SSJ
V. 2.6.

## umontreal.iro.lecuyer.probdist Class WeibullDist

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

`public class WeibullDistextends 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.