umontreal.iro.lecuyer.probdist

Class GammaDist

java.lang.Object

umontreal.iro.lecuyer.probdist.ContinuousDistribution

umontreal.iro.lecuyer.probdist.GammaDist

All Implemented Interfaces:

Distribution

Direct Known Subclasses:

ErlangDist, GammaDistFromMoments

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

Constructors

Constructor and Description
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

Modifier and Type	Method and Description
double	barF(double x) Returns the complementary distribution function.
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) 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 evaluated at x.
static double	density(double alpha, double lambda, double x) Computes the density function at x.
double	getAlpha() Return the parameter α for this object.
static GammaDist	getInstanceFromMLE(double[] x, int n) Creates a new instance of a gamma distribution with parameters α and λ estimated using the maximum likelihood method based on the n observations x[i], i = 0, 1,…, n - 1.
double	getLambda() Return the parameter λ for this object.
double	getMean() Returns the mean.
static double	getMean(double alpha, double lambda) Computes and returns the mean E[X] = α/λ of the gamma distribution with parameters α and λ.
static double[]	getMLE(double[] x, int n) Estimates the parameters (α, λ) of the gamma distribution using the maximum likelihood method, from the n observations x[i], i = 0, 1,…, n - 1.
double[]	getParams() Return a table containing the parameters of the current distribution.
double	getStandardDeviation() Returns the standard deviation.
static double	getStandardDeviation(double alpha, double lambda) Computes and returns the standard deviation of the gamma distribution with parameters α and λ.
double	getVariance() Returns the variance.
static double	getVariance(double alpha, double lambda) Computes and returns the variance Var[X] = α/λ2 of the gamma distribution with parameters α and λ.
double	inverseF(double u) Returns the inverse distribution function x = F-1(u).
static double	inverseF(double alpha, double lambda, int d, double u) Computes the inverse distribution function.
static double	inverseF(double alpha, int d, double u) Same as inverseF (alpha, 1, d, u).
void	setParams(double alpha, double lambda, int d)
java.lang.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

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 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 alpha,
                             double lambda,
                             double x)

Computes the density function at x.

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.

inverseF

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

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

getMLE

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

Estimates the parameters (α, λ) of the gamma distribution using the maximum likelihood method, from the n observations x[i], i = 0, 1,…, n - 1. The estimates are returned in a two-element array, in regular order: [α, λ].

Parameters:

x - the list of observations to use to evaluate parameters

n - the number of observations to use to evaluate parameters

Returns:

returns the parameters [ hat(α), hat(λ)]

getInstanceFromMLE

public static GammaDist getInstanceFromMLE(double[] x,
                                           int n)

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

Parameters:

x - the list of observations to use to evaluate parameters

n - the number of observations to use to evaluate parameters

getMean

public static double getMean(double alpha,
                             double lambda)

Computes and returns the mean E[X] = α/λ of the gamma distribution with parameters α and λ.

Returns:

the mean of the gamma distribution E[X] = α/λ

getVariance

public static double getVariance(double alpha,
                                 double lambda)

Computes and returns the variance Var[X] = α/λ² of the gamma distribution with parameters α and λ.

Returns:

the variance of the gamma distribution Var[X] = α/λ²

getStandardDeviation

public static double getStandardDeviation(double alpha,
                                          double lambda)

Computes and returns the standard deviation of the gamma distribution with parameters α and λ.

Returns:

the standard deviation of the gamma distribution

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)

getParams

public double[] getParams()

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

toString

public java.lang.String toString()

Overrides:

toString in class java.lang.Object