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

Inheritance diagram for LogarithmicDist:

Collaboration diagram for LogarithmicDist:

Public Member Functions
	LogarithmicDist (double theta)
	Constructs a logarithmic distribution with parameter \(\theta= \) `theta`.

double	prob (int x)

double	cdf (int x)

double	barF (int x)

int	inverseFInt (double u)

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 DiscreteDistributionInt
abstract double	prob (int x)
	Returns \(p(x)\), the probability of \(x\). More...

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

abstract double	cdf (int x)
	Returns the distribution function \(F\) evaluated at \(x\) (see ( FDistDisc )). More...

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

double	barF (int x)
	Returns \(\bar{F}(x)\), the complementary distribution function. More...

int	getXinf ()
	Returns the lower limit \(x_a\) of the support of the probability mass function. More...

int	getXsup ()
	Returns the upper limit \(x_b\) of the support of the probability mass function. More...

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

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

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. More...

static int	inverseF (double theta, double u)

static double []	getMLE (int[] x, int n)
	Estimates the parameter \(\theta\) of the logarithmic distribution using the maximum likelihood method, from the \(n\) observations \(x[i]\), \(i = 0, 1, …, n-1\). More...

static LogarithmicDist	getInstanceFromMLE (int[] x, int n)
	Creates a new instance of a logarithmic distribution with parameter \(\theta\) estimated using the maximum likelihood method based on the \(n\) observations \(x[i]\), \(i = 0, 1, …, n-1\). More...

static double	getMean (double theta)
	Computes and returns the mean \[ E[X] = \frac{-\theta}{(1 - \theta)\ln(1 - \theta)} \] of the logarithmic distribution with parameter \(\theta= \) `theta`. More...

static double	getVariance (double theta)
	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`. More...

static double	getStandardDeviation (double theta)
	Computes and returns the standard deviation of the logarithmic distribution with parameter \(\theta= \) `theta`. More...

Additional Inherited Members
Static Public Attributes inherited from 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. More...

Protected Attributes inherited from DiscreteDistributionInt
double	cdf [] = null

double	pdf [] = null

int	xmin = 0

int	xmax = 0

int	xmed = 0

int	supportA = Integer.MIN_VALUE

int	supportB = Integer.MAX_VALUE

Static Protected Attributes inherited from DiscreteDistributionInt
static final double	EPS_EXTRA = 1.0e-6

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 \( x\le0\).

Member Function Documentation

◆ barF()

static double barF	(	double	theta,
		int	x
	)

static

Computes the complementary distribution function.

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

◆ getInstanceFromMLE()

static 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

◆ getMean()

static double 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 \(E[X] = -\theta/ ((1 - \theta) ln(1 - \theta))\)

◆ getMLE()

static double [] 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 [56] (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 [ \(\hat{\theta}\)]

◆ getStandardDeviation()

static double 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

◆ getVariance()

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

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

LogarithmicDist.java

Public Member Functions

Static Public Member Functions

Additional Inherited Members

Detailed Description

Member Function Documentation

◆ barF()

◆ getInstanceFromMLE()

◆ getMean()

◆ getMLE()

◆ getStandardDeviation()

◆ getVariance()