umontreal.iro.lecuyer.probdist
Class LognormalDist

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

All Implemented Interfaces:

Distribution

Direct Known Subclasses:

LognormalDistFromMoments

public class LognormalDist
extends ContinuousDistribution

Extends the class ContinuousDistribution for the lognormal distribution. (See also the Johnson S_L distribution JohnsonSLDist in this package.) It has scale parameter μ and shape parameter σ > 0. The density is

f (x) = ((2π)^1/2σx)^-1e^{-(ln(x)-μ)2/(2σ2)} for x > 0,

and 0 elsewhere. The distribution function is

F(x) = Φ((ln(x)-μ)/σ) for x > 0,

where Φ is the standard normal distribution function. Its inverse is given by

F^-1(u) = e^μ+σΦ-1(u) for 0 <= u < 1.

If ln(Y) has a normal distribution, then Y has a lognormal distribution with the same parameters.

This class relies on the methods NormalDist.cdf01 and NormalDist.inverseF01 of NormalDist to approximate Φ and Φ^-1.

Field Summary

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

decPrec

Constructor Summary

LognormalDist()
Constructs a LognormalDist object with default parameters μ = 0 and σ = 1.

LognormalDist(double mu, double sigma)
Constructs a LognormalDist object with parameters μ = mu and σ = sigma.

Method Summary

double	barF(double x) Returns the complementary distribution function.
static double	barF(double mu, double sigma, double x) Computes the lognormal complementary distribution function bar(F)(x), using NormalDist.barF01.
double	cdf(double x) Returns the distribution function F(x).
static double	cdf(double mu, double sigma, double x) Computes the lognormal distribution function, using NormalDist.cdf01.
double	density(double x) Returns f (x), the density evaluated at x.
static double	density(double mu, double sigma, double x) Computes the lognormal density function f (x).
static LognormalDist	getInstanceFromMLE(double[] x, int n) Creates a new instance of a lognormal distribution with parameters μ and σ 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 mu, double sigma) Computes and returns the mean E[X] = eμ+σ2/2 of the lognormal distribution with parameters μ and σ.
static double[]	getMLE(double[] x, int n) Estimates the parameters (μ, σ) of the lognormal distribution using the maximum likelihood method, from the n observations x[i], i = 0, 1,…, n - 1.
double	getMu() Returns the parameter μ of this object.
double[]	getParams() Returns a table containing the parameters of the current distribution, in the order: [μ, σ].
double	getSigma() Returns the parameter σ of this object.
double	getStandardDeviation() Returns the standard deviation.
static double	getStandardDeviation(double mu, double sigma) Computes and returns the standard deviation of the lognormal distribution with parameters μ and σ.
double	getVariance() Returns the variance.
static double	getVariance(double mu, double sigma) Computes and returns the variance Var[X] = e2μ+σ2(eσ2 - 1) of the lognormal distribution with parameters μ and σ.
double	inverseF(double u) Returns the inverse distribution function x = F-1(u).
static double	inverseF(double mu, double sigma, double u) Computes the inverse of the lognormal distribution function, using NormalDist.inverseF01.
void	setParams(double mu, double sigma) Sets the parameters μ and σ 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

LognormalDist

public LognormalDist()

Constructs a LognormalDist object with default parameters μ = 0 and σ = 1.

LognormalDist

public LognormalDist(double mu,
                     double sigma)

Constructs a LognormalDist object with parameters μ = mu and σ = sigma.

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 mu,
                             double sigma,
                             double x)

Computes the lognormal density function f (x).

cdf

public static double cdf(double mu,
                         double sigma,
                         double x)

Computes the lognormal distribution function, using NormalDist.cdf01.

barF

public static double barF(double mu,
                          double sigma,
                          double x)

Computes the lognormal complementary distribution function bar(F)(x), using NormalDist.barF01.

inverseF

public static double inverseF(double mu,
                              double sigma,
                              double u)

Computes the inverse of the lognormal distribution function, using NormalDist.inverseF01.

getMLE

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

Estimates the parameters (μ, σ) of the lognormal 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: [μ, σ].

Parameters:

x - the list of observations used to evaluate parameters

n - the number of observations used to evaluate parameters

Returns:

returns the parameters [hat(μ), hat(σ)]

getInstanceFromMLE

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

Creates a new instance of a lognormal distribution with parameters μ and σ 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 mu,
                             double sigma)

Computes and returns the mean E[X] = e^μ+σ2/2 of the lognormal distribution with parameters μ and σ.

Returns:

the mean of the lognormal distribution

getVariance

public static double getVariance(double mu,
                                 double sigma)

Computes and returns the variance Var[X] = e^2μ+σ2(e^σ2 - 1) of the lognormal distribution with parameters μ and σ.

Returns:

the variance of the lognormal distribution

getStandardDeviation

public static double getStandardDeviation(double mu,
                                          double sigma)

Computes and returns the standard deviation of the lognormal distribution with parameters μ and σ.

Returns:

the standard deviation of the lognormal distribution

getMu

public double getMu()

Returns the parameter μ of this object.

getSigma

public double getSigma()

Returns the parameter σ of this object.

setParams

public void setParams(double mu,
                      double sigma)

Sets the parameters μ and σ of this object.

getParams

public double[] getParams()

Returns a table containing the parameters of the current distribution, in the order: [μ, σ].

toString

public String toString()

Overrides:

toString in class Object