GammaDist (Java Libraries for Stochastic Simulation)

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

Methods
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.
`static double[]`	`getMaximumLikelihoodEstimate(double[] x, int n)` Deprecated.
`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 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)`
`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 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).
  - 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)]
  - getMaximumLikelihoodEstimate
```
@Deprecated
public static double[] getMaximumLikelihoodEstimate(double[] x,
                                               int n)
```
    Deprecated.
    
    Same as getMLE.
  - 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

Class GammaDist

Field Summary

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

Constructor Summary

Method Summary

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

Methods inherited from class java.lang.Object

Constructor Detail

GammaDist

GammaDist

GammaDist

Method Detail

density

cdf

barF

inverseF

getMean

getVariance

getStandardDeviation

density

cdf

cdf

barF

barF

inverseF

inverseF

getMLE

getMaximumLikelihoodEstimate

getInstanceFromMLE

getMean

getVariance

getStandardDeviation

getAlpha

getLambda

setParams

getParams

toString