BetaSymmetricalDist (Java Libraries for Stochastic Simulation)

java.lang.Object
- umontreal.iro.lecuyer.probdist.ContinuousDistribution
- - umontreal.iro.lecuyer.probdist.BetaDist
  - - umontreal.iro.lecuyer.probdist.BetaSymmetricalDist

All Implemented Interfaces:

Distribution
```
public class BetaSymmetricalDist
extends BetaDist
```
Specializes the class BetaDist to the case of a symmetrical beta distribution over the interval [0, 1], with shape parameters α = β. A faster inversion method is implemented here for this special case. Because of the symmetry around 1/2, four series are used to compute the cdf, two around x = 0 and two around x = 1/2. Given u, one then solves each series for x by using the Newton-Raphson method which shows quadratic convergence when the starting iterate is close enough to the solution x.

Field Summary
- Fields inherited from class umontreal.iro.lecuyer.probdist.ContinuousDistribution
  decPrec

Constructor Summary

Constructors
Constructor and Description
`BetaSymmetricalDist(double alpha)` Constructs a `BetaSymmetricalDist` object with parameters α = β = `alpha`, over the unit interval (0, 1).
`BetaSymmetricalDist(double alpha, int d)` Same as `BetaSymmetricalDist (alpha)`, but using approximations of roughly `d` decimal digits of precision when computing the distribution, complementary distribution, and inverse functions.

Method Summary

Methods
Modifier and Type	Method and Description
`static double`	`barF(double alpha, int d, double x)` Equivalent to `1 - cdf(alpha, d, x)`.
`double`	`cdf(double x)` Returns the distribution function F(x).
`static double`	`cdf(double alpha, int d, double x)` Same as `cdf` `(alpha, alpha, d, x)`.
`static double`	`density(double alpha, double x)` Returns the density evaluated at x.
`static BetaSymmetricalDist`	`getInstanceFromMLE(double[] x, int n)` Creates a new instance of a symmetrical beta distribution with parameter α estimated using the maximum likelihood method based on the n observations x[i], i = 0, 1,…, n - 1.
`static double[]`	`getMaximumLikelihoodEstimate(double[] x, int n)` Deprecated.
`double`	`getMean()` Returns the mean.
`static double`	`getMean(double alpha)` Computes and returns the mean E[X] = 1/2 of the symmetrical beta distribution with parameter α.
`static double[]`	`getMLE(double[] x, int n)` Estimates the parameter α of the symmetrical beta distribution over the interval [0, 1] using the maximum likelihood method, from the n observations x[i], i = 0, 1,…, n - 1.
`double[]`	`getParams()` Return a table containing the parameter of the current distribution.
`double`	`getStandardDeviation()` Returns the standard deviation.
`static double`	`getStandardDeviation(double alpha)` Computes and returns the standard deviation of the symmetrical beta distribution with parameter α.
`double`	`getVariance()` Returns the variance.
`static double`	`getVariance(double alpha)` Computes and returns the variance, Var[X] = 1/(8α + 4), of the symmetrical beta distribution with parameter α.
`double`	`inverseF(double u)` Returns the inverse distribution function x = F^-1(u).
`static double`	`inverseF(double alpha, double u)` Returns the inverse distribution function evaluated at u, for the symmetrical beta distribution over the interval [0, 1], with shape parameters 0 < α = β = `alpha`.
`void`	`setParams(double alpha, double beta, double a, double b, int d)`
`java.lang.String`	`toString()`

Methods inherited from class umontreal.iro.lecuyer.probdist.BetaDist
barF, barF, cdf, cdf, density, density, density, getA, getAlpha, getB, getBeta, getMean, getMean, getStandardDeviation, getStandardDeviation, getVariance, getVariance, inverseF, inverseF

Methods inherited from class umontreal.iro.lecuyer.probdist.ContinuousDistribution
barF, getXinf, getXsup, inverseBisection, inverseBrent, setXinf, setXsup

Methods inherited from class java.lang.Object
equals, getClass, hashCode, notify, notifyAll, wait, wait, wait

- Constructor Detail
  - BetaSymmetricalDist
```
public BetaSymmetricalDist(double alpha)
```
    Constructs a BetaSymmetricalDist object with parameters α = β = alpha, over the unit interval (0, 1).
  - BetaSymmetricalDist
```
public BetaSymmetricalDist(double alpha,
                   int d)
```
    Same as BetaSymmetricalDist (alpha), but using approximations of roughly d decimal digits of precision when computing the distribution, complementary distribution, and inverse functions.
- Method Detail
  - cdf
```
public double cdf(double x)
```
    Description copied from interface: Distribution
    
    Returns the distribution function F(x).
    
    Specified by:
    
    cdf in interface Distribution
    
    Overrides:
    
    cdf in class BetaDist
    
    Parameters:
    x - value at which the distribution function is evaluated
    
    Returns:
    distribution function evaluated at x
  - inverseF
```
public double inverseF(double u)
```
    Description copied from class: ContinuousDistribution
    
    Returns the inverse distribution function x = F^-1(u). Restrictions: u∈[0, 1].
    
    Specified by:
    
    inverseF in interface Distribution
    
    Overrides:
    
    inverseF in class BetaDist
    
    Parameters:
    u - value at which the inverse distribution function is evaluated
    
    Returns:
    the inverse distribution function evaluated at u
  - density
```
public static double density(double alpha,
             double x)
```
    Returns the density evaluated at x.
  - cdf
```
public static double cdf(double alpha,
         int d,
         double x)
```
    Same as cdf (alpha, alpha, d, x).
  - barF
```
public static double barF(double alpha,
          int d,
          double x)
```
    Equivalent to 1 - cdf(alpha, d, x).
  - inverseF
```
public static double inverseF(double alpha,
              double u)
```
    Returns the inverse distribution function evaluated at u, for the symmetrical beta distribution over the interval [0, 1], with shape parameters 0 < α = β = alpha. Uses four different hypergeometric series to compute the distribution u = F(x) (for the four cases x close to 0 and α < 1, x close to 0 and α > 1, x close to 1/2 and α < 1, and x close to 1/2 and α > 1), which are then solved by Newton's method for the solution of equations. For α > 100000, uses a normal approximation given in.
  - getMean
```
public double getMean()
```
    Description copied from class: ContinuousDistribution
    
    Returns the mean.
    
    Specified by:
    
    getMean in interface Distribution
    
    Overrides:
    
    getMean in class BetaDist
    
    Returns:
    the mean
  - getVariance
```
public double getVariance()
```
    Description copied from class: ContinuousDistribution
    
    Returns the variance.
    
    Specified by:
    
    getVariance in interface Distribution
    
    Overrides:
    
    getVariance in class BetaDist
    
    Returns:
    the variance
  - getStandardDeviation
```
public double getStandardDeviation()
```
    Description copied from class: ContinuousDistribution
    
    Returns the standard deviation.
    
    Specified by:
    
    getStandardDeviation in interface Distribution
    
    Overrides:
    
    getStandardDeviation in class BetaDist
    
    Returns:
    the standard deviation
  - getMLE
```
public static double[] getMLE(double[] x,
              int n)
```
    Estimates the parameter α of the symmetrical beta distribution over the interval [0, 1] 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.
    
    Parameters:
    x - the list of observations to use to evaluate parameters
    n - the number of observations to use to evaluate parameters
    
    Returns:
    returns the parameter [ hat(α)]
  - getMaximumLikelihoodEstimate
```
@Deprecated
public static double[] getMaximumLikelihoodEstimate(double[] x,
                                               int n)
```
    Deprecated.
    
    Same as getMLE.
  - getInstanceFromMLE
```
public static BetaSymmetricalDist getInstanceFromMLE(double[] x,
                                     int n)
```
    Creates a new instance of a symmetrical beta distribution with parameter α 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
```
public static double getMean(double alpha)
```
    Computes and returns the mean E[X] = 1/2 of the symmetrical beta distribution with parameter α.
    
    Returns:
    the mean of the symmetrical beta distribution E[X] = 1/2
  - getVariance
```
public static double getVariance(double alpha)
```
    Computes and returns the variance, Var[X] = 1/(8α + 4), of the symmetrical beta distribution with parameter α.
    
    Returns:
    the variance of the symmetrical beta distribution Var[X] = 1/[4(2α + 1)]
  - getStandardDeviation
```
public static double getStandardDeviation(double alpha)
```
    Computes and returns the standard deviation of the symmetrical beta distribution with parameter α.
    
    Returns:
    the standard deviation of the symmetrical beta distribution
  - setParams
```
public void setParams(double alpha,
             double beta,
             double a,
             double b,
             int d)
```
    Overrides:
    
    setParams in class BetaDist
  - getParams
```
public double[] getParams()
```
    Return a table containing the parameter of the current distribution.
    
    Specified by:
    
    getParams in interface Distribution
    
    Overrides:
    
    getParams in class BetaDist
  - toString
```
public java.lang.String toString()
```
    Overrides:
    
    toString in class BetaDist

Class BetaSymmetricalDist

Field Summary

Fields inherited from class umontreal.iro.lecuyer.probdist.ContinuousDistribution

Constructor Summary

Method Summary

Methods inherited from class umontreal.iro.lecuyer.probdist.BetaDist

Methods inherited from class umontreal.iro.lecuyer.probdist.ContinuousDistribution

Methods inherited from class java.lang.Object

Constructor Detail

BetaSymmetricalDist

BetaSymmetricalDist

Method Detail

cdf

inverseF

density

cdf

barF

inverseF

getMean

getVariance

getStandardDeviation

getMLE

getMaximumLikelihoodEstimate

getInstanceFromMLE

getMean

getVariance

getStandardDeviation

setParams

getParams

toString