Extends the class ContinuousDistribution for the Student. More...

Inheritance diagram for umontreal.ssj.probdist.StudentDist:

Public Member Functions
	StudentDist (int n)
	Constructs a StudentDist object with n degrees of freedom.
double	density (double x)
	Returns \(f(x)\), the density evaluated at \(x\).
double	cdf (double x)
	Returns the distribution function \(F(x)\).
double	barF (double x)
	Returns the complementary distribution function.
double	inverseF (double u)
	Returns the inverse distribution function \(x = F^{-1}(u)\).
double	getMean ()
	Returns the mean.
double	getVariance ()
	Returns the variance.
double	getStandardDeviation ()
	Returns the standard deviation.
int	getN ()
	Returns the parameter \(n\) associated with this object.
void	setN (int n)
	Sets the parameter \(n\) associated with this object.
double[]	getParams ()
	Return a table containing the parameter of the current distribution.
String	toString ()
	Returns a String containing information about the current distribution.
Public Member Functions inherited from umontreal.ssj.probdist.ContinuousDistribution
double	inverseBrent (double a, double b, double u, double tol)
	Computes the inverse distribution function \(x = F^{-1}(u)\), using the Brent-Dekker method.
double	inverseBisection (double u)
	Computes and returns the inverse distribution function \(x = F^{-1}(u)\), using bisection.
double	getXinf ()
	Returns \(x_a\) such that the probability density is 0 everywhere outside the interval \([x_a, x_b]\).
double	getXsup ()
	Returns \(x_b\) such that the probability density is 0 everywhere outside the interval \([x_a, x_b]\).
void	setXinf (double xa)
	Sets the value \(x_a=\) xa, such that the probability density is 0 everywhere outside the interval \([x_a, x_b]\).
void	setXsup (double xb)
	Sets the value \(x_b=\) xb, such that the probability density is 0 everywhere outside the interval \([x_a, x_b]\).

Static Public Member Functions
static double	density (int n, double x)
	Computes the density function ( `fstudent` ) of a Student \(t\)-distribution with \(n\) degrees of freedom.
static double	cdf (int n, double x)
	Computes the Student \(t\)-distribution function \(u=F(x)\) with.
static double	cdf2 (int n, int d, double x)
	Same as `cdf(n, x)`.
static double	barF (int n, double x)
	Computes the complementary distribution function \(v = \bar{F}(x)\) with \(n\) degrees of freedom.
static double	inverseF (int n, double u)
	Returns the inverse \(x = F^{-1}(u)\) of Student.
static double[]	getMLE (double[] x, int m)
	Estimates the parameter \(n\) of the Student \(t\)-distribution using the maximum likelihood method, from the \(m\) observations.
static StudentDist	getInstanceFromMLE (double[] x, int m)
	Creates a new instance of a Student \(t\)-distribution with parameter \(n\) estimated using the maximum likelihood method based on the \(m\) observations \(x[i]\), \(i = 0, 1, …, m-1\).
static double	getMean (int n)
	Returns the mean \(E[X] = 0\) of the Student \(t\)-distribution with parameter \(n\).
static double	getVariance (int n)
	Computes and returns the variance \(\mbox{Var}[X] = n/(n - 2)\) of the Student \(t\)-distribution with parameter \(n\).
static double	getStandardDeviation (int n)
	Computes and returns the standard deviation of the Student.

Detailed Description

Extends the class ContinuousDistribution for the Student.

\(t\)-distribution [96] (page 362) with \(n\) degrees of freedom, where \(n\) is a positive integer. Its density is

\[ f (x) = \frac{\Gamma\left((n + 1)/2 \right)}{\Gamma(n/2) \sqrt{\pi n}} \left(1 + \frac{x^2}{n}\right)^{-(n+1)/2} \qquad\mbox{for } -\infty< x < \infty, \tag{fstudent} \]

where \(\Gamma(x)\) is the gamma function defined in ( Gamma ).

Definition at line 47 of file StudentDist.java.

Constructor & Destructor Documentation

◆ StudentDist()

umontreal.ssj.probdist.StudentDist.StudentDist ( int n )

Constructs a StudentDist object with n degrees of freedom.

Definition at line 112 of file StudentDist.java.

Member Function Documentation

◆ barF() [1/2]

double umontreal.ssj.probdist.StudentDist.barF ( double x )

Returns the complementary distribution function.

The default implementation computes \(\bar{F}(x) = 1 - F(x)\).

Parameters

x	value at which the complementary distribution function is evaluated

Returns: complementary distribution function evaluated at x

Reimplemented from umontreal.ssj.probdist.ContinuousDistribution.

Reimplemented in umontreal.ssj.probdist.StudentDistQuick.

Definition at line 124 of file StudentDist.java.

◆ barF() [2/2]

double umontreal.ssj.probdist.StudentDist.barF	(	int	n,
		double	x )

static

Computes the complementary distribution function \(v = \bar{F}(x)\) with \(n\) degrees of freedom.

Gives 13 decimal digits of precision for \(n \le10^5\). For \(n > 10^5\), gives at least 6 decimal digits of precision everywhere, and at least 9 decimal digits of precision for all \(v > 10^{-15}\).

Reimplemented in umontreal.ssj.probdist.StudentDistQuick.

Definition at line 205 of file StudentDist.java.

◆ cdf() [1/2]

double umontreal.ssj.probdist.StudentDist.cdf ( double x )

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

Parameters

x	value at which the distribution function is evaluated

Returns: distribution function evaluated at x

Implements umontreal.ssj.probdist.Distribution.

Reimplemented in umontreal.ssj.probdist.StudentDistQuick.

Definition at line 120 of file StudentDist.java.

◆ cdf() [2/2]

double umontreal.ssj.probdist.StudentDist.cdf	(	int	n,
		double	x )

static

Computes the Student \(t\)-distribution function \(u=F(x)\) with.

\(n\) degrees of freedom. Gives 13 decimal digits of precision for \(n \le10^5\). For \(n > 10^5\), gives at least 6 decimal digits of precision everywhere, and at least 9 decimal digits of precision for all \(u > 10^{-15}\).

Reimplemented in umontreal.ssj.probdist.StudentDistQuick.

Definition at line 162 of file StudentDist.java.

◆ cdf2()

double umontreal.ssj.probdist.StudentDist.cdf2	(	int	n,
		int	d,
		double	x )

static

Same as cdf(n, x).

Definition at line 192 of file StudentDist.java.

◆ density() [1/2]

double umontreal.ssj.probdist.StudentDist.density ( double x )

Returns \(f(x)\), the density evaluated at \(x\).

Parameters

x	value at which the density is evaluated

Returns: density function evaluated at x

Reimplemented from umontreal.ssj.probdist.ContinuousDistribution.

Definition at line 116 of file StudentDist.java.

◆ density() [2/2]

double umontreal.ssj.probdist.StudentDist.density	(	int	n,
		double	x )

static

Computes the density function ( fstudent ) of a Student \(t\)-distribution with \(n\) degrees of freedom.

Definition at line 149 of file StudentDist.java.

◆ getInstanceFromMLE()

StudentDist umontreal.ssj.probdist.StudentDist.getInstanceFromMLE	(	double[]	x,
		int	m )

static

Creates a new instance of a Student \(t\)-distribution with parameter \(n\) estimated using the maximum likelihood method based on the \(m\) observations \(x[i]\), \(i = 0, 1, …, m-1\).

Parameters

x	the list of observations to use to evaluate parameters
m	the number of observations to use to evaluate parameters

Definition at line 316 of file StudentDist.java.

◆ getMean() [1/2]

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

Returns the mean.

Returns: the mean

Reimplemented from umontreal.ssj.probdist.ContinuousDistribution.

Definition at line 132 of file StudentDist.java.

◆ getMean() [2/2]

double umontreal.ssj.probdist.StudentDist.getMean ( int n )

static

Returns the mean \(E[X] = 0\) of the Student \(t\)-distribution with parameter \(n\).

Returns: the mean of the Student \(t\)-distribution \(E[X] = 0\)

Definition at line 327 of file StudentDist.java.

◆ getMLE()

double[] umontreal.ssj.probdist.StudentDist.getMLE	(	double[]	x,
		int	m )

static

Estimates the parameter \(n\) of the Student \(t\)-distribution using the maximum likelihood method, from the \(m\) observations.

\(x[i]\), \(i = 0, 1,…, m-1\). The estimate is returned in a one-element array.

Parameters

x	the list of observations to use to evaluate parameters
m	the number of observations to use to evaluate parameters

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

Definition at line 265 of file StudentDist.java.

◆ getN()

int umontreal.ssj.probdist.StudentDist.getN ( )

Returns the parameter \(n\) associated with this object.

Definition at line 359 of file StudentDist.java.

◆ getParams()

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

Return a table containing the parameter of the current distribution.

Implements umontreal.ssj.probdist.Distribution.

Definition at line 376 of file StudentDist.java.

◆ getStandardDeviation() [1/2]

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

Returns the standard deviation.

Returns: the standard deviation

Reimplemented from umontreal.ssj.probdist.ContinuousDistribution.

Definition at line 140 of file StudentDist.java.

◆ getStandardDeviation() [2/2]

double umontreal.ssj.probdist.StudentDist.getStandardDeviation ( int n )

static

Computes and returns the standard deviation of the Student.

\(t\)-distribution with parameter \(n\).

Returns: the standard deviation of the Student \(t\)-distribution

Definition at line 352 of file StudentDist.java.

◆ getVariance() [1/2]

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

Returns the variance.

Returns: the variance

Reimplemented from umontreal.ssj.probdist.ContinuousDistribution.

Definition at line 136 of file StudentDist.java.

◆ getVariance() [2/2]

double umontreal.ssj.probdist.StudentDist.getVariance ( int n )

static

Computes and returns the variance \(\mbox{Var}[X] = n/(n - 2)\) of the Student \(t\)-distribution with parameter \(n\).

Returns: the variance of the Student \(t\)-distribution \(\mbox{Var}[X] = n / (n - 2)\)

Definition at line 340 of file StudentDist.java.

◆ inverseF() [1/2]

double umontreal.ssj.probdist.StudentDist.inverseF ( double u )

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

Restrictions: \(u \in[0,1]\).

Parameters

u	value at which the inverse distribution function is evaluated

Returns: the inverse distribution function evaluated at u

Exceptions

IllegalArgumentException if \(u\) is not in the interval \([0,1]\)

Reimplemented from umontreal.ssj.probdist.ContinuousDistribution.

Reimplemented in umontreal.ssj.probdist.StudentDistQuick.

Definition at line 128 of file StudentDist.java.

◆ inverseF() [2/2]

double umontreal.ssj.probdist.StudentDist.inverseF	(	int	n,
		double	u )

static

Returns the inverse \(x = F^{-1}(u)\) of Student.

\(t\)-distribution function with \(n\) degrees of freedom. Gives 13 decimal digits of precision for \(n \le10^5\), and at least 9 decimal digits of precision for \(n > 10^5\).

Reimplemented in umontreal.ssj.probdist.StudentDistQuick.

Definition at line 233 of file StudentDist.java.

◆ setN()

void umontreal.ssj.probdist.StudentDist.setN ( int n )

Sets the parameter \(n\) associated with this object.

Definition at line 366 of file StudentDist.java.

◆ toString()

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

Returns a String containing information about the current distribution.

Definition at line 384 of file StudentDist.java.

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

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

SSJ API Documentation Stochastic Simulation in Java

Public Member Functions

Static Public Member Functions

Detailed Description

Constructor & Destructor Documentation

◆ StudentDist()

Member Function Documentation

◆ barF() [1/2]

◆ barF() [2/2]

◆ cdf() [1/2]

◆ cdf() [2/2]

◆ cdf2()

◆ density() [1/2]

◆ density() [2/2]

◆ getInstanceFromMLE()

◆ getMean() [1/2]

◆ getMean() [2/2]

◆ getMLE()

◆ getN()

◆ getParams()

◆ getStandardDeviation() [1/2]

◆ getStandardDeviation() [2/2]

◆ getVariance() [1/2]

◆ getVariance() [2/2]

◆ inverseF() [1/2]

◆ inverseF() [2/2]

◆ setN()

◆ toString()