FisherZDistribution

FisherZDistribution[n,m]

represents a Fisher distribution with n numerator and m denominator degrees of freedom.

Background & Context

• FisherZDistribution[n,m] represents a continuous statistical distribution defined over the set of real numbers and parameterized by two positive real numbers n and m that represent degrees of freedom. The probability density function (PDF) of the Fisher distribution is smooth and unimodal, and the parameters m and n determine its overall height and steepness. In addition, the tails of the PDF are "thin", in the sense that the PDF decreases exponentially for large values of . (This behavior can be made quantitatively precise by analyzing the SurvivalFunction of the distribution.)
• Fisher's distribution is obtained as the distribution of the random variable , where , and was first presented in the mid-1920s by English statistician Ronald Fisher. Fisher's distribution was devised as a tool for analyzing the errors of widely used distributions of the time and can be utilized in many of the same applications as FRatioDistribution. However, because FRatioDistribution is mathematically simpler, it has become somewhat uncommon to use FisherZDistribution to model real-world scenarios, though one example in which FisherZDistribution is preferable is in modeling the so-called -statistic among variates that are binormally (BinormalDistribution) distributed.
• RandomVariate can be used to give one or more machine- or arbitrary-precision (the latter via the WorkingPrecision option) pseudorandom variates from Fisher's distribution. Distributed[x,FisherZDistribution[n,m]], written more concisely as xFisherZDistribution[n,m], can be used to assert that a random variable x is distributed according to Fisher's distribution. Such an assertion can then be used in functions such as Probability, NProbability, Expectation, and NExpectation.
• The probability density and cumulative distribution functions may be given using PDF[FisherZDistribution[n,m],x] and CDF[FisherZDistribution[n,m],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 Fisher's distribution, EstimatedDistribution to estimate a Fisher's parametric distribution from given data, and FindDistributionParameters to fit data to a Fisher's distribution. ProbabilityPlot can be used to generate a plot of the CDF of given data against the CDF of a symbolic Fisher's distribution and QuantilePlot to generate a plot of the quantiles of given data against the quantiles of a symbolic Fisher's distribution
• TransformedDistribution can be used to represent a transformed Fisher's 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 Fisher's distribution, and ProductDistribution can be used to compute a joint distribution with independent component distributions involving Fisher's distributions.
• Fisher's distribution is related to a number of other distributions. As previously noted, FisherZDistribution is obtained as a transformation of FRatioDistribution, so that the PDF of FisherZDistribution[n,m] is precisely the same as that of TransformedDistribution[Log[u]/2,u FRatioDistribution[n,m]]. FisherZDistribution approaches normality, in the sense that FisherZDistribution[n,m] approaches NormalDistribution[μ,σ] as m and n tend to Infinity for and . Finally, ChiSquareDistribution and StudentTDistribution can be derived as transformations of FisherZDistribution. FisherZDistribution is also related to ParetoDistribution, BinormalDistribution, and LogNormalDistribution.

Examples

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Basic Examples(4)

Probability density function:

Cumulative distribution function:

Mean:

Median:

Scope(7)

Generate a sample of pseudorandom numbers from a Fisher 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:

Kurtosis:

Different moments with closed forms as functions of parameters, including Moment:

Hazard function:

Quantile function:

Applications(1)

Given a binormal sample, the -statistic follows a shifted FisherZDistribution:

Generate the distribution of -statistics for binormal samples of size :

Visually compare the -statistic distribution to a shifted FisherZDistribution:

DistributionFitTest confirms the result:

Properties & Relations(2)

Relationships to other distributions:

Fisher distribution is a transformation of FRatioDistribution:

Neat Examples(1)

PDFs for different n values with CDF contours:

Wolfram Research (2010), FisherZDistribution, Wolfram Language function, https://reference.wolfram.com/language/ref/FisherZDistribution.html (updated 2016).

Text

Wolfram Research (2010), FisherZDistribution, Wolfram Language function, https://reference.wolfram.com/language/ref/FisherZDistribution.html (updated 2016).

CMS

Wolfram Language. 2010. "FisherZDistribution." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2016. https://reference.wolfram.com/language/ref/FisherZDistribution.html.

APA

Wolfram Language. (2010). FisherZDistribution. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/FisherZDistribution.html

BibTeX

@misc{reference.wolfram_2024_fisherzdistribution, author="Wolfram Research", title="{FisherZDistribution}", year="2016", howpublished="\url{https://reference.wolfram.com/language/ref/FisherZDistribution.html}", note=[Accessed: 23-June-2024 ]}

BibLaTeX

@online{reference.wolfram_2024_fisherzdistribution, organization={Wolfram Research}, title={FisherZDistribution}, year={2016}, url={https://reference.wolfram.com/language/ref/FisherZDistribution.html}, note=[Accessed: 23-June-2024 ]}