Class FDistribution

java.lang.Object
org.apache.commons.statistics.distribution.AbstractContinuousDistribution
org.apache.commons.statistics.distribution.FDistribution
All Implemented Interfaces:
ContinuousDistribution
public final class FDistribution extends AbstractContinuousDistribution
Implementation of the F-distribution.

The probability density function of \( X \) is:

\[ \begin{aligned} f(x; n, m) &= \frac{1}{\operatorname{B}\left(\frac{n}{2},\frac{m}{2}\right)} \left(\frac{n}{m}\right)^{n/2} x^{n/2 - 1} \left(1+\frac{n}{m} \, x \right)^{-(n+m)/2} \\ &= \frac{n^{\frac n 2} m^{\frac m 2} x^{\frac{n}{2}-1} }{ (nx+m)^{\frac{(n+m)}{2}} \operatorname{B}\left(\frac{n}{2},\frac{m}{2}\right)} \end{aligned} \]

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

See Also:

  • Field Details

    • SUPPORT_LO

      private static final double SUPPORT_LO
      Support lower bound.
      See Also:

    • SUPPORT_HI

      private static final double SUPPORT_HI
      Support upper bound.
      See Also:

    • MIN_DENOMINATOR_DF_FOR_MEAN

      private static final double MIN_DENOMINATOR_DF_FOR_MEAN
      The minimum degrees of freedom for the denominator when computing the mean.
      See Also:

    • MIN_DENOMINATOR_DF_FOR_VARIANCE

      private static final double MIN_DENOMINATOR_DF_FOR_VARIANCE
      The minimum degrees of freedom for the denominator when computing the variance.
      See Also:

    • numeratorDegreesOfFreedom

      private final double numeratorDegreesOfFreedom
      The numerator degrees of freedom.

    • denominatorDegreesOfFreedom

      private final double denominatorDegreesOfFreedom
      The denominator degrees of freedom.

    • nHalfLogNmHalfLogM

      private final double nHalfLogNmHalfLogM
      n/2 * log(n) + m/2 * log(m) with n = numerator DF and m = denominator DF.

    • logBetaNhalfMhalf

      private final double logBetaNhalfMhalf
      LogBeta(n/2, n/2) with n = numerator DF.

    • mean

      private final double mean
      Cached value for inverse probability function.

    • variance

      private final double variance
      Cached value for inverse probability function.

  • Constructor Details

    • FDistribution

      private FDistribution(double numeratorDegreesOfFreedom, double denominatorDegreesOfFreedom)
      Parameters:
      numeratorDegreesOfFreedom - Numerator degrees of freedom.
      denominatorDegreesOfFreedom - Denominator degrees of freedom.

  • Method Details

    • 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.

    • computeDensity

      private double computeDensity(double x, boolean log)
      Compute the density at point x. Assumes x is within the support bound.
      Parameters:
      x - Value
      log - true to compute the log density
      Returns:
      pdf(x) or logpdf(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:

      \[ \mathbb{E}[X] = \begin{cases} \frac{m}{m-2} & \text{for } m \gt 2 \\ \text{undefined} & \text{otherwise} \end{cases} \]

      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:

      \[ \operatorname{var}[X] = \begin{cases} \frac{2m^2 (n+m-2)}{n (m-2)^2 (m-4)} & \text{for } m \gt 4 \\ \text{undefined} & \text{otherwise} \end{cases} \]

      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. \( \inf \{ x \in \mathbb R : P(X \le x) \gt 0 \} \).

      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.