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Statistics[Distributions][NonCentralStudentT] - noncentral t-distribution

Calling Sequence

NonCentralStudentT(nu, delta)

NonCentralStudentTDistribution(nu, delta)

Parameters

nu

-

degrees of freedom

delta

-

noncentrality parameter

Description

• 

The noncentral Student-t distribution is a continuous probability distribution with probability density function given by:

ft=Γν2+124δ2t22t2+2ν+4δ4t42t2+2ν2+4δ2t2ν2t2+2ν1+νΓν2+32πLν2+1212δ2t22t2+2ν21+νΓν2+2δ2t21+2δ2t22t2+2ν+νΓν2+32πLν2+1232δ2t22t2+2ν2t2+2ν1+νΓν2+2+Γν2+1δt21t2+ν2δ2t22t2+2ν+νΓν2+1πLν212δ2t22t2+2ν2νΓν2+32δ2t2Γν2+1πLν232δ2t22t2+2ν2t2+2ννΓν2+32νν2ⅇδ22Γν2πt2+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.

Notes

• 

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.

Notes

• 

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.

Examples

withStatistics&colon;

X:=RandomVariableNonCentralStudentT3&comma;&delta;&colon;

PDFX&comma;u

&lcub;233&pi;1&plus;13u22&delta;&equals;0322u2&plus;6&ExponentialE;12u2&delta;2u2&plus;3&pi;1u2&plus;32&delta;3u3&plus;32u2&plus;6&ExponentialE;12u2&delta;2u2&plus;3&pi;1u2&plus;32&delta;u3&plus;2&ExponentialE;12u2&delta;2u2&plus;3erfu&delta;2u2&plus;6&pi;&delta;3u3&plus;9&pi;u&delta;21u2&plus;3&ExponentialE;12u2&delta;2u2&plus;32u2&plus;6&plus;6&ExponentialE;12u2&delta;2u2&plus;3erfu&delta;2u2&plus;6&pi;&delta;u3&plus;2u2&delta;22u2&plus;6&plus;18u&delta;&pi;&ExponentialE;12u2&delta;2u2&plus;3erfu&delta;2u2&plus;6&plus;42u2&plus;6u2&plus;122u2&plus;63&ExponentialE;12&delta;22u2&plus;6u2&plus;33&pi;otherwise

(1)

PDFX&comma;13

&lcub;2433923&pi;&delta;&equals;021872458624175656&ExponentialE;156&delta;2&pi;282&delta;3&plus;1956&ExponentialE;156&delta;2&pi;282&delta;&plus;227&ExponentialE;156&delta;2erf1168&delta;569&pi;&delta;3&plus;569&ExponentialE;156&delta;2erf1168&delta;569&pi;&delta;&plus;281&delta;2569&plus;112815695693&ExponentialE;12&delta;2&pi;otherwise

(2)

MeanX

&delta;23&pi;

(3)

VarianceX

3&delta;2&plus;36&delta;2&pi;

(4)

See Also

Statistics, Statistics[Distributions], Statistics[RandomVariable]

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.


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