for the gamma distribution with shape parameter α > 0 and scale parameter λ > 0. The density is

f (x) = λ^αx^α-1e^-λx/Γ(α), for x > 0,

where Γ is the gamma function, defined by

Γ(α) = ∫₀^∞x^α-1e^-xdx.

In particular, Γ(n) = (n - 1)! when n is a positive integer.

Field Summary

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

decPrec

Constructor Summary

GammaDist(double alpha)
Constructs a GammaDist object with parameters α = alpha and λ = 1.

GammaDist(double alpha, double lambda)
Constructs a GammaDist object with parameters α = alpha and λ = lambda.

GammaDist(double alpha, double lambda, int d)
Constructs a GammaDist object with parameters α = alpha and λ = lambda, and approximations of roughly d decimal digits of precision when computing functions.

Method Summary

static double	barF(double alpha, double lambda, int d, double x) Computes the complementary distribution function.
static double	barF(double alpha, int d, double x) Same as barF (alpha, 1.0, d, x).
double	cdf(double x) Computes and returns the distribution function F(x).
static double	cdf(double alpha, double lambda, int d, double x) Returns an approximation of the gamma distribution function with parameters α = alpha and λ = lambda.
static double	cdf(double alpha, int d, double x) Equivalent to cdf (alpha, 1.0, d, x).
double	density(double x) Returns f (x), the density of X evaluated at x.
static double	density(double alpha, double lambda, double x) Computes the density function.
double	getAlpha() Return the parameter α for this object.
double	getLambda() Return the parameter λ for this object.
double	inverseF(double u) Computes and returns the inverse distribution function F-1(u), defined in.
static double	inverseF(double alpha, double lambda, int d, double u) Computes the inverse distribution function using the algorithm implemented in the Cephes Math Library.
static double	inverseF(double alpha, int d, double u) Same as inverseF (alpha, 1, d, u).
void	setParams(double alpha, double lambda, int d)

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

barF, inverseBisection, inverseBrent

Methods inherited from class java.lang.Object

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

Constructor Detail

GammaDist

public GammaDist(double alpha)

Constructs a GammaDist object with parameters α = alpha and λ = 1.

GammaDist

public GammaDist(double alpha,
                 double lambda)

Constructs a GammaDist object with parameters α = alpha and λ = lambda.

GammaDist

public GammaDist(double alpha,
                 double lambda,
                 int d)

Constructs a GammaDist object with parameters α = alpha and λ = lambda, and approximations of roughly d decimal digits of precision when computing functions.

Method Detail

density

public double density(double x)

Description copied from class: ContinuousDistribution

Returns f (x), the density of X 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

Computes and returns the distribution function F(x).

Parameters:

x - value at which the distribution function is evaluated

Returns:

distribution function evaluated at x

inverseF

public double inverseF(double u)

Description copied from interface: Distribution

Computes and returns the inverse distribution function F^-1(u), defined in.

Specified by:

inverseF in interface Distribution

Overrides:

inverseF in class ContinuousDistribution

Parameters:

u - value in the interval (0, 1) for which the inverse distribution function is evaluated

Returns:

the inverse distribution function evaluated at u

density

public static double density(double alpha,
                             double lambda,
                             double x)

Computes the density function.

cdf

public static double cdf(double alpha,
                         double lambda,
                         int d,
                         double x)

Returns an approximation of the gamma distribution function with parameters α = alpha and λ = lambda. The function tries to return d decimals digits of precision. For α not too large (e.g., α <= 1000), d gives a good idea of the precision attained.

cdf

public static double cdf(double alpha,
                         int d,
                         double x)

Equivalent to cdf (alpha, 1.0, d, x).

barF

public static double barF(double alpha,
                          double lambda,
                          int d,
                          double x)

Computes the complementary distribution function.

barF

public static double barF(double alpha,
                          int d,
                          double x)

Same as barF (alpha, 1.0, d, x).

inverseF

public static double inverseF(double alpha,
                              double lambda,
                              int d,
                              double u)

Computes the inverse distribution function using the algorithm implemented in the Cephes Math Library. Starting with the approximation x = αt³, where t = 1 - z - Φ^-1(u)(z)^1/2, z = α/9, and Φ^-1 is the inverse of the standard normal distribution, the method uses Newton iterations to estimate the inverse. The precision of the algorithm depends on the accuracy of the barF function. The argument d gives a good idea of the precision attained.

inverseF

public static double inverseF(double alpha,
                              int d,
                              double u)

Same as inverseF (alpha, 1, d, u).

getAlpha

public double getAlpha()

Return the parameter α for this object.

getLambda

public double getLambda()

Return the parameter λ for this object.

setParams

public void setParams(double alpha,
                      double lambda,
                      int d)