SSJ  3.3.1
Stochastic Simulation in Java
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Pearson6Dist Class Reference

Extends the class ContinuousDistribution for the Pearson type VI distribution with shape parameters \(\alpha_1 > 0\) and \(\alpha_2 > 0\), and scale parameter \(\beta> 0\). More...

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Public Member Functions

 Pearson6Dist (double alpha1, double alpha2, double beta)
 Constructs a Pearson6Dist object with parameters \(\alpha_1\) = alpha1, \(\alpha_2\) = alpha2 and \(\beta\) = beta.
 
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.
 
double getAlpha1 ()
 Returns the \(\alpha_1\) parameter of this object.
 
double getAlpha2 ()
 Returns the \(\alpha_2\) parameter of this object.
 
double getBeta ()
 Returns the \(\beta\) parameter of this object.
 
void setParam (double alpha1, double alpha2, double beta)
 Sets the parameters \(\alpha_1\), \(\alpha_2\) and \(\beta\) of this object.
 
double [] getParams ()
 Return a table containing the parameters of the current distribution. More...
 
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 (double alpha1, double alpha2, double beta, double x)
 Computes the density function of a Pearson VI distribution with shape parameters \(\alpha_1\) and \(\alpha_2\), and scale parameter \(\beta\).
 
static double cdf (double alpha1, double alpha2, double beta, double x)
 Computes the distribution function of a Pearson VI distribution with shape parameters \(\alpha_1\) and \(\alpha_2\), and scale parameter \(\beta\).
 
static double barF (double alpha1, double alpha2, double beta, double x)
 Computes the complementary distribution function of a Pearson VI distribution with shape parameters \(\alpha_1\) and \(\alpha_2\), and scale parameter \(\beta\).
 
static double inverseF (double alpha1, double alpha2, double beta, double u)
 Computes the inverse distribution function of a Pearson VI distribution with shape parameters \(\alpha_1\) and \(\alpha_2\), and scale parameter \(\beta\).
 
static double [] getMLE (double[] x, int n)
 Estimates the parameters \((\alpha_1,\alpha_2,\beta)\) of the Pearson VI distribution using the maximum likelihood method, from the \(n\) observations \(x[i]\), \(i = 0, 1,…, n-1\). More...
 
static Pearson6Dist getInstanceFromMLE (double[] x, int n)
 Creates a new instance of a Pearson VI distribution with parameters \(\alpha_1\), \(\alpha_2\) and \(\beta\), estimated using the maximum likelihood method based on the \(n\) observations \(x[i]\), \(i = 0, 1, …, n-1\). More...
 
static double getMean (double alpha1, double alpha2, double beta)
 Computes and returns the mean \(E[X] = (\beta\alpha_1) / (\alpha_2 - 1)\) of a Pearson VI distribution with shape parameters \(\alpha_1\) and \(\alpha_2\), and scale parameter \(\beta\).
 
static double getVariance (double alpha1, double alpha2, double beta)
 Computes and returns the variance \(\mbox{Var}[X] = [\beta^2 \alpha_1 (\alpha_1 + \alpha_2 - 1)] / [(\alpha_2 - 1)^2(\alpha_2 - 2)]\) of a Pearson VI distribution with shape parameters \(\alpha_1\) and \(\alpha_2\), and scale parameter \(\beta\).
 
static double getStandardDeviation (double alpha1, double alpha2, double beta)
 Computes and returns the standard deviation of a Pearson VI distribution with shape parameters \(\alpha_1\) and \(\alpha_2\), and scale parameter \(\beta\).
 

Protected Attributes

double alpha1
 
double alpha2
 
double beta
 
double logBeta
 
- 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 Pearson type VI distribution with shape parameters \(\alpha_1 > 0\) and \(\alpha_2 > 0\), and scale parameter \(\beta> 0\).

The density function is given by

\[ f(x) =\left\{\begin{array}{ll} \displaystyle\frac{\left(x/{\beta}\right)^{\alpha_1 - 1}}{\beta\mathcal{B}(\alpha_1, \alpha_2)(1 + x/{\beta})^{\alpha_1 + \alpha_2}} & \quad\mbox{for } x > 0, \\ 0 & \quad\mbox{otherwise,} \end{array} \right. \tag{fpearson6} \]

where \(\mathcal{B}\) is the beta function. The distribution function is given by

\[ F(x) = F_B\left(\frac{x}{x + \beta}\right) \qquad\mbox{for } x > 0, \tag{Fpearson6} \]

and \(F(x) = 0\) otherwise, where \(F_B(x)\) is the distribution function of a beta distribution with shape parameters \(\alpha_1\) and \(\alpha_2\).

Member Function Documentation

◆ barF()

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.

◆ cdf()

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.

◆ getInstanceFromMLE()

static Pearson6Dist getInstanceFromMLE ( double []  x,
int  n 
)
static

Creates a new instance of a Pearson VI distribution with parameters \(\alpha_1\), \(\alpha_2\) and \(\beta\), estimated using the maximum likelihood method based on the \(n\) observations \(x[i]\), \(i = 0, 1, …, n-1\).

Parameters
xthe list of observations to use to evaluate parameters
nthe number of observations to use to evaluate parameters

◆ getMLE()

static double [] getMLE ( double []  x,
int  n 
)
static

Estimates the parameters \((\alpha_1,\alpha_2,\beta)\) of the Pearson VI distribution using the maximum likelihood method, from the \(n\) observations \(x[i]\), \(i = 0, 1,…, n-1\).

The estimates are returned in a three-element array, in regular order: [ \(\alpha_1, \alpha_2\), \(\beta\)]. The estimate of the parameters is given by maximizing numerically the log-likelihood function, using the Uncmin package [211], [233] .

Parameters
xthe list of observations to use to evaluate parameters
nthe number of observations to use to evaluate parameters
Returns
returns the parameters [ \(\hat{\alpha_1}, \hat{\alpha_2}, \hat{\beta}\)]

◆ getParams()

double [] getParams ( )

Return a table containing the parameters of the current distribution.

This table is put in regular order: [ \(\alpha_1\), \(\alpha_2\), \(\beta\)].

Implements Distribution.

◆ inverseF()

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.


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