Specializes the class BetaDist to the case of a symmetrical beta distribution over the interval \([0,1]\), with shape parameters \(\alpha= \beta\). More...
Inheritance diagram for BetaSymmetricalDist:
Collaboration diagram for BetaSymmetricalDist:
Public Member Functions | |
| BetaSymmetricalDist (double alpha) | |
Constructs a BetaSymmetricalDist object with parameters \(\alpha= \beta=\) 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. | |
| 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. | |
| void | setParams (double alpha, double beta, double a, double b, int d) |
| 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 BetaDist | |
| BetaDist (double alpha, double beta) | |
Constructs a BetaDist object with parameters \(\alpha=\) alpha, \(\beta=\) beta and default domain \([0,1]\). | |
| BetaDist (double alpha, double beta, double a, double b) | |
Constructs a BetaDist object with parameters \(\alpha=\) alpha, \(\beta=\) beta and domain \([\)a \(,\) b \(]\). | |
| BetaDist (double alpha, double beta, int d) | |
| BetaDist (double alpha, double beta, double a, double b, int d) | |
| 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 | getAlpha () |
| Returns the parameter \(\alpha\) of this object. | |
| double | getBeta () |
| Returns the parameter \(\beta\) of this object. | |
| double | getA () |
| Returns the parameter \(a\) of this object. | |
| double | getB () |
| Returns the parameter \(b\) of this object. | |
| void | setParams (double alpha, double beta, double a, double b, int d) |
| void | setParams (double alpha, double beta, double a, double b) |
| Sets the parameters of the current distribution. More... | |
| double [] | getParams () |
| Return an array containing the parameters of the current distribution as [ \(\alpha\), \(\beta\), \(a\), \(b\)]. | |
| 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 alpha, double x) |
| Returns the density evaluated at \(x\). | |
| static double | cdf (double alpha, int d, double x) |
| Same as cdf(alpha, alpha, d, x). | |
| static double | barF (double alpha, int d, double x) |
| Returns the complementary distribution function. | |
| 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 \(0 < \alpha= \beta\) = alpha. More... | |
| static double [] | getMLE (double[] x, int n) |
| Estimates the parameter \(\alpha\) of the symmetrical beta distribution over the interval [0, 1] using the maximum likelihood method, from the \(n\) observations \(x[i]\), \(i = 0, 1, …, n-1\). More... | |
| static BetaSymmetricalDist | getInstanceFromMLE (double[] x, int n) |
| Creates a new instance of a symmetrical beta distribution with parameter \(\alpha\) estimated using the maximum likelihood method based on the \(n\) observations \(x[i]\), \(i = 0, 1, …, n-1\). More... | |
| static double | getMean (double alpha) |
| Computes and returns the mean \(E[X] = 1/2\) of the symmetrical beta distribution with parameter \(\alpha\). More... | |
| static double | getVariance (double alpha) |
| Computes and returns the variance, \(\mbox{Var}[X] = 1/(8\alpha+ 4)\), of the symmetrical beta distribution with parameter \(\alpha\). More... | |
| static double | getStandardDeviation (double alpha) |
| Computes and returns the standard deviation of the symmetrical beta distribution with parameter \(\alpha\). More... | |
| Static Public Member Functions inherited from BetaDist | |
| static double | density (double alpha, double beta, double x) |
| Same as density(alpha, beta, 0, 1, x). | |
| static double | density (double alpha, double beta, double a, double b, double x) |
| Computes the density function of the beta distribution. | |
| static double | cdf (double alpha, double beta, int d, double x) |
| static double | cdf (double alpha, double beta, double a, double b, int d, double x) |
| static double | barF (double alpha, double beta, int d, double x) |
| static double | barF (double alpha, double beta, double a, double b, int d, double x) |
| static double | cdf (double alpha, double beta, double x) |
| Same as cdf(alpha, beta, 0, 1, x). | |
| static double | cdf (double alpha, double beta, double a, double b, double x) |
| Computes the distribution function. More... | |
| static double | barF (double alpha, double beta, double x) |
| Same as barF(alpha, beta, 0, 1, x). | |
| static double | barF (double alpha, double beta, double a, double b, double x) |
| Computes the complementary distribution function. | |
| static double | inverseF (double alpha, double beta, int d, double u) |
| static double | inverseF (double alpha, double beta, double u) |
| Same as inverseF(alpha, beta, 0, 1, u). | |
| static double | inverseF (double alpha, double beta, double a, double b, int d, double u) |
| static double | inverseF (double alpha, double beta, double a, double b, double u) |
| Returns the inverse beta distribution function using the algorithm implemented in [183] . More... | |
| static double [] | getMLE (double[] x, int n) |
| Estimates the parameters \((\alpha,\beta)\) of the beta distribution over the interval \([0,1]\) using the maximum likelihood method, from the \(n\) observations \(x[i]\), \(i = 0, 1,…, n-1\). More... | |
| static BetaDist | getInstanceFromMLE (double[] x, int n) |
| Creates a new instance of a beta distribution with parameters \(\alpha\) and \(\beta\) over the interval \([0,1]\) estimated using the maximum likelihood method based on the \(n\) observations \(x[i]\), \(i = 0, 1, …, n-1\). More... | |
| static double | getMean (double alpha, double beta) |
| Computes and returns the mean \(E[X] = \alpha/ (\alpha+ \beta)\) of the beta distribution with parameters \(\alpha\) and \(\beta\), over the interval \([0, 1]\). More... | |
| static double | getMean (double alpha, double beta, double a, double b) |
| Computes and returns the mean \(E[X] = (b\alpha+ a\beta)/ (\alpha+ \beta)\) of the beta distribution with parameters \(\alpha\) and \(\beta\) over the interval \([a, b]\). More... | |
| static double | getVariance (double alpha, double beta) |
| Computes and returns the variance \(\mbox{Var}[X] = \frac{\alpha\beta}{(\alpha+ \beta)^2 (\alpha+ \beta+ 1)}\) of the beta distribution with parameters \(\alpha\) and \(\beta\), over the interval \([0, 1]\). More... | |
| static double | getVariance (double alpha, double beta, double a, double b) |
| Computes and returns the variance \(\mbox{Var}[X] = \frac{\alpha\beta(b-a)^2}{(\alpha+ \beta)^2 (\alpha+ \beta+ 1)}\) of the beta distribution with parameters \(\alpha\) and \(\beta\), over the interval \([a, b]\). More... | |
| static double | getStandardDeviation (double alpha, double beta) |
| Computes the standard deviation of the beta distribution with parameters \(\alpha\) and \(\beta\), over the interval \([0, 1]\). More... | |
| static double | getStandardDeviation (double alpha, double beta, double a, double b) |
| Computes the standard deviation of the beta distribution with parameters \(\alpha\) and \(\beta\), over the interval \([a, b]\). More... | |
Additional Inherited Members | |
| Public Attributes inherited from ContinuousDistribution | |
| int | decPrec = 15 |
| Protected Attributes inherited from BetaDist | |
| double | alpha |
| double | beta |
| double | a |
| double | bminusa |
| double | logFactor |
| double | Beta |
| double | logBeta |
| Protected Attributes inherited from ContinuousDistribution | |
| double | supportA = Double.NEGATIVE_INFINITY |
| double | supportB = Double.POSITIVE_INFINITY |
| Static Protected Attributes inherited from ContinuousDistribution | |
| static final double | XBIG = 100.0 |
| static final double | XBIGM = 1000.0 |
| static final double [] | EPSARRAY |
| Static Package Functions inherited from BetaDist | |
| static double | beta_g (double x) |
| Package Attributes inherited from BetaDist | |
| double | b |
Detailed Description
Specializes the class BetaDist to the case of a symmetrical beta distribution over the interval \([0,1]\), with shape parameters \(\alpha= \beta\).
Faster methods are implemented here for this special case [134] . Because of the symmetry around 1/2, four series are used to compute the cdf, two around \(x = 0\) and two around \(x = 1/2\).
Member Function Documentation
◆ barF()
| 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.
◆ cdf()
| 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.
◆ getInstanceFromMLE()
|
static |
Creates a new instance of a symmetrical beta distribution with parameter \(\alpha\) 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()
|
static |
Computes and returns the mean \(E[X] = 1/2\) of the symmetrical beta distribution with parameter \(\alpha\).
- Returns
- the mean of the symmetrical beta distribution \(E[X] = 1/2\)
◆ getMLE()
|
static |
Estimates the parameter \(\alpha\) 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. The maximum likelihood estimator \(\hat{\alpha}\) satisfies the equation
\begin{align*} \psi(\hat{\alpha}) - \psi(2\hat{\alpha}) = \frac{1}{2n} \sum_{i=1}^n \ln(x_i(1 - x_i)) \end{align*}
where \(\bar{x}_n\) is the average of \(x[0], …, x[n-1]\), and \(\psi\) is the logarithmic derivative of the Gamma function \(\psi(x) = \Gamma’(x) / \Gamma(x)\).
- 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 [ \(\hat{\alpha}\)]
◆ getStandardDeviation()
|
static |
Computes and returns the standard deviation of the symmetrical beta distribution with parameter \(\alpha\).
- Returns
- the standard deviation of the symmetrical beta distribution
◆ getVariance()
|
static |
Computes and returns the variance, \(\mbox{Var}[X] = 1/(8\alpha+ 4)\), of the symmetrical beta distribution with parameter \(\alpha\).
- Returns
- the variance of the symmetrical beta distribution \(\mbox{Var}[X] = 1 / [4 (2\alpha+ 1)]\)
◆ 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 |
Returns the inverse distribution function evaluated at \(u\), for the symmetrical beta distribution over the interval \([0,1]\), with shape parameters \(0 < \alpha= \beta\) = alpha.
Uses four different hypergeometric series to compute the distribution \(u = F(x)\) (for the four cases \(x\) close to 0 and \(\alpha< 1\), \(x\) close to 0 and \(\alpha> 1\), \(x\) close to 1/2 and \(\alpha< 1\), and \(x\) close to 1/2 and \(\alpha> 1\)), which are then solved by Newton’s method for the solution of equations. For \(\alpha> 100000\), uses a normal approximation given in [201] .
The documentation for this class was generated from the following file:
- BetaSymmetricalDist.java
