Statistics[Distributions]
NonCentralFRatio
noncentral f-distribution
Calling Sequence
Parameters
Description
Notes
Examples
References
NonCentralFRatio(nu, omega, delta)
NonCentralFRatioDistribution(nu, omega, delta)
nu
-
first degrees of freedom parameter
omega
second degrees of freedom parameter
delta
noncentrality parameter
The noncentral f-ratio distribution is a continuous probability distribution with probability density function given by:
ft=0t<01+tδhypergeom1,ν2+ω2+1,2,ν2+1,δtν2νt+ων+ω2νt+ωⅇ−δ2νν2ωω2tν2−1Βν2,ω2νt+ων2+ω2otherwise
subject to the following conditions:
0<ν,0<ω,0<δ
The NonCentralFRatio variate with degrees of freedom nu and omega and noncentrality parameter delta=0 is equivalent to the FRatio variate with degrees of freedom nu and omega.
The NonCentralFRatio variate with degrees of freedom nu and omega and noncentrality parameter delta is related to the independent NonCentralChiSquare variate and ChiSquare variate by NonCentralFRatio(nu,omega,delta) ~ (NonCentralChiSquare(nu,delta)*omega)/(ChiSquare(omega)*nu)
Note that the NonCentralFRatio command is inert and should be used in combination with the RandomVariable command.
The Quantile and CDF functions applied to a noncentral F-ratio distribution use a sequence of iterations in order to converge on the desired output point. The maximum number of iterations to perform is equal to 100 by default, but this value can be changed by setting the environment variable _EnvStatisticsIterations to the desired number of iterations.
withStatistics:
X≔RandomVariableNonCentralFRatioν,ω,δ:
PDFX,u
0u<01+uδhypergeom1,ν2+ω2+1,2,ν2+1,δuν2νu+ων+ω2νu+ωⅇ−δ2νν2ωω2uν2−1Βν2,ω2νu+ων2+ω2otherwise
PDFX,0.5
1.+0.2500000000δhypergeom1.,0.5000000000ν+0.5000000000ω+1.,2.,0.5000000000ν+1.,0.2500000000δνω+0.5νν+ωω+0.5νⅇ−0.5000000000δν0.5000000000νω0.5000000000ω0.50.5000000000ν−1.Β0.5000000000ν,0.5000000000ωω+0.5ν0.5000000000ν+0.5000000000ω
MeanX
undefinedω≤2ων+δνω−2otherwise
VarianceX
undefinedω≤42ω2ν+δ2+ν+2δω−2ν2ω−22ω−4otherwise
Evans, Merran; Hastings, Nicholas; and Peacock, Brian. Statistical Distributions. 3rd ed. Hoboken: Wiley, 2000.
Johnson, Norman L.; Kotz, Samuel; and Balakrishnan, N. Continuous Univariate Distributions. 2nd ed. 2 vols. Hoboken: Wiley, 1995.
Stuart, Alan, and Ord, Keith. Kendall's Advanced Theory of Statistics. 6th ed. London: Edward Arnold, 1998. Vol. 1: Distribution Theory.
See Also
Statistics
Statistics[RandomVariable]
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