SSJ
V. 2.6.

## umontreal.iro.lecuyer.probdist Class ParetoDist

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

`public class ParetoDistextends ContinuousDistribution`

Extends the class `ContinuousDistribution` for a distribution from the Pareto family, with shape parameter α > 0 and location parameter β > 0. The density for this type of Pareto distribution is

f (x) = αβα/xα+1        for x >= β,

and 0 otherwise. The distribution function is

F(x) = 1 - (β/x)α        for x >= β,

and the inverse distribution function is

F-1(u) = β(1 - u)-1/α        for 0 <= u < 1.

Field Summary

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

Constructor Summary
`ParetoDist(double alpha)`
Constructs a ParetoDist object with parameters α = alpha and β = 1.
```ParetoDist(double alpha, double beta)```
Constructs a ParetoDist object with parameters α = alpha and β = beta.

Method Summary
` double` `barF(double x)`
Returns the complementary distribution function.
`static double` ```barF(double alpha, double beta, double x)```
Computes the complementary distribution function.
` double` `cdf(double x)`
Returns the distribution function F(x).
`static double` ```cdf(double alpha, double beta, double x)```
Computes the distribution function.
` double` `density(double x)`
Returns f (x), the density evaluated at x.
`static double` ```density(double alpha, double beta, double x)```
Computes the density function.
` double` `getAlpha()`
Returns the parameter α.
` double` `getBeta()`
Returns the parameter β.
`static ParetoDist` ```getInstanceFromMLE(double[] x, int n)```
Creates a new instance of a Pareto distribution with parameters α and β estimated using the maximum likelihood method based on the n observations x[i], i = 0, 1,…, n - 1.
` double` `getMean()`
Returns the mean.
`static double` ```getMean(double alpha, double beta)```
Computes and returns the mean E[X] = αβ/(α - 1) of the Pareto distribution with parameters α and β.
`static double[]` ```getMLE(double[] x, int n)```
Estimates the parameters (α, β) of the Pareto 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 beta)```
Computes and returns the standard deviation of the Pareto distribution with parameters α and β.
` double` `getVariance()`
Returns the variance.
`static double` ```getVariance(double alpha, double beta)```
Computes and returns the variance of the Pareto distribution with parameters α and β.
` double` `inverseF(double u)`
Returns the inverse distribution function x = F-1(u).
`static double` ```inverseF(double alpha, double beta, double u)```
Computes the inverse of the distribution function.
` void` ```setParams(double alpha, double beta)```
Sets the parameter α and β 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

### ParetoDist

`public ParetoDist(double alpha)`
Constructs a ParetoDist object with parameters α = alpha and β = 1.

### ParetoDist

```public ParetoDist(double alpha,
double beta)```
Constructs a ParetoDist object with parameters α = alpha and β = beta.

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 beta,
double x)```
Computes the density function.

### cdf

```public static double cdf(double alpha,
double beta,
double x)```
Computes the distribution function.

### barF

```public static double barF(double alpha,
double beta,
double x)```
Computes the complementary distribution function.

### inverseF

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

### getMLE

```public static double[] getMLE(double[] x,
int n)```
Estimates the parameters (α, β) of the Pareto 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, β].

Parameters:
`x` - the list of observations used to evaluate parameters
`n` - the number of observations used to evaluate parameters
Returns:
returns the parameters [ hat(α), hat(β)]

### getInstanceFromMLE

```public static ParetoDist getInstanceFromMLE(double[] x,
int n)```
Creates a new instance of a Pareto distribution with parameters α and β 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 beta)```
Computes and returns the mean E[X] = αβ/(α - 1) of the Pareto distribution with parameters α and β.

Returns:
the mean of the Pareto distribution

### getVariance

```public static double getVariance(double alpha,
double beta)```
Computes and returns the variance of the Pareto distribution with parameters α and β.

Returns:
the variance of the Pareto distribution Var[X] = αβ2/[(α -2)(α - 1)]

### getStandardDeviation

```public static double getStandardDeviation(double alpha,
double beta)```
Computes and returns the standard deviation of the Pareto distribution with parameters α and β.

Returns:
the standard deviation of the Pareto distribution

### getAlpha

`public double getAlpha()`
Returns the parameter α.

### getBeta

`public double getBeta()`
Returns the parameter β.

### setParams

```public void setParams(double alpha,
double beta)```
Sets the parameter α and β 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, β].

### 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.