SSJ  3.3.1
Stochastic Simulation in Java
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StudentDist Class Reference

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
xvalue 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
xvalue 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
xthe list of observations to use to evaluate parameters
mthe 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
xthe list of observations to use to evaluate parameters
mthe 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
uvalue 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: