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Statistics[Distributions]

  

NonCentralFRatio

  

noncentral f-distribution

 

Calling Sequence

Parameters

Description

Notes

Examples

References

Calling Sequence

NonCentralFRatio(nu, omega, delta)

NonCentralFRatioDistribution(nu, omega, delta)

Parameters

nu

-

first degrees of freedom parameter

omega

-

second degrees of freedom parameter

delta

-

noncentrality parameter

Description

• 

The noncentral f-ratio distribution is a continuous probability distribution with probability density function given by:

ft=0t<01+tδF221,ν2+ω2+1;2,ν2+1;δtν2νt+ων+ω2νt+ω&ExponentialE;δ2νν2ωω2tν21Βν2&comma;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.

Notes

• 

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.

Examples

withStatistics&colon;

XRandomVariableNonCentralFRatio&nu;&comma;&omega;&comma;&delta;&colon;

PDFX&comma;u

&lcub;0u<01&plus;12u&delta;hypergeom1&comma;12&nu;&plus;12&omega;&plus;1&comma;2&comma;12&nu;&plus;1&comma;12&delta;u&nu;&nu;u&plus;&omega;&nu;&plus;&omega;&nu;u&plus;&omega;&ExponentialE;12&delta;&nu;12&nu;&omega;12&omega;u12&nu;1&Beta;12&nu;&comma;12&omega;&nu;u&plus;&omega;12&nu;&plus;12&omega;otherwise

(1)

PDFX&comma;0.5

1.&plus;0.2500000000&delta;hypergeom1.&comma;0.5000000000&nu;&plus;0.5000000000&omega;&plus;1.&comma;2.&comma;0.5000000000&nu;&plus;1.&comma;0.2500000000&delta;&nu;&omega;&plus;0.5&nu;&nu;&plus;&omega;&omega;&plus;0.5&nu;&ExponentialE;0.5000000000&delta;&nu;0.5000000000&nu;&omega;0.5000000000&omega;0.50.5000000000&nu;1.&Beta;0.5000000000&nu;&comma;0.5000000000&omega;&omega;&plus;0.5&nu;0.5000000000&nu;&plus;0.5000000000&omega;

(2)

MeanX

&lcub;undefined&omega;2&omega;&nu;&plus;&delta;&nu;2&plus;&omega;otherwise

(3)

VarianceX

&lcub;undefined&omega;42&omega;2&nu;&plus;&delta;2&plus;&nu;&plus;2&delta;2&plus;&omega;&nu;22&plus;&omega;24&plus;&omega;otherwise

(4)

References

  

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[Distributions]

Statistics[RandomVariable]

 


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