Package org.apache.commons.statistics.distribution

Class TDistribution

java.lang.Object

org.apache.commons.statistics.distribution.TDistribution

All Implemented Interfaces:

ContinuousDistribution

public abstract class TDistribution
extends Object

Implementation of Student's t-distribution.

The probability density function of $X$ is:

$$f(x;v)=\frac{\mathrm{\Gamma }(\frac{\nu +1}{2})}{\sqrt{\nu \pi }\mathrm{\Gamma }(\frac{\nu }{2})}{(1+\frac{t^2}{\nu })}^{-\frac{\nu +1}{2}}$$

for $v>0$ the degrees of freedom, $\mathrm{\Gamma }$ is the gamma function, and $x\in (-\mathrm{\infty },\mathrm{\infty })$.

See Also:

Student's t-distribution (Wikipedia), Student's t-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	Description
ContinuousDistribution.Sampler	createSampler(org.apache.commons.rng.UniformRandomProvider rng)	Creates a sampler.
double	getDegreesOfFreedom()	Gets the degrees of freedom parameter of this distribution.
abstract double	getMean()	Gets the mean of this distribution.
double	getSupportLowerBound()	Gets the lower bound of the support.
double	getSupportUpperBound()	Gets the upper bound of the support.
abstract 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.
static TDistribution	of(double degreesOfFreedom)	Creates a Student's t-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

Methods inherited from interface org.apache.commons.statistics.distribution.ContinuousDistribution

cumulativeProbability, density, logDensity

Method Detail

of

public static TDistribution of(double degreesOfFreedom)

Creates a Student's t-distribution.

Parameters:

degreesOfFreedom - Degrees of freedom.

Returns:

the distribution

Throws:

IllegalArgumentException - if degreesOfFreedom <= 0

getDegreesOfFreedom

public double getDegreesOfFreedom()

Gets the degrees of freedom parameter of this distribution.

Returns:

the degrees of freedom.

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.

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:

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

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).

getMean

public abstract double getMean()

Gets the mean of this distribution.

For degrees of freedom parameter $v$, the mean is:

$$\mathbb{E}[X]=\{\begin{array}{ll}0& \text{for}v>1\\ \text{undefined}& \text{otherwise}\end{array}$$

Returns:

the mean, or NaN if it is not defined.

getVariance

public abstract double getVariance()

Gets the variance of this distribution.

For degrees of freedom parameter $v$, the variance is:

$$\mathrm{var}[X]=\{\begin{array}{ll}\frac{v}{v-2}& \text{for}v>2\\ \mathrm{\infty }& \text{for}1<v\le 2\\ \text{undefined}& \text{otherwise}\end{array}$$

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 negative infinity.

Returns:

negative infinity.

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

createSampler

public ContinuousDistribution.Sampler createSampler(org.apache.commons.rng.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.