Extends the class DiscreteDistributionInt for the logarithmic distribution. More...

Inheritance diagram for umontreal.ssj.probdist.LogarithmicDist:

Public Member Functions
	LogarithmicDist (double theta)
	Constructs a logarithmic distribution with parameter \(\theta= \) theta.
double	prob (int x)
	Returns \(p(x)\), the probability of \(x\).
double	cdf (int x)
	Returns the distribution function \(F\) evaluated at \(x\) (see ( `FDistDisc` )).
double	barF (int x)
	Returns \(\bar{F}(x)\), the complementary distribution function.
int	inverseFInt (double u)
	Returns the inverse distribution function \(F^{-1}(u)\), where.
double	getMean ()
	Returns the mean of the distribution function.
double	getVariance ()
	Returns the variance of the distribution function.
double	getStandardDeviation ()
	Returns the standard deviation of the distribution function.
double	getTheta ()
	Returns the \(\theta\) associated with this object.
void	setTheta (double theta)
	Sets the \(\theta\) associated with this object.
double[]	getParams ()
	Return a table containing the parameters of the current distribution.
String	toString ()
	Returns a String containing information about the current distribution.
Public Member Functions inherited from umontreal.ssj.probdist.DiscreteDistributionInt
double	cdf (double x)
	Returns the distribution function \(F\) evaluated at \(x\) (see ( `FDistDisc` )).
double	barF (double x)
	Returns \(\bar{F}(x)\), the complementary distribution function.
int	getXinf ()
	Returns the lower limit \(x_a\) of the support of the probability mass function.
int	getXsup ()
	Returns the upper limit \(x_b\) of the support of the probability mass function.
double	inverseF (double u)
	Returns the inverse distribution function \(F^{-1}(u)\), where.

Static Public Member Functions
static double	prob (double theta, int x)
	Computes the logarithmic probability \(p(x)\) given in ( `flogar` ) .
static double	cdf (double theta, int x)
	Computes the distribution function \(F(x)\).
static double	barF (double theta, int x)
	Computes the complementary distribution function.
static double[]	getMLE (int[] x, int n)
	Estimates the parameter \(\theta\) of the logarithmic distribution using the maximum likelihood method, from the \(n\) observations.
static LogarithmicDist	getInstanceFromMLE (int[] x, int n)
	Creates a new instance of a logarithmic distribution with parameter.
static double	getMean (double theta)
	Computes and returns the mean.
static double	getVariance (double theta)
	Computes and returns the variance.
static double	getStandardDeviation (double theta)
	Computes and returns the standard deviation of the logarithmic distribution with parameter \(\theta= \) theta.

Additional Inherited Members
Static Public Attributes inherited from umontreal.ssj.probdist.DiscreteDistributionInt
static double	EPSILON = 1.0e-16
	Environment variable that determines what probability terms can be considered as negligible when building precomputed tables for distribution and mass functions.

Detailed Description

Extends the class DiscreteDistributionInt for the logarithmic distribution.

It has shape parameter \(\theta\), where \(0 < \theta<1\). Its mass function is

\[ p(x) = \frac{-\theta^x}{x\log(1- \theta)} \qquad\mbox{for } x = 1,2,3,…\tag{flogar} \]

Its distribution function is

\[ F(x) = \frac{-1}{\log(1 - \theta)}\sum_{i=1}^x \frac{\theta^i}{i}, \qquad\mbox{ for } x = 1, 2, 3, … \]

and is 0 for \( x\le0\).

Definition at line 46 of file LogarithmicDist.java.

Constructor & Destructor Documentation

◆ LogarithmicDist()

umontreal.ssj.probdist.LogarithmicDist.LogarithmicDist ( double theta )

Constructs a logarithmic distribution with parameter \(\theta= \) theta.

Definition at line 68 of file LogarithmicDist.java.

Member Function Documentation

◆ barF() [1/2]

double umontreal.ssj.probdist.LogarithmicDist.barF	(	double	theta,
		int	x )

static

Computes the complementary distribution function.

WARNING: The complementary distribution function is defined as \(\bar{F}(x) = P[X \ge x]\).

Definition at line 153 of file LogarithmicDist.java.

◆ barF() [2/2]

double umontreal.ssj.probdist.LogarithmicDist.barF ( int x )

Returns \(\bar{F}(x)\), the complementary distribution function.

See the WARNING above. The default implementation returns 1.0 - cdf(x - 1), which is not accurate when \(F(x)\) is near 1.

Parameters

x	value at which the complementary distribution function must be evaluated

Returns: the complementary distribution function evaluated at x

Reimplemented from umontreal.ssj.probdist.DiscreteDistributionInt.

Definition at line 90 of file LogarithmicDist.java.

◆ cdf() [1/2]

double umontreal.ssj.probdist.LogarithmicDist.cdf	(	double	theta,
		int	x )

static

Computes the distribution function \(F(x)\).

Definition at line 134 of file LogarithmicDist.java.

◆ cdf() [2/2]

double umontreal.ssj.probdist.LogarithmicDist.cdf ( int x )

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

Parameters

x	value at which the distribution function must be evaluated

Returns: the distribution function evaluated at x

Reimplemented from umontreal.ssj.probdist.DiscreteDistributionInt.

Definition at line 78 of file LogarithmicDist.java.

◆ getInstanceFromMLE()

LogarithmicDist umontreal.ssj.probdist.LogarithmicDist.getInstanceFromMLE	(	int[]	x,
		int	n )

static

Creates a new instance of a logarithmic distribution with parameter.

\(\theta\) 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

Definition at line 214 of file LogarithmicDist.java.

◆ getMean() [1/2]

double umontreal.ssj.probdist.LogarithmicDist.getMean ( )

Returns the mean of the distribution function.

Implements umontreal.ssj.probdist.Distribution.

Definition at line 107 of file LogarithmicDist.java.

◆ getMean() [2/2]

double umontreal.ssj.probdist.LogarithmicDist.getMean ( double theta )

static

Computes and returns the mean.

\[ E[X] = \frac{-\theta}{(1 - \theta)\ln(1 - \theta)} \]

of the logarithmic distribution with parameter \(\theta= \) theta.

Returns: the mean of the logarithmic distribution \(E[X] = -\theta/ ((1 - \theta) ln(1 - \theta))\)

Definition at line 227 of file LogarithmicDist.java.

◆ getMLE()

double[] umontreal.ssj.probdist.LogarithmicDist.getMLE	(	int[]	x,
		int	n )

static

Estimates the parameter \(\theta\) 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. The maximum likelihood estimator \(\hat{\theta}\) satisfies the equation (see [55] (page 122))

\begin{align*} \bar{x}_n = \frac{-\hat{\theta}}{(1 - \hat{\theta}) \ln(1 - \hat{\theta})} \end{align*}

where \(\bar{x}_n\) is the average of \(x[0], …, x[n-1]\).

Parameters

x	the list of observations used to evaluate parameters
n	the number of observations used to evaluate parameters

Returns: returns the parameter [ \(\hat{\theta}\)]

Definition at line 187 of file LogarithmicDist.java.

◆ getParams()

double[] umontreal.ssj.probdist.LogarithmicDist.getParams ( )

Return a table containing the parameters of the current distribution.

Implements umontreal.ssj.probdist.Distribution.

Definition at line 282 of file LogarithmicDist.java.

◆ getStandardDeviation() [1/2]

double umontreal.ssj.probdist.LogarithmicDist.getStandardDeviation ( )

Returns the standard deviation of the distribution function.

Implements umontreal.ssj.probdist.Distribution.

Definition at line 115 of file LogarithmicDist.java.

◆ getStandardDeviation() [2/2]

double umontreal.ssj.probdist.LogarithmicDist.getStandardDeviation ( double theta )

static

Computes and returns the standard deviation of the logarithmic distribution with parameter \(\theta= \) theta.

Returns: the standard deviation of the logarithmic distribution

Definition at line 257 of file LogarithmicDist.java.

◆ getTheta()

double umontreal.ssj.probdist.LogarithmicDist.getTheta ( )

Returns the \(\theta\) associated with this object.

Definition at line 264 of file LogarithmicDist.java.

◆ getVariance() [1/2]

double umontreal.ssj.probdist.LogarithmicDist.getVariance ( )

Returns the variance of the distribution function.

Implements umontreal.ssj.probdist.Distribution.

Definition at line 111 of file LogarithmicDist.java.

◆ getVariance() [2/2]

double umontreal.ssj.probdist.LogarithmicDist.getVariance ( double theta )

static

Computes and returns the variance.

\[ \mbox{Var}[X] = \frac{-\theta(\theta+ \ln(1 - \theta))}{[(1 - \theta) \ln(1 - \theta)]^2} \]

of the logarithmic distribution with parameter \(\theta=\) theta.

Returns: the variance of the logarithmic distribution \(\mbox{Var}[X] = -\theta(\theta+ ln(1 - \theta)) / ((1 - \theta)^2 (ln(1 - \theta))^2)\)

Definition at line 243 of file LogarithmicDist.java.

◆ inverseFInt()

int umontreal.ssj.probdist.LogarithmicDist.inverseFInt ( double u )

Returns the inverse distribution function \(F^{-1}(u)\), where.

\(0\le u\le1\). The default implementation uses binary search.

Parameters

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

Returns: the inverse distribution function evaluated at u

Exceptions

IllegalArgumentException	if \(u\) is not in the interval \((0,1)\)
ArithmeticException	if the inverse cannot be computed, for example if it would give infinity in a theoritical context

Reimplemented from umontreal.ssj.probdist.DiscreteDistributionInt.

Definition at line 103 of file LogarithmicDist.java.

◆ prob() [1/2]

double umontreal.ssj.probdist.LogarithmicDist.prob	(	double	theta,
		int	x )

static

Computes the logarithmic probability \(p(x)\) given in ( flogar ) .

Definition at line 123 of file LogarithmicDist.java.

◆ prob() [2/2]

double umontreal.ssj.probdist.LogarithmicDist.prob ( int x )

Returns \(p(x)\), the probability of \(x\).

Parameters

x	value at which the mass function must be evaluated

Returns: the mass function evaluated at x

Reimplemented from umontreal.ssj.probdist.DiscreteDistributionInt.

Definition at line 72 of file LogarithmicDist.java.

◆ setTheta()

void umontreal.ssj.probdist.LogarithmicDist.setTheta ( double theta )

Sets the \(\theta\) associated with this object.

Definition at line 271 of file LogarithmicDist.java.

◆ toString()

String umontreal.ssj.probdist.LogarithmicDist.toString ( )

Returns a String containing information about the current distribution.

Definition at line 290 of file LogarithmicDist.java.

The documentation for this class was generated from the following file:

src/main/java/umontreal/ssj/probdist/LogarithmicDist.java

SSJ API Documentation Stochastic Simulation in Java

Public Member Functions

Static Public Member Functions

Additional Inherited Members

Detailed Description

Constructor & Destructor Documentation

◆ LogarithmicDist()

Member Function Documentation

◆ barF() [1/2]

◆ barF() [2/2]

◆ cdf() [1/2]

◆ cdf() [2/2]

◆ getInstanceFromMLE()

◆ getMean() [1/2]

◆ getMean() [2/2]

◆ getMLE()

◆ getParams()

◆ getStandardDeviation() [1/2]

◆ getStandardDeviation() [2/2]

◆ getTheta()

◆ getVariance() [1/2]

◆ getVariance() [2/2]

◆ inverseFInt()

◆ prob() [1/2]

◆ prob() [2/2]

◆ setTheta()

◆ toString()