Extends the class ContinuousDistribution for the noncentral chi-square distribution with \(\nu\) degrees of freedom and noncentrality parameter \(\lambda\), where \(\nu> 0\) and \(\lambda> 0\) [100] (page 436). More...
Public Member Functions | |
| ChiSquareNoncentralDist (double nu, double lambda) | |
Constructs a noncentral chi-square distribution with \(\nu= \) nu degrees of freedom and noncentrality parameter \(\lambda= \) lambda. | |
| 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 | getNu () |
| Returns the parameter \(\nu\) of this object. | |
| double | getLambda () |
| Returns the parameter \(\lambda\) of this object. | |
| void | setParams (double nu, double lambda) |
Sets the parameters \(\nu=\) nu and \(\lambda= \) lambda of this object. | |
| double [] | getParams () |
| Returns a table containing the parameters 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 (double nu, double lambda, double x) |
Computes the density function ( nc-fchi2 ) for a noncentral chi-square distribution with \(\nu= \) nu degrees of freedom and parameter \(\lambda= \) lambda. | |
| static double | cdf (double nu, double lambda, double x) |
Computes the noncentral chi-square distribution function ( nc-cdfchi2 ) with \(\nu= \) nu degrees of freedom and parameter \(\lambda= \) lambda. More... | |
| static double | barF (double nu, double lambda, double x) |
Computes the complementary noncentral chi-square distribution function with \(\nu= \) nu degrees of freedom and parameter \(\lambda= \) lambda. | |
| static double | inverseF (double nu, double lambda, double u) |
Computes the inverse of the noncentral chi-square distribution with \(\nu= \) nu degrees of freedom and parameter \(\lambda= \) lambda. | |
| static double | getMean (double nu, double lambda) |
Computes and returns the mean \(E[X] = \nu+ \lambda\) of the noncentral chi-square distribution with parameters \(\nu= \) nu and \(\lambda= \) lambda. More... | |
| static double | getVariance (double nu, double lambda) |
Computes and returns the variance \(\mbox{Var}[X] = 2(\nu+ 2\lambda)\) of the noncentral chi-square distribution with parameters \(\nu=\) nu and \(\lambda= \) lambda. More... | |
| static double | getStandardDeviation (double nu, double lambda) |
Computes and returns the standard deviation of the noncentral chi-square distribution with parameters \(\nu=\) nu and \(\lambda= \) lambda. More... | |
Static Protected Attributes | |
| static final int | PREC = 15 |
| static final double | EPS = Num.TEN_NEG_POW[PREC] |
| Static Protected Attributes inherited from ContinuousDistribution | |
| static final double | XBIG = 100.0 |
| static final double | XBIGM = 1000.0 |
| static final double [] | EPSARRAY |
Additional Inherited Members | |
| Public Attributes inherited from ContinuousDistribution | |
| int | decPrec = 15 |
| Protected Attributes inherited from ContinuousDistribution | |
| double | supportA = Double.NEGATIVE_INFINITY |
| double | supportB = Double.POSITIVE_INFINITY |
Detailed Description
Extends the class ContinuousDistribution for the noncentral chi-square distribution with \(\nu\) degrees of freedom and noncentrality parameter \(\lambda\), where \(\nu> 0\) and \(\lambda> 0\) [100] (page 436).
\[ f(x) =\frac{e^{-(x + \lambda)/2}}{2} \left(\frac{x}{\lambda}\right)^{(\nu-2)/4} I_{\nu/2 - 1}\left(\sqrt{\lambda x}\right) \qquad\mbox{for } x > 0, \tag{nc-fchi2} \]
where \(I_{\nu}(x)\) is the modified Bessel function of the first kind of order \(\nu\) given by
\[ I_{\nu}(z) = \sum_{j=0}^{\infty}\frac{(z/2)^{\nu+ 2j}}{j!\; \Gamma(\nu+ j +1)}, \]
where \(\Gamma(x)\) is the gamma function. Notice that this distribution is more general than the chi-square distribution since its number of degrees of freedom can be any positive real number. For \(\lambda= 0\) and \(\nu\) a positive integer, we have the ordinary chi-square distribution.
The cumulative probability function can be written as
\[ P[X \le x] = \sum_{j=0}^{\infty}\frac{e^{-\lambda/2}(\lambda/2)^j}{j!} P[\chi^2_{\nu+ 2j} \le x], \tag{nc-cdfchi2} \]
where \(\chi^2_{\nu+ 2j}\) is the central chi-square distribution with \(\nu+ 2j\) degrees of freedom.
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() [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 |
Computes the noncentral chi-square distribution function ( nc-cdfchi2 ) with \(\nu= \) nu degrees of freedom and parameter \(\lambda= \) lambda.
For \(\lambda< 1000\), we use a reasonnably efficient method of summing the cumulative function ( nc-cdfchi2 ) by using recurrence relations to compute both the Poisson and the chi-square probabilities in terms of neighboring probability terms. For \(\lambda\ge1000\), we use the normal approximation given in [203] , except very near the mean where we use the central chi square approximation from [200] . This function returns at least 6 decimal degits of precision nearly everywhere.
◆ getMean()
|
static |
Computes and returns the mean \(E[X] = \nu+ \lambda\) of the noncentral chi-square distribution with parameters \(\nu= \) nu and \(\lambda= \) lambda.
- Returns
- the mean of the Noncentral noncentral chi-square distribution
◆ getStandardDeviation()
|
static |
Computes and returns the standard deviation of the noncentral chi-square distribution with parameters \(\nu=\) nu and \(\lambda= \) lambda.
- Returns
- the standard deviation of the noncentral chi-square distribution
◆ getVariance()
|
static |
Computes and returns the variance \(\mbox{Var}[X] = 2(\nu+ 2\lambda)\) of the noncentral chi-square distribution with parameters \(\nu=\) nu and \(\lambda= \) lambda.
- Returns
- the variance of the noncentral chi-square distribution
◆ inverseF()
| 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.
The documentation for this class was generated from the following file:
- ChiSquareNoncentralDist.java
