umontreal.iro.lecuyer.probdist
Class LogarithmicDist

java.lang.Object
  umontreal.iro.lecuyer.probdist.DiscreteDistributionInt
      umontreal.iro.lecuyer.probdist.LogarithmicDist

All Implemented Interfaces:

Distribution

public class LogarithmicDist
extends DiscreteDistributionInt

Extends the class DiscreteDistributionInt for the logarithmic distribution. It has shape parameter θ, where 0 < θ < 1. Its mass function is

p(x) = - θ^x/(x log(1 - θ) for x = 1, 2, 3,...

Its distribution function is

F(x) = -1/log(1-θ)∑_i=1^xθⁱ/i, & for x > 0.

Field Summary

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

EPSILON

Constructor Summary

LogarithmicDist(double theta)
Constructs a logarithmic distribution with parameter θ = theta.

Method Summary

static double	barF(double theta, int x) Computes the complementary distribution function.
double	barF(int x) Returns bar(F)(x), the complementary distribution function.
static double	cdf(double theta, int x) Computes the distribution function F(x).
double	cdf(int x) Returns the distribution function F evaluated at x (see).
static LogarithmicDist	getInstanceFromMLE(int[] x, int n) Creates a new instance of a logarithmic distribution with parameter θ estimated using the maximum likelihood method based on the n observations x[i], i = 0, 1,…, n - 1.
double	getMean() Returns the mean of the distribution function.
static double	getMean(double theta) Computes and returns the mean of the logarithmic distribution with parameter θ = theta.
static double[]	getMLE(int[] x, int n) Estimates the parameter θ of the logarithmic 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 of the distribution function.
static double	getStandardDeviation(double theta) Computes and returns the standard deviation of the logarithmic distribution with parameter θ = theta.
double	getTheta() Returns the θ associated with this object.
double	getVariance() Returns the variance of the distribution function.
static double	getVariance(double theta) Computes and returns the variance of the logarithmic distribution with parameter θ = theta.
static int	inverseF(double theta, double u)
int	inverseFInt(double u) Returns the inverse distribution function F-1(u), where 0 <= u <= 1.
static double	prob(double theta, int x) Computes the logarithmic probability p(x).
double	prob(int x) Returns p(x), the probability of x.
void	setTheta(double theta) Sets the θ associated with this object.
String	toString()

Methods inherited from class umontreal.iro.lecuyer.probdist.DiscreteDistributionInt

barF, cdf, getXinf, getXsup, inverseF

Methods inherited from class java.lang.Object

equals, getClass, hashCode, notify, notifyAll, wait, wait, wait

Constructor Detail

LogarithmicDist

public LogarithmicDist(double theta)

Constructs a logarithmic distribution with parameter θ = theta.

Method Detail

prob

public double prob(int x)

Description copied from class: DiscreteDistributionInt

Returns p(x), the probability of x.

Specified by:

prob in class DiscreteDistributionInt

Parameters:

x - value at which the mass function must be evaluated

Returns:

the mass function evaluated at x

cdf

public double cdf(int x)

Description copied from class: DiscreteDistributionInt

Returns the distribution function F evaluated at x (see).

Specified by:

cdf in class DiscreteDistributionInt

Parameters:

x - value at which the distribution function must be evaluated

Returns:

the distribution function evaluated at x

barF

public double barF(int x)

Description copied from class: DiscreteDistributionInt

Returns bar(F)(x), the complementary distribution function. See the WARNING above.

Overrides:

barF in class DiscreteDistributionInt

Parameters:

x - value at which the complementary distribution function must be evaluated

Returns:

the complementary distribution function evaluated at x

inverseFInt

public int inverseFInt(double u)

Description copied from class: DiscreteDistributionInt

Returns the inverse distribution function F^-1(u), where 0 <= u <= 1. The default implementation uses binary search.

Overrides:

inverseFInt in class DiscreteDistributionInt

Parameters:

u - value in the interval (0, 1) for which the inverse distribution function is evaluated

Returns:

the inverse distribution function evaluated at u

getMean

public double getMean()

Description copied from interface: Distribution

Returns the mean of the distribution function.

getVariance

public double getVariance()

Description copied from interface: Distribution

Returns the variance of the distribution function.

getStandardDeviation

public double getStandardDeviation()

Description copied from interface: Distribution

Returns the standard deviation of the distribution function.

prob

public static double prob(double theta,
                          int x)

Computes the logarithmic probability p(x).

cdf

public static double cdf(double theta,
                         int x)

Computes the distribution function F(x).

barF

public static double barF(double theta,
                          int x)

Computes the complementary distribution function. WARNING: The complementary distribution function is defined as bar(F)(x) = P[X >= x].

inverseF

public static int inverseF(double theta,
                           double u)

getMLE

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

Estimates the parameter θ of the logarithmic distribution using the maximum likelihood method, from the n observations x[i], i = 0, 1,…, n - 1. The estimate is returned in element 0 of the returned array.

Parameters:

x - the list of observations used to evaluate parameters

n - the number of observations used to evaluate parameters

Returns:

returns the parameter [ hat(&thetas;)]

getInstanceFromMLE

public static LogarithmicDist getInstanceFromMLE(int[] x,
                                                 int n)

Creates a new instance of a logarithmic distribution with parameter θ 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 theta)

Computes and returns the mean of the logarithmic distribution with parameter θ = theta.

Returns:

the mean of the logarithmic distribution E[X] = - θ/((1 - θ)ln(1 - θ))

getVariance

public static double getVariance(double theta)

Computes and returns the variance of the logarithmic distribution with parameter θ = theta.

Returns:

the variance of the logarithmic distribution Var[X] = - θ(θ + ln(1 - θ))/((1 - θ)²(ln(1 - θ))²)

getStandardDeviation

public static double getStandardDeviation(double theta)

Computes and returns the standard deviation of the logarithmic distribution with parameter θ = theta.

Returns:

the standard deviation of the logarithmic distribution

getTheta

public double getTheta()

Returns the θ associated with this object.

setTheta

public void setTheta(double theta)

Sets the θ associated with this object.

getParams

public double[] getParams()

Return a table containing the parameters of the current distribution.

toString

public String toString()

Overrides:

toString in class Object