SSJ
V. 2.6.

umontreal.iro.lecuyer.probdist
Class WeibullDist

java.lang.Object
  extended by umontreal.iro.lecuyer.probdist.ContinuousDistribution
      extended by umontreal.iro.lecuyer.probdist.WeibullDist
All Implemented Interfaces:
Distribution

public class WeibullDist
extends ContinuousDistribution

This class extends the class ContinuousDistribution for the Weibull distribution with shape parameter α > 0, location parameter δ, and scale parameter λ > 0. The density function is

f (x) = αλα(x - δ)α-1e-(λ(x-δ))α        for x > δ.

the distribution function is

F(x) = 1 - e-(λ(x-δ))α        for x > δ,

and the inverse distribution function is

F-1(u) = (- ln(1 - u))1/α/λ + δ        for 0 <= u < 1.


Field Summary
 
Fields inherited from class umontreal.iro.lecuyer.probdist.ContinuousDistribution
decPrec
 
Constructor Summary
WeibullDist(double alpha)
          Constructs a WeibullDist object with parameters α = alpha, λ = 1, and δ = 0.
WeibullDist(double alpha, double lambda, double delta)
          Constructs a WeibullDist object with parameters α = alpha, λ = lambda, and δ = delta.
 
Method Summary
 double barF(double x)
          Returns the complementary distribution function.
static double barF(double alpha, double x)
          Same as barF (alpha, 1, 0, x).
static double barF(double alpha, double lambda, double delta, double x)
          Computes the complementary distribution function.
 double cdf(double x)
          Returns the distribution function F(x).
static double cdf(double alpha, double x)
          Same as cdf (alpha, 1, 0, x).
static double cdf(double alpha, double lambda, double delta, double x)
          Computes the distribution function.
 double density(double x)
          Returns f (x), the density evaluated at x.
static double density(double alpha, double x)
          Same as density (alpha, 1, 0, x).
static double density(double alpha, double lambda, double delta, double x)
          Computes the density function.
 double getAlpha()
          Returns the parameter α.
 double getDelta()
          Returns the parameter δ.
static WeibullDist getInstanceFromMLE(double[] x, int n)
          Creates a new instance of a Weibull distribution with parameters α, λ and δ = 0 estimated using the maximum likelihood method based on the n observations x[i], i = 0, 1,…, n - 1.
 double getLambda()
          Returns the parameter λ.
 double getMean()
          Returns the mean.
static double getMean(double alpha, double lambda, double delta)
          Computes and returns the mean of the Weibull distribution with parameters α, λ and δ.
static double[] getMLE(double[] x, int n)
          Estimates the parameters (α, λ) of the Weibull distribution, assuming that δ = 0, 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, double delta)
          Computes and returns the standard deviation of the Weibull distribution with parameters α, λ and δ.
 double getVariance()
          Returns the variance.
static double getVariance(double alpha, double lambda, double delta)
          Computes and returns the variance of the Weibull distribution with parameters α, λ and δ.
 double inverseF(double u)
          Returns the inverse distribution function x = F-1(u).
static double inverseF(double alpha, double x)
          Same as inverseF (alpha, 1, 0, x).
static double inverseF(double alpha, double lambda, double delta, double u)
          Computes the inverse of the distribution function.
 void setParams(double alpha, double lambda, double delta)
          Sets the parameters α, λ and δ 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

WeibullDist

public WeibullDist(double alpha)
Constructs a WeibullDist object with parameters α = alpha, λ = 1, and δ = 0.


WeibullDist

public WeibullDist(double alpha,
                   double lambda,
                   double delta)
Constructs a WeibullDist object with parameters α = alpha, λ = lambda, and δ = delta.

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 delta,
                             double x)
Computes the density function.


density

public static double density(double alpha,
                             double x)
Same as density (alpha, 1, 0, x).


cdf

public static double cdf(double alpha,
                         double lambda,
                         double delta,
                         double x)
Computes the distribution function.


cdf

public static double cdf(double alpha,
                         double x)
Same as cdf (alpha, 1, 0, x).


barF

public static double barF(double alpha,
                          double lambda,
                          double delta,
                          double x)
Computes the complementary distribution function.


barF

public static double barF(double alpha,
                          double x)
Same as barF (alpha, 1, 0, x).


inverseF

public static double inverseF(double alpha,
                              double lambda,
                              double delta,
                              double u)
Computes the inverse of the distribution function.


inverseF

public static double inverseF(double alpha,
                              double x)
Same as inverseF (alpha, 1, 0, x).


getMLE

public static double[] getMLE(double[] x,
                              int n)
Estimates the parameters (α, λ) of the Weibull distribution, assuming that δ = 0, 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 parameter [ hat(α), hat(λ), hat(δ) = 0]

getInstanceFromMLE

public static WeibullDist getInstanceFromMLE(double[] x,
                                             int n)
Creates a new instance of a Weibull distribution with parameters α, λ and δ = 0 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,
                             double delta)
Computes and returns the mean of the Weibull distribution with parameters α, λ and δ.

Returns:
the mean of the Weibull distribution E[X] = δ + Γ(1 + 1/α)/λ

getVariance

public static double getVariance(double alpha,
                                 double lambda,
                                 double delta)
Computes and returns the variance of the Weibull distribution with parameters α, λ and δ.

Returns:
the variance of the Weibull distribution Var[X] = 1/λ2| Γ(2/α +1) - Γ2(1/α + 1)|

getStandardDeviation

public static double getStandardDeviation(double alpha,
                                          double lambda,
                                          double delta)
Computes and returns the standard deviation of the Weibull distribution with parameters α, λ and δ.

Returns:
the standard deviation of the Weibull distribution

getAlpha

public double getAlpha()
Returns the parameter α.


getLambda

public double getLambda()
Returns the parameter λ.


getDelta

public double getDelta()
Returns the parameter δ.


setParams

public void setParams(double alpha,
                      double lambda,
                      double delta)
Sets the parameters α, λ and δ for this object.


getParams

public double[] getParams()
Return a table containing the parameters of the current distribution. This table is put in regular order: [α, λ, δ].


toString

public String toString()
Overrides:
toString in class Object

SSJ
V. 2.6.

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