represents a noncentral F-ratio distribution with n numerator degrees of freedom, m denominator degrees of freedom, and numerator noncentrality parameter λ.

NoncentralFRatioDistribution[n,m,λ, η]

represents a doubly noncentral F-ratio distribution with numerator noncentrality parameter λ and denominator noncentrality parameter η.


  • The noncentral F-ratio distribution is the distribution of the ratio of a noncentral random variable and a random variable divided by their respective degrees of freedom.
  • The doubly noncentral F-ratio distribution is the distribution of a ratio of two noncentral -distributed random variables divided by their respective degrees of freedom.
  • NoncentralFRatioDistribution allows n and m to be any positive real numbers, and λ and η to be any non-negative real numbers.
  • NoncentralFRatioDistribution allows n, m, λ, and η to be dimensionless quantities.
  • NoncentralFRatioDistribution can be used with such functions as Mean, CDF, and RandomVariate.

Background & Context

  • NoncentralFRatioDistribution[n,m,λ,η] represents a continuous statistical distribution supported over the interval and defined as the distribution of the ratio where Y1NoncentralChiSquareDistribution[n,λ] and Y2NoncentralChiSquareDistribution[m,η] are independent variates with n and m degrees of freedom, respectively, and with parameters λ and η of noncentrality, respectively. Depending on the values of n, m, λ, and η, the probability density function (PDF) may be either unimodal or monotonically decreasing, with a potential singularity nearing the lower endpoint of its domain. In addition, the tails of the PDF are "fat" in the sense that the PDF decreases algebraically rather than decreasing exponentially for large values of . (This behavior can be made quantitatively precise by analyzing the SurvivalFunction of the distribution.) The four-parameter form NoncentralFRatioDistribution[n,m,λ,η] is commonly called the doubly noncentral F-ratio distribution, while the three-argument form NoncentralFRatioDistribution[n,m,λ] (which is most often referred to as "the" noncentral F-ratio distribution) is equivalent to NoncentralFRatioDistribution[n,m,λ,λ] and is sometimes referred to as the noncentral FisherSnedecor distribution or Snedecor's noncentral F-distribution.
  • The noncentral F-ratio distribution was first derived in the late 1930s, though its properties remained largely uninvestigated until the late-1940s work of Patnaik. Named by Patnaik, the noncentral F-ratio distribution has been used to study the properties of analysis of variance tests under so-called nonstandard conditions and has itself been the catalyst for much research in the fields of computer science, numerical analysis, and approximation theory. Many of the most well-known applications of the distribution are in statistics, where it is used e.g. to compute the powers of a test based on a central F-statistic (for example, in tests based on the Hotelling test). The noncentral F-ratio distribution is also used to study multivariate calibration problems via multiple-use confidence estimation.
  • RandomVariate can be used to give one or more machine- or arbitrary-precision (the latter via the WorkingPrecision option) pseudorandom variates from a noncentral beta distribution. Distributed[x,NoncentralFRatioDistribution[n,m,λ,η]], written more concisely as xNoncentralFRatioDistribution[n,m,λ,η], can be used to assert that a random variable x is distributed according to a noncentral beta distribution. Such an assertion can then be used in functions such as Probability, NProbability, Expectation, and NExpectation.
  • The probability density and cumulative distribution functions for noncentral beta distributions may be given using PDF[NoncentralFRatioDistribution[n,m,λ,η],x] and CDF[NoncentralFRatioDistribution[n,m,λ,η],x]. The mean, median, variance, raw moments, and central moments may be computed using Mean, Median, Variance, Moment, and CentralMoment, respectively.
  • DistributionFitTest can be used to test if a given dataset is consistent with a noncentral beta distribution, EstimatedDistribution to estimate a noncentral beta parametric distribution from given data, and FindDistributionParameters to fit data to a noncentral beta distribution. ProbabilityPlot can be used to generate a plot of the CDF of given data against the CDF of a symbolic noncentral beta distribution, and QuantilePlot to generate a plot of the quantiles of given data against the quantiles of a symbolic noncentral beta distribution.
  • TransformedDistribution can be used to represent a transformed noncentral beta distribution, CensoredDistribution to represent the distribution of values censored between upper and lower values, and TruncatedDistribution to represent the distribution of values truncated between upper and lower values. CopulaDistribution can be used to build higher-dimensional distributions that contain a noncentral beta distribution, and ProductDistribution can be used to compute a joint distribution with independent component distributions involving noncentral beta distributions.
  • NoncentralFRatioDistribution is related to a number of other distributions. It is an immediate generalization of FRatioDistribution, in the sense that the PDF of both NoncentralFRatioDistribution[n,m,0,0] and NoncentralFRatioDistribution[n,m,0] are precisely the same as that of FRatioDistribution[n,m]. NoncentralFRatioDistribution can be realized as a transformation (TransformedDistribution) of both NoncentralChiSquareDistribution and NoncentralBetaDistribution and is also closely related to ChiDistribution, ChiSquareDistribution, StudentTDistribution, LaplaceDistribution, and FisherZDistribution.


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Basic Examples  (5)

Probability density function:

Probability density function for a doubly noncentral F-ratio distribution:

Cumulative distribution function:

Cumulative distribution function for a doubly noncentral F-ratio distribution:

Mean and variance:

Mean and variance for doubly noncentral:

Scope  (8)

Generate a sample of pseudorandom numbers from a noncentral F-ratio distribution:

Compare its histogram to the PDF:

Distribution parameters estimation:

Estimate the distribution parameters from sample data:

Compare the density histogram of the sample with the PDF of the estimated distribution:

Skewness depends on degrees of freedom m, n as well as noncentrality λ:

Kurtosis depends on degrees of freedom m, n as well as noncentrality λ:

Different moments with closed forms as functions of parameters:


Closed form for symbolic order:




Hazard function:

Hazard function of a doubly noncentral F-ratio distribution:

Quantile function:

Applications  (1)

NoncentralFRatioDistribution appears in computation of the power function of a hypothesis test about coefficients of linear model fit. The following 21 sample points were measured in an experiment:

Construct the linear model for the data in the form :

Hypothesis testing about coefficients and having simultaneously particular values is done using the statistic that follows FRatioDistribution with 2 and 19 degrees of freedom, respectively:

Compute the value of the statistic under the null hypothesis that and :

The critical value at the 5% significance level:

Hence the alternative hypothesis cannot be rejected:

Assuming that the true values are actually 1.37 and 2.88, statistic will follow a NoncentralFRatioDistribution with noncentrality parameter :

The power of the test, assuming true values of and :

Plot the power as a function of the noncentrality parameter:

Properties & Relations  (6)

Relationships to other distributions:

Noncentral F-ratio distribution simplifies to FRatioDistribution:

Doubly noncentral F-ratio distribution simplifies to FRatioDistribution:

Doubly noncentral F-ratio distribution simplifies to noncentral F-ratio distribution:

The ratio of two NoncentralChiSquareDistribution follows a noncentral F-ratio distribution:

NoncentralBetaDistribution is a transformation of NoncentralFRatioDistribution:

Possible Issues  (4)

NoncentralFRatioDistribution is not defined when n or m is not a positive real number:

NoncentralFRatioDistribution is not defined when λ is not a non-negative real number:

The characteristic function of a noncentral F-ratio distribution has no closed-form representation:

Substitution of invalid parameters into symbolic outputs gives results that are not meaningful:

Neat Examples  (1)

PDFs for different λ values with CDF contours:

Introduced in 2007
Updated in 2010