org.apache.commons.statistics.distribution

Class FDistribution

java.lang.Object

org.apache.commons.statistics.distribution.FDistribution

All Implemented Interfaces:

ContinuousDistribution

public final class FDistribution
extends Object

Implementation of the F-distribution.

The probability density function of $X$ is:

$$\begin{array}{rl}f(x;n,m)& =\frac{1}{\mathrm{B}(\frac{n}{2},\frac{m}{2})}{\left(\frac{n}{m}\right)}^{n/2}{x}^{n/2-1}{(1+\frac{n}{m}x)}^{-(n+m)/2}\\ & =\frac{n^{\frac{n}{2}}{m}^{\frac{m}{2}}{x}^{\frac{n}{2}-1}}{(nx+m{)}^{\frac{(n+m)}{2}}\mathrm{B}(\frac{n}{2},\frac{m}{2})}\end{array}$$

for $n,m>0$ the degrees of freedom, $\mathrm{B}(a,b)$ is the beta function, and $x\in [0,\mathrm{\infty })$.

See Also:

F-distribution (Wikipedia), F-distribution (MathWorld)

Nested Class Summary

Nested classes/interfaces inherited from interface org.apache.commons.statistics.distribution.ContinuousDistribution

ContinuousDistribution.Sampler

Method Summary

Modifier and Type	Method and Description
ContinuousDistribution.Sampler	createSampler(UniformRandomProvider rng) Creates a sampler.
double	cumulativeProbability(double x) For a random variable X whose values are distributed according to this distribution, this method returns P(X <= x).
double	density(double x) Returns the probability density function (PDF) of this distribution evaluated at the specified point x.
double	getDenominatorDegreesOfFreedom() Gets the denominator degrees of freedom parameter of this distribution.
double	getMean() Gets the mean of this distribution.
double	getNumeratorDegreesOfFreedom() Gets the numerator degrees of freedom parameter of this distribution.
double	getSupportLowerBound() Gets the lower bound of the support.
double	getSupportUpperBound() Gets the upper bound of the support.
double	getVariance() Gets the variance of this distribution.
double	inverseCumulativeProbability(double p) Computes the quantile function of this distribution.
double	inverseSurvivalProbability(double p) Computes the inverse survival probability function of this distribution.
double	logDensity(double x) Returns the natural logarithm of the probability density function (PDF) of this distribution evaluated at the specified point x.
static FDistribution	of(double numeratorDegreesOfFreedom, double denominatorDegreesOfFreedom) Creates an F-distribution.
double	probability(double x0, double x1) For a random variable X whose values are distributed according to this distribution, this method returns P(x0 < X <= x1).
double	survivalProbability(double x) For a random variable X whose values are distributed according to this distribution, this method returns P(X > x).

Methods inherited from class java.lang.Object

clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

Method Detail

of

public static FDistribution of(double numeratorDegreesOfFreedom,
                               double denominatorDegreesOfFreedom)

Creates an F-distribution.

Parameters:

numeratorDegreesOfFreedom - Numerator degrees of freedom.

denominatorDegreesOfFreedom - Denominator degrees of freedom.

Returns:

the distribution

Throws:

IllegalArgumentException - if numeratorDegreesOfFreedom <= 0 or denominatorDegreesOfFreedom <= 0.

getNumeratorDegreesOfFreedom

public double getNumeratorDegreesOfFreedom()

Gets the numerator degrees of freedom parameter of this distribution.

Returns:

the numerator degrees of freedom.

getDenominatorDegreesOfFreedom

public double getDenominatorDegreesOfFreedom()

Gets the denominator degrees of freedom parameter of this distribution.

Returns:

the denominator degrees of freedom.

density

public double density(double x)

Returns the probability density function (PDF) of this distribution evaluated at the specified point x. In general, the PDF is the derivative of the CDF. If the derivative does not exist at x, then an appropriate replacement should be returned, e.g. Double.POSITIVE_INFINITY, Double.NaN, or the limit inferior or limit superior of the difference quotient.

Returns the limit when x = 0:

• df1 < 2: Infinity
• df1 == 2: 1
• df1 > 2: 0

Where df1 is the numerator degrees of freedom.

Parameters:

x - Point at which the PDF is evaluated.

Returns:

the value of the probability density function at x.

logDensity

public double logDensity(double x)

Returns the natural logarithm of the probability density function (PDF) of this distribution evaluated at the specified point x.

Returns the limit when x = 0:

• df1 < 2: Infinity
• df1 == 2: 0
• df1 > 2: -Infinity

Where df1 is the numerator degrees of freedom.

Parameters:

x - Point at which the PDF is evaluated.

Returns:

the logarithm of the value of the probability density function at x.

cumulativeProbability

public double cumulativeProbability(double x)

For a random variable X whose values are distributed according to this distribution, this method returns P(X <= x). In other words, this method represents the (cumulative) distribution function (CDF) for this distribution.

Parameters:

x - Point at which the CDF is evaluated.

Returns:

the probability that a random variable with this distribution takes a value less than or equal to x.

survivalProbability

public double survivalProbability(double x)

For a random variable X whose values are distributed according to this distribution, this method returns P(X > x). In other words, this method represents the complementary cumulative distribution function.

By default, this is defined as 1 - cumulativeProbability(x), but the specific implementation may be more accurate.

Parameters:

x - Point at which the survival function is evaluated.

Returns:

the probability that a random variable with this distribution takes a value greater than x.

getMean

public double getMean()

Gets the mean of this distribution.

For denominator degrees of freedom parameter $m$, the mean is:

$$\mathbb{E}[X]=\{\begin{array}{ll}\frac{m}{m-2}& \text{for}m>2\\ \text{undefined}& \text{otherwise}\end{array}$$

Returns:

the mean, or NaN if it is not defined.

getVariance

public double getVariance()

Gets the variance of this distribution.

For numerator degrees of freedom parameter $n$ and denominator degrees of freedom parameter $m$, the variance is:

$$\mathrm{var}[X]=\{\begin{array}{ll}\frac{2m^2(n+m-2)}{n(m-2{)}^2(m-4)}& \text{for}m>4\\ \text{undefined}& \text{otherwise}\end{array}$$

Returns:

the variance, or NaN if it is not defined.

getSupportLowerBound

public double getSupportLowerBound()

Gets the lower bound of the support. It must return the same value as inverseCumulativeProbability(0), i.e. $inf\{x\in \mathbb{R}:P(X\le x)>0\}$.

The lower bound of the support is always 0.

Returns:

0.

getSupportUpperBound

public double getSupportUpperBound()

Gets the upper bound of the support. It must return the same value as inverseCumulativeProbability(1), i.e. $inf\{x\in \mathbb{R}:P(X\le x)=1\}$.

The upper bound of the support is always positive infinity.

Returns:

positive infinity.

probability

public double probability(double x0,
                          double x1)

For a random variable X whose values are distributed according to this distribution, this method returns P(x0 < X <= x1). The default implementation uses the identity P(x0 < X <= x1) = P(X <= x1) - P(X <= x0)

Specified by:

probability in interface ContinuousDistribution

Parameters:

x0 - Lower bound (exclusive).

x1 - Upper bound (inclusive).

Returns:

the probability that a random variable with this distribution takes a value between x0 and x1, excluding the lower and including the upper endpoint.

inverseCumulativeProbability

public double inverseCumulativeProbability(double p)

Computes the quantile function of this distribution. For a random variable X distributed according to this distribution, the returned value is:

$$x=\{\begin{array}{ll}inf\{x\in \mathbb{R}:P(X\le x)\ge p\}& \text{for}0<p\le 1\\ inf\{x\in \mathbb{R}:P(X\le x)>0\}& \text{for}p=0\end{array}$$

The default implementation returns:

• ContinuousDistribution.getSupportLowerBound() for p = 0,
• ContinuousDistribution.getSupportUpperBound() for p = 1, or
• the result of a search for a root between the lower and upper bound using cumulativeProbability(x) - p. The bounds may be bracketed for efficiency.

Specified by:

inverseCumulativeProbability in interface ContinuousDistribution

Parameters:

p - Cumulative probability.

Returns:

the smallest p-quantile of this distribution (largest 0-quantile for p = 0).

Throws:

IllegalArgumentException - if p < 0 or p > 1

inverseSurvivalProbability

public double inverseSurvivalProbability(double p)

Computes the inverse survival probability function of this distribution. For a random variable X distributed according to this distribution, the returned value is:

$$x=\{\begin{array}{ll}inf\{x\in \mathbb{R}:P(X\ge x)\le p\}& \text{for}0\le p<1\\ inf\{x\in \mathbb{R}:P(X\ge x)<1\}& \text{for}p=1\end{array}$$

By default, this is defined as inverseCumulativeProbability(1 - p), but the specific implementation may be more accurate.

The default implementation returns:

• ContinuousDistribution.getSupportLowerBound() for p = 1,
• ContinuousDistribution.getSupportUpperBound() for p = 0, or
• the result of a search for a root between the lower and upper bound using survivalProbability(x) - p. The bounds may be bracketed for efficiency.

Specified by:

inverseSurvivalProbability in interface ContinuousDistribution

Parameters:

p - Survival probability.

Returns:

the smallest (1-p)-quantile of this distribution (largest 0-quantile for p = 1).

Throws:

IllegalArgumentException - if p < 0 or p > 1

createSampler

public ContinuousDistribution.Sampler createSampler(UniformRandomProvider rng)

Creates a sampler.

Specified by:

createSampler in interface ContinuousDistribution

Parameters:

rng - Generator of uniformly distributed numbers.

Returns:

a sampler that produces random numbers according this distribution.