Class 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:
    F-distribution (Wikipedia), F-distribution (MathWorld)
    • Field Detail

      • SUPPORT_LO

        private static final double SUPPORT_LO
        Support lower bound.
        See Also:
        Constant Field Values
      • SUPPORT_HI

        private static final double SUPPORT_HI
        Support upper bound.
        See Also:
        Constant Field Values
      • 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:
        Constant Field Values
      • 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:
        Constant Field Values
      • 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 Detail

      • FDistribution

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