SSJ
V. 1.2.5.

umontreal.iro.lecuyer.probdist
Class GammaDist

java.lang.Object
  extended by umontreal.iro.lecuyer.probdist.ContinuousDistribution
      extended by umontreal.iro.lecuyer.probdist.GammaDist
All Implemented Interfaces:
Distribution
Direct Known Subclasses:
ErlangDist

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) = λαxα-1e-λx/Γ(α),        for x > 0,

where Γ is the gamma function, defined by

Γ(α) = ∫0xα-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.
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 in table x[i], i = 0, 1,…, n - 1.
 double getLambda()
          Return the parameter λ for this object.
static double[] getMaximumLikelihoodEstimate(double[] x, int n)
          Estimates and returns the parameters [ hat(α), hat(λ)] of the gamma distribution using the maximum likelihood method based on the n observations in table x[i], i = 0, 1,…, n - 1.
 double getMean()
          Returns the mean of the distribution function.
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.
 double getStandardDeviation()
          Returns the standard deviation of the distribution function.
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 of the distribution function.
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)
          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

getMean

public double getMean()
Description copied from interface: Distribution
Returns the mean of the distribution function.


getVariance

public double getVariance()
Description copied from interface: Distribution
Returns the variance of the distribution function.


getStandardDeviation

public double getStandardDeviation()
Description copied from interface: Distribution
Returns the standard deviation of the distribution function.


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).


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 in table 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

getMaximumLikelihoodEstimate

public static double[] getMaximumLikelihoodEstimate(double[] x,
                                                    int n)
Estimates and returns the parameters [ hat(α), hat(λ)] of the gamma distribution using the maximum likelihood method based on the n observations in table 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
Returns:
returns the parameters [ hat(α), hat(λ)]

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)

SSJ
V. 1.2.5.

To submit a bug or ask questions, send an e-mail to Pierre L'Ecuyer.