Statistics[Distributions]
NonCentralStudentT
noncentral t-distribution
Calling Sequence
Parameters
Description
Notes
Examples
References
NonCentralStudentT(nu, delta)
NonCentralStudentTDistribution(nu, delta)
nu
-
degrees of freedom
delta
noncentrality parameter
The noncentral Student-t distribution is a continuous probability distribution with probability density function given by:
ft=Γν2+12−4δ2t22t2+2ν+4δ4t42t2+2ν2+4δ2t2ν2t2+2ν−1+νΓ−ν2+32πLaguerreL−ν2+12,12,δ2t22t2+2ν2−1+νΓ−ν2+2−δ2t2−1+2δ2t22t2+2ν+νΓ−ν2+32πLaguerreL−ν2+12,32,δ2t22t2+2ν2t2+2ν−1+νΓ−ν2+2+Γν2+1δt21t2+ν2δ2t22t2+2ν+νΓ−ν2+1πLaguerreL−ν2,12,δ2t22t2+2ν2νΓ−ν2+32−δ2t2Γ−ν2+1πLaguerreL−ν2,32,δ2t22t2+2ν2t2+2ννΓ−ν2+32νν2ⅇ−δ22Γν2πt2+νν2+12
subject to the following conditions:
0<ν,δ::real
The NonCentralStudentT variate with noncentrality parameter delta=0 and degrees of freedom nu is equivalent to the StudentT variate with degrees of freedom nu.
The NonCentralStudentT variate with noncentrality parameter delta and degrees of freedom nu is related to the Normal variate and the ChiSquare variate by StudentTnu,delta`~`Normaldelta,1sqrtChiSquarenunu.
Note that the NonCentralStudentT command is inert and should be used in combination with the RandomVariable command.
Quantile calculations for the non-central student-t distribution are very sensitive to small perturbations when delta is large. As a result, numeric methods for calculating quantiles will often not converge unless Digits is set to 25 or higher.
The Quantile and CDF functions applied to a noncentral Student-t 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≔RandomVariableNonCentralStudentT3,δ:
PDFX,u
233π1+u232δ=03π21u2+3u2+3ⅇu2δ22u2+3δ3u3+π2ⅇu2δ22u2+3erf2uδ2u2+3δ3u3+3π21u2+3u2+3ⅇu2δ22u2+3δu3+3π2ⅇu2δ22u2+3erf2uδ2u2+3δu3+9πuδ21u2+3ⅇu2δ22u2+3u2+3+92uδπⅇu2δ22u2+3erf2uδ2u2+3+2u2δ2u2+3+4u2+3u2+12u2+33ⅇ−δ222u2+372πotherwise
PDFX,13
2433392πδ=02187π2ⅇδ256δ327+π2ⅇδ256erf2δ289168δ327+28π2ⅇδ256δ9+28π2ⅇδ256erf2δ289168δ9+2δ228981+112289812893ⅇ−δ221229312πotherwise
MeanX
δ23π
VarianceX
3δ2+3−6δ2π
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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