umontreal.iro.lecuyer.probdist
Class LoglogisticDist

java.lang.Object
  umontreal.iro.lecuyer.probdist.ContinuousDistribution
      umontreal.iro.lecuyer.probdist.LoglogisticDist

All Implemented Interfaces:

Distribution

public class LoglogisticDist
extends ContinuousDistribution

Extends the class ContinuousDistribution for the Log-Logistic distribution with shape parameter α > 0 and scale parameter β > 0. Its density is

f (x) = (α(x/β)^α-1)/(β[1 + (x/β)^α]²) for x > 0

and its distribution function is

F(x) = 1/(1 + (x/β)^-α) for x > 0.

The complementary distribution is

bar(F)(x) = 1/(1 + (x/β)^α) for x > 0.

Field Summary

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

decPrec

Constructor Summary

LoglogisticDist(double alpha, double beta)
Constructs a log-logistic distribution with parameters α and β.

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 of the log-logistic distribution with parameters α and β.
double	cdf(double x) Returns the distribution function F(x).
static double	cdf(double alpha, double beta, double x) Computes the distribution function of the log-logistic distribution with parameters α and β.
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 for a log-logisitic distribution with parameters α and β.
double	getAlpha() Return the parameter α of this object.
double	getBeta() Returns the parameter β of this object.
static LoglogisticDist	getInstanceFromMLE(double[] x, int n) Creates a new instance of a log-logistic 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 of the log-logistic distribution with parameters α and β.
static double[]	getMLE(double[] x, int n) Estimates the parameters (α, β) of the log-logistic 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 log-logistic 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 log-logistic 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 log-logistic distribution with parameters α and β.
void	setParams(double alpha, double beta) Sets the parameters α and β of 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

LoglogisticDist

public LoglogisticDist(double alpha,
                       double beta)

Constructs a log-logistic distribution with parameters α and β.

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 for a log-logisitic distribution with parameters α and β.

cdf

public static double cdf(double alpha,
                         double beta,
                         double x)

Computes the distribution function of the log-logistic distribution with parameters α and β.

barF

public static double barF(double alpha,
                          double beta,
                          double x)

Computes the complementary distribution function of the log-logistic distribution with parameters α and β.

inverseF

public static double inverseF(double alpha,
                              double beta,
                              double u)

Computes the inverse of the log-logistic distribution with parameters α and β.

getMLE

public static double[] getMLE(double[] x,
                              int n)

Estimates the parameters (α, β) of the log-logistic 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 to use to evaluate parameters

n - the number of observations to use to evaluate parameters

Returns:

returns the parameters [ hat(α), hat(β)]

getInstanceFromMLE

public static LoglogisticDist getInstanceFromMLE(double[] x,
                                                 int n)

Creates a new instance of a log-logistic 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 of the log-logistic distribution with parameters α and β.

Returns:

the mean of the log-logistic distribution E[X] = βθ cosec(θ), where θ = π/α

getVariance

public static double getVariance(double alpha,
                                 double beta)

Computes and returns the variance of the log-logistic distribution with parameters α and β.

Returns:

the variance of the log-logistic distribution Var[X] = β²θ(2cosec(2θ) - θ[cosec(θ)]²), where θ = π/α

getStandardDeviation

public static double getStandardDeviation(double alpha,
                                          double beta)

Computes and returns the standard deviation of the log-logistic distribution with parameters α and β.

Returns:

the standard deviation of the log-logistic distribution

getAlpha

public double getAlpha()

Return the parameter α of this object.

getBeta

public double getBeta()

Returns the parameter β of this object.

setParams

public void setParams(double alpha,
                      double beta)

Sets the parameters α and β of 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