Extends the class ContinuousDistribution for the Student \(t\)-distribution [100] (page 362) with \(n\) degrees of freedom, where \(n\) is a positive integer. More...

Inheritance diagram for StudentDist:

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Collaboration diagram for StudentDist:

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Public Member Functions
	StudentDist (int n)
	Constructs a `StudentDist` object with `n` degrees of freedom.

double	density (double x)

double	cdf (double x)
	Returns the distribution function \(F(x)\). More...

double	barF (double x)
	Returns \(\bar{F}(x) = 1 - F(x)\). More...

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

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.

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 ContinuousDistribution
abstract double	density (double x)
	Returns \(f(x)\), the density evaluated at \(x\). More...

double	barF (double x)
	Returns the complementary distribution function. More...

double	inverseBrent (double a, double b, double u, double tol)
	Computes the inverse distribution function \(x = F^{-1}(u)\), using the Brent-Dekker method. More...

double	inverseBisection (double u)
	Computes and returns the inverse distribution function \(x = F^{-1}(u)\), using bisection. More...

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

double	getMean ()
	Returns the mean. More...

double	getVariance ()
	Returns the variance. More...

double	getStandardDeviation ()
	Returns the standard deviation. More...

double	getXinf ()
	Returns \(x_a\) such that the probability density is 0 everywhere outside the interval \([x_a, x_b]\). More...

double	getXsup ()
	Returns \(x_b\) such that the probability density is 0 everywhere outside the interval \([x_a, x_b]\). More...

void	setXinf (double xa)
	Sets the value \(x_a=\) `xa`, such that the probability density is 0 everywhere outside the interval \([x_a, x_b]\). More...

void	setXsup (double xb)
	Sets the value \(x_b=\) `xb`, such that the probability density is 0 everywhere outside the interval \([x_a, x_b]\). More...

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 \(n\) degrees of freedom. More...

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. More...

static double	inverseF (int n, double u)
	Returns the inverse \(x = F^{-1}(u)\) of Student \(t\)-distribution function with \(n\) degrees of freedom. More...

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 \(x[i]\), \(i = 0, 1,…, m-1\). More...

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 \(x[i]\), \(i = 0, 1, …, m-1\). More...

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

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

static double	getStandardDeviation (int n)
	Computes and returns the standard deviation of the Student \(t\)-distribution with parameter \(n\). More...

Protected Attributes
int	n

Protected Attributes inherited from ContinuousDistribution
double	supportA = Double.NEGATIVE_INFINITY

double	supportB = Double.POSITIVE_INFINITY

Additional Inherited Members
Public Attributes inherited from ContinuousDistribution
int	decPrec = 15

Static Protected Attributes inherited from ContinuousDistribution
static final double	XBIG = 100.0

static final double	XBIGM = 1000.0

static final double []	EPSARRAY

Detailed Description

Extends the class ContinuousDistribution for the Student \(t\)-distribution [100] (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 ).

Member Function Documentation

◆ barF() [1/2]

double barF ( double x )

Returns \(\bar{F}(x) = 1 - F(x)\).

Parameters

x	value at which the complementary distribution function is evaluated

Returns: complementary distribution function evaluated at x

Implements Distribution.

◆ barF() [2/2]

static double 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 \(v > 10^{-15}\).

◆ cdf() [1/2]

double 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 Distribution.

◆ cdf() [2/2]

static double 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 \(u > 10^{-15}\).

◆ getInstanceFromMLE()

static 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

◆ getMean()

static double 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 \(E[X] = 0\)

◆ getMLE()

static double [] 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 [ \(\hat{n}\)]

◆ getStandardDeviation()

static double 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

◆ getVariance()

static double 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 \(\mbox{Var}[X] = n / (n - 2)\)

◆ inverseF() [1/2]

double inverseF ( double u )

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

Parameters

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

Returns: the inverse distribution function evaluated at u

Implements Distribution.

◆ inverseF() [2/2]

static double 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 \(n > 10^5\).

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

StudentDist.java

Public Member Functions

Static Public Member Functions

Protected Attributes

Additional Inherited Members

Detailed Description

Member Function Documentation

◆ barF() [1/2]

◆ barF() [2/2]

◆ cdf() [1/2]

◆ cdf() [2/2]

◆ getInstanceFromMLE()

◆ getMean()

◆ getMLE()

◆ getStandardDeviation()

◆ getVariance()

◆ inverseF() [1/2]

◆ inverseF() [2/2]