This class extends the class ContinuousDistribution for the Weibull distribution [95] (page 628) with shape parameter \(\alpha> 0\), location parameter \(\delta\), and scale parameter \(\lambda> 0\). More...

Inheritance diagram for umontreal.ssj.probdist.WeibullDist:

Public Member Functions
	WeibullDist (double alpha)
	Constructs a WeibullDist object with parameters \(\alpha\) = alpha, \(\lambda\) = 1, and \(\delta\) = 0.
	WeibullDist (double alpha, double lambda, double delta)
	Constructs a WeibullDist object with parameters \(\alpha=\) alpha, \(\lambda\) = lambda, and \(\delta\) = delta.
double	density (double x)
	Returns \(f(x)\), the density evaluated at \(x\).
double	cdf (double x)
	Returns the distribution function \(F(x)\).
double	barF (double x)
	Returns the complementary distribution function.
double	inverseF (double u)
	Returns the inverse distribution function \(x = F^{-1}(u)\).
double	getMean ()
	Returns the mean.
double	getVariance ()
	Returns the variance.
double	getStandardDeviation ()
	Returns the standard deviation.
double	getAlpha ()
	Returns the parameter \(\alpha\).
double	getLambda ()
	Returns the parameter \(\lambda\).
double	getDelta ()
	Returns the parameter \(\delta\).
void	setParams (double alpha, double lambda, double delta)
	Sets the parameters \(\alpha\), \(\lambda\) and \(\delta\) for this object.
double[]	getParams ()
	Return a table containing the parameters of the current distribution.
String	toString ()
	Returns a String containing information about the current distribution.
Public Member Functions inherited from umontreal.ssj.probdist.ContinuousDistribution
double	inverseBrent (double a, double b, double u, double tol)
	Computes the inverse distribution function \(x = F^{-1}(u)\), using the Brent-Dekker method.
double	inverseBisection (double u)
	Computes and returns the inverse distribution function \(x = F^{-1}(u)\), using bisection.
double	getXinf ()
	Returns \(x_a\) such that the probability density is 0 everywhere outside the interval \([x_a, x_b]\).
double	getXsup ()
	Returns \(x_b\) such that the probability density is 0 everywhere outside the interval \([x_a, x_b]\).
void	setXinf (double xa)
	Sets the value \(x_a=\) xa, such that the probability density is 0 everywhere outside the interval \([x_a, x_b]\).
void	setXsup (double xb)
	Sets the value \(x_b=\) xb, such that the probability density is 0 everywhere outside the interval \([x_a, x_b]\).

Static Public Member Functions
static double	density (double alpha, double lambda, double delta, double x)
	Computes the density function.
static double	density (double alpha, double x)
	Same as density (alpha, 1, 0, x).
static double	cdf (double alpha, double lambda, double delta, double x)
	Computes the distribution function.
static double	cdf (double alpha, double x)
	Same as cdf (alpha, 1, 0, x).
static double	barF (double alpha, double lambda, double delta, double x)
	Computes the complementary distribution function.
static double	barF (double alpha, double x)
	Same as barF (alpha, 1, 0, x).
static double	inverseF (double alpha, double lambda, double delta, double u)
	Computes the inverse of the distribution function.
static double	inverseF (double alpha, double x)
	Same as inverseF (alpha, 1, 0, x).
static double[]	getMLE (double[] x, int n)
	Estimates the parameters \((\alpha, \lambda)\) of the Weibull distribution, assuming that \(\delta= 0\), using the maximum likelihood method, from the \(n\) observations \(x[i]\), \(i = 0, 1, …, n-1\).
static WeibullDist	getInstanceFromMLE (double[] x, int n)
	Creates a new instance of a Weibull distribution with parameters.
static double	getMean (double alpha, double lambda, double delta)
	Computes and returns the mean \(E[X] = \delta+ \Gamma(1 + 1/\alpha)/\lambda\) of the Weibull distribution with parameters \(\alpha\),.
static double	getVariance (double alpha, double lambda, double delta)
	Computes and returns the variance.
static double	getStandardDeviation (double alpha, double lambda, double delta)
	Computes and returns the standard deviation of the Weibull distribution with parameters \(\alpha\), \(\lambda\) and.

Detailed Description

This class extends the class ContinuousDistribution for the Weibull distribution [95] (page 628) with shape parameter \(\alpha> 0\), location parameter \(\delta\), and scale parameter \(\lambda> 0\).

The density function is

\[ f(x) = \alpha\lambda^{\alpha}(x-\delta)^{\alpha-1} e^{-(\lambda(x-\delta))^{\alpha}} \qquad\mbox{for }x>\delta, \tag{fweibull} \]

the distribution function is

\[ F(x) = 1 - e^{-(\lambda(x - \delta))^{\alpha}} \qquad\mbox{for }x>\delta, \tag{Fweibull} \]

and the inverse distribution function is

\[ F^{-1}(u) = (-\ln(1-u))^{1/\alpha}/\lambda+ \delta\qquad\mbox{for } 0 \le u < 1. \]

Definition at line 49 of file WeibullDist.java.

Constructor & Destructor Documentation

◆ WeibullDist() [1/2]

umontreal.ssj.probdist.WeibullDist.WeibullDist ( double alpha )

Constructs a WeibullDist object with parameters \(\alpha\) = alpha, \(\lambda\) = 1, and \(\delta\) = 0.

Definition at line 97 of file WeibullDist.java.

◆ WeibullDist() [2/2]

umontreal.ssj.probdist.WeibullDist.WeibullDist	(	double	alpha,
		double	lambda,
		double	delta )

Constructs a WeibullDist object with parameters \(\alpha=\) alpha, \(\lambda\) = lambda, and \(\delta\) = delta.

Definition at line 105 of file WeibullDist.java.

Member Function Documentation

◆ barF() [1/3]

double umontreal.ssj.probdist.WeibullDist.barF	(	double	alpha,
		double	lambda,
		double	delta,
		double	x )

static

Computes the complementary distribution function.

Definition at line 191 of file WeibullDist.java.

◆ barF() [2/3]

double umontreal.ssj.probdist.WeibullDist.barF	(	double	alpha,
		double	x )

static

Same as barF (alpha, 1, 0, x).

Definition at line 211 of file WeibullDist.java.

◆ barF() [3/3]

double umontreal.ssj.probdist.WeibullDist.barF ( double x )

Returns the complementary distribution function.

The default implementation computes \(\bar{F}(x) = 1 - F(x)\).

Parameters

x	value at which the complementary distribution function is evaluated

Returns: complementary distribution function evaluated at x

Reimplemented from umontreal.ssj.probdist.ContinuousDistribution.

Definition at line 117 of file WeibullDist.java.

◆ cdf() [1/3]

double umontreal.ssj.probdist.WeibullDist.cdf	(	double	alpha,
		double	lambda,
		double	delta,
		double	x )

static

Computes the distribution function.

Definition at line 165 of file WeibullDist.java.

◆ cdf() [2/3]

double umontreal.ssj.probdist.WeibullDist.cdf	(	double	alpha,
		double	x )

static

Same as cdf (alpha, 1, 0, x).

Definition at line 184 of file WeibullDist.java.

◆ cdf() [3/3]

double umontreal.ssj.probdist.WeibullDist.cdf ( double x )

Returns the distribution function \(F(x)\).

Parameters

x	value at which the distribution function is evaluated

Returns: distribution function evaluated at x

Implements umontreal.ssj.probdist.Distribution.

Definition at line 113 of file WeibullDist.java.

◆ density() [1/3]

double umontreal.ssj.probdist.WeibullDist.density	(	double	alpha,
		double	lambda,
		double	delta,
		double	x )

static

Computes the density function.

Definition at line 140 of file WeibullDist.java.

◆ density() [2/3]

double umontreal.ssj.probdist.WeibullDist.density	(	double	alpha,
		double	x )

static

Same as density (alpha, 1, 0, x).

Definition at line 158 of file WeibullDist.java.

◆ density() [3/3]

double umontreal.ssj.probdist.WeibullDist.density ( double x )

Returns \(f(x)\), the density evaluated at \(x\).

Parameters

x	value at which the density is evaluated

Returns: density function evaluated at x

Reimplemented from umontreal.ssj.probdist.ContinuousDistribution.

Definition at line 109 of file WeibullDist.java.

◆ getAlpha()

double umontreal.ssj.probdist.WeibullDist.getAlpha ( )

Returns the parameter \(\alpha\).

Definition at line 375 of file WeibullDist.java.

◆ getDelta()

double umontreal.ssj.probdist.WeibullDist.getDelta ( )

Returns the parameter \(\delta\).

Definition at line 389 of file WeibullDist.java.

◆ getInstanceFromMLE()

WeibullDist umontreal.ssj.probdist.WeibullDist.getInstanceFromMLE	(	double[]	x,
		int	n )

static

Creates a new instance of a Weibull distribution with parameters.

\(\alpha\), \(\lambda\) and \(\delta= 0\) 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

Definition at line 316 of file WeibullDist.java.

◆ getLambda()

double umontreal.ssj.probdist.WeibullDist.getLambda ( )

Returns the parameter \(\lambda\).

Definition at line 382 of file WeibullDist.java.

◆ getMean() [1/2]

double umontreal.ssj.probdist.WeibullDist.getMean ( )

Returns the mean.

Returns: the mean

Reimplemented from umontreal.ssj.probdist.ContinuousDistribution.

Definition at line 125 of file WeibullDist.java.

◆ getMean() [2/2]

double umontreal.ssj.probdist.WeibullDist.getMean	(	double	alpha,
		double	lambda,
		double	delta )

static

Computes and returns the mean \(E[X] = \delta+ \Gamma(1 + 1/\alpha)/\lambda\) of the Weibull distribution with parameters \(\alpha\),.

\(\lambda\) and \(\delta\).

Returns: the mean of the Weibull distribution \(E[X] = \delta+ \Gamma(1 + 1/\alpha) / \lambda\)

Definition at line 330 of file WeibullDist.java.

◆ getMLE()

double[] umontreal.ssj.probdist.WeibullDist.getMLE	(	double[]	x,
		int	n )

static

Estimates the parameters \((\alpha, \lambda)\) of the Weibull distribution, assuming that \(\delta= 0\), 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: [ \(\alpha\), \(\lambda\)]. The maximum likelihood estimators are the values \((\hat{\alpha}\), \(\hat{\lambda})\) that satisfy the equations

\begin{align*} \frac{\sum_{i=1}^n x_i^{\hat{\alpha}} \ln(x_i)}{\sum_{i=1}^n x_i^{\hat{\alpha}}} - \frac{1}{\hat{\alpha}} & = \frac{\sum_{i=1}^n \ln(x_i)}{n} \\ \hat{\lambda} & = \left( \frac{n}{\sum_{i=1}^n x_i^{\hat{\alpha}}} \right)^{1/\hat{\alpha}} \end{align*}

See [114] (page 303).

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 [ \(\hat{\alpha}\), \(\hat{\lambda}\), \(\hat{\delta}\) = 0]

Definition at line 303 of file WeibullDist.java.

◆ getParams()

double[] umontreal.ssj.probdist.WeibullDist.getParams ( )

Return a table containing the parameters of the current distribution.

This table is put in regular order: [ \(\alpha\), \(\lambda\), \(\delta\)].

Implements umontreal.ssj.probdist.Distribution.

Definition at line 413 of file WeibullDist.java.

◆ getStandardDeviation() [1/2]

double umontreal.ssj.probdist.WeibullDist.getStandardDeviation ( )

Returns the standard deviation.

Returns: the standard deviation

Reimplemented from umontreal.ssj.probdist.ContinuousDistribution.

Definition at line 133 of file WeibullDist.java.

◆ getStandardDeviation() [2/2]

double umontreal.ssj.probdist.WeibullDist.getStandardDeviation	(	double	alpha,
		double	lambda,
		double	delta )

static

Computes and returns the standard deviation of the Weibull distribution with parameters \(\alpha\), \(\lambda\) and.

\(\delta\).

Returns: the standard deviation of the Weibull distribution

Definition at line 368 of file WeibullDist.java.

◆ getVariance() [1/2]

double umontreal.ssj.probdist.WeibullDist.getVariance ( )

Returns the variance.

Returns: the variance

Reimplemented from umontreal.ssj.probdist.ContinuousDistribution.

Definition at line 129 of file WeibullDist.java.

◆ getVariance() [2/2]

double umontreal.ssj.probdist.WeibullDist.getVariance	(	double	alpha,
		double	lambda,
		double	delta )

static

Computes and returns the variance.

\(\mbox{Var}[X] = | \Gamma(2/\alpha+ 1) - \Gamma^2(1/\alpha+ 1) | /\lambda^2\) of the Weibull distribution with parameters \(\alpha\), \(\lambda\) and \(\delta\).

Returns: the variance of the Weibull distribution \(\mbox{Var}[X] = 1 / \lambda^2 | \Gamma(2/\alpha+ 1) - \Gamma^2(1/\alpha+ 1) |\)

Definition at line 349 of file WeibullDist.java.

◆ inverseF() [1/3]

double umontreal.ssj.probdist.WeibullDist.inverseF	(	double	alpha,
		double	lambda,
		double	delta,
		double	u )

static

Computes the inverse of the distribution function.

Definition at line 218 of file WeibullDist.java.

◆ inverseF() [2/3]

double umontreal.ssj.probdist.WeibullDist.inverseF	(	double	alpha,
		double	x )

static

Same as inverseF (alpha, 1, 0, x).

Definition at line 242 of file WeibullDist.java.

◆ inverseF() [3/3]

double umontreal.ssj.probdist.WeibullDist.inverseF ( double u )

Returns the inverse distribution function \(x = F^{-1}(u)\).

Restrictions: \(u \in[0,1]\).

Parameters

u	value at which the inverse distribution function is evaluated

Returns: the inverse distribution function evaluated at u

Exceptions

IllegalArgumentException if \(u\) is not in the interval \([0,1]\)

Reimplemented from umontreal.ssj.probdist.ContinuousDistribution.

Definition at line 121 of file WeibullDist.java.

◆ setParams()

void umontreal.ssj.probdist.WeibullDist.setParams	(	double	alpha,
		double	lambda,
		double	delta )

Sets the parameters \(\alpha\), \(\lambda\) and \(\delta\) for this object.

Definition at line 397 of file WeibullDist.java.

◆ toString()

String umontreal.ssj.probdist.WeibullDist.toString ( )

Returns a String containing information about the current distribution.

Definition at line 421 of file WeibullDist.java.

The documentation for this class was generated from the following file:

src/main/java/umontreal/ssj/probdist/WeibullDist.java

SSJ API Documentation Stochastic Simulation in Java

Public Member Functions

Static Public Member Functions

Detailed Description

Constructor & Destructor Documentation

◆ WeibullDist() [1/2]

◆ WeibullDist() [2/2]

Member Function Documentation

◆ barF() [1/3]

◆ barF() [2/3]

◆ barF() [3/3]

◆ cdf() [1/3]

◆ cdf() [2/3]

◆ cdf() [3/3]

◆ density() [1/3]

◆ density() [2/3]

◆ density() [3/3]

◆ getAlpha()

◆ getDelta()

◆ getInstanceFromMLE()

◆ getLambda()

◆ getMean() [1/2]

◆ getMean() [2/2]

◆ getMLE()

◆ getParams()

◆ getStandardDeviation() [1/2]

◆ getStandardDeviation() [2/2]

◆ getVariance() [1/2]

◆ getVariance() [2/2]

◆ inverseF() [1/3]

◆ inverseF() [2/3]

◆ inverseF() [3/3]

◆ setParams()

◆ toString()