PLAYGROUND

DISTRIBUTIONS / CONTINUOUS / NON CENTRAL T STUDENT

Non Central T Student distribution

The idea in one sentence

The noncentral Student t adds a shift to the normal numerator of Student’s t. It is the distribution of a t statistic under an alternative, with lambda expressing effect size.

Support and interpretation

It spans the real line. n is degrees of freedom, lambda noncentrality, and loc/scale additional implementation transforms.

Its place in the family

At lambda=0 it is central t; squaring connects it to noncentral F. Central t approaches normal as n grows, and noncentral t retains that link with a suitable shift.

An applied reading

In experiment planning, noncentral t maps a standardized difference to rejection probability. lambda depends on the alternative and standard error, not merely a visual centre.

Do not treat lambda as loc or compare noncentral-t and central-t means without checking moment existence.

The t distribution shifts because the alternative is true

Central t assumes that the numerator mean equals the value stated by the null hypothesis. When a real difference exists, that Normal numerator is shifted before division by the estimated scale. The result is Noncentral t, and lambda represents a standardized effect amplified by sample size.

This law underpins power, sample-size, and effect-size interval calculations. Its square is related to a Noncentral F variable with one numerator degree of freedom. It must not be confused with Student t 3P: adding location after generating a t variable does not reproduce the dependence between signal and variance estimation that defines noncentrality.

Decision guide

A good candidate when: a t statistic is studied under an alternative with nonzero standardized mean, for example in power or coverage calculations.

Compare it with: central t under the null. Noncentrality is not the location parameter of an ordinary t distribution and should not be interpreted as one.

References

Non Central T Student Distribution: equations and calculator

Distribution defintion

X∼NonCentralTStudent(λ,n,Loc,Sc)X\sim\mathrm{NonCentralTStudent}\left(\lambda,n,\text{Loc},\text{Sc}\right)

Distribution domain

x∈(−∞,∞)x\in\left(-\infty,\infty\right)

Parameters domain and parameters constraints

λ∈R,n∈R+,Sc∈R+,Loc∈R\lambda\in\mathbb{R},n\in\mathbb{R}^{+},\text{Sc}\in\mathbb{R}^{+},\text{Loc}\in\mathbb{R}

Cumulative distribution function

FX(x)={12∑j=0∞1j!(−λ2)je−λ22Γ(j+12)πIn/(n+z(x)2)(n2,j+12)if  z(x)≥01−12∑j=0∞1j!(−λ2)je−λ22Γ(j+12)πIn/(n+z(x)2)(n2,j+12)if  z(x)<0F_{X}\left(x\right)=\left\{\begin{array}{cl}\frac{1}{2}\sum_{j=0}^\infty\frac{1}{j!}(-\lambda\sqrt{2})^je^{\frac{-\lambda^2}{2}}\frac{\Gamma(\frac{j+1}{2})}{\sqrt{\pi}}I_{n/(n+z(x)^2)}\left (\frac{n}{2},\frac{j+1}{2}\right ) & \text{if } \ z(x)\ge 0 \\ 1-\frac{1}{2}\sum_{j=0}^\infty\frac{1}{j!}(-\lambda\sqrt{2})^je^{\frac{-\lambda^2}{2}}\frac{\Gamma(\frac{j+1}{2})}{\sqrt{\pi}}I_{n/(n+z(x)^2)}\left (\frac{n}{2},\frac{j+1}{2}\right ) & \text{if } \ z(x) < 0\end{array} \right.

Probability density function

fX(x)=1Scnn/2Γ(n+1)2neλ2/2(n+z(x)2)n/2Γ(n/2)×{2λz(x)1F1(n2+1,32,λ2z(x)22(n+z(x)2))(n+z(x)2)Γ(n+12)−1F1(n+12,12,λ2z(x)22(n+z(x)2))n+z(x)2Γ(n2+1)}f_{X}\left(x\right)=\frac{1}{\text{Sc}}\frac{n^{n/2}\Gamma\left(n+1\right)}{2^{n}e^{\lambda^{2}/2}\left(n+z(x)^{2}\right)^{n/2}\Gamma\left(n/2\right)}\times \\ \left\{ \frac{\sqrt{2}\lambda z(x)_{1}F_{1}\left(\frac{n}{2}+1,\frac{3}{2},\frac{\lambda^{2}z(x)^{2}}{2\left(n+z(x)^{2}\right)}\right)}{\left(n+z(x)^{2}\right)\Gamma\left(\frac{n+1}{2}\right)} - \frac{_{1}F_{1}\left(\frac{n+1}{2},\frac{1}{2},\frac{\lambda^{2}z(x)^{2}}{2\left(n+z(x)^{2}\right)}\right)}{\sqrt{n+z(x)^{2}}\Gamma\left(\frac{n}{2}+1\right)} \right\}

Percent point function/Sample

SampleX=Loc+Sc(λ+Φ−1(u))(2P−1(n2,u))/n\text{Sample}_{X}=\text{Loc}+\text{Sc}\frac{\left(\lambda+\Phi^{-1}\left(u\right)\right)}{\left(\sqrt{2\text{P}^{-1}\left(\frac{n}{2},u\right)}\right)/n}

Non-central parametric moments

μ~k′=E[X~k]=∫0∞xkfX~(x)dx=e−λ2/2nπΓ(n/2)Γ(n−k2)nk/2∑r=0∞λr2r/2r!Γ(r+k+12)\tilde{\mu}'_{k}=E[\tilde{X}^k]=\int_{0}^{\infty}x^{k}f_{\tilde{X}}\left(x\right)dx=\frac{e^{-\lambda^{2}/2}}{\sqrt{n\pi}\Gamma\left(n/2\right)}\Gamma\left(\frac{n-k}{2}\right)n^{k/2}\sum_{r=0}^{\infty }\frac{\lambda^{r}2^{r/2}}{r!}\Gamma\left(\frac{r+k+1}{2}\right)

Parametric mean

Mean(X)=Loc+Scμ~1′\mathrm{Mean}(X)=\text{Loc}+\text{Sc}\tilde{\mu}'_{1}

Parametric variance

Variance(X)=Sc2(μ~2′−μ~1′2)\mathrm{Variance}(X)=\text{Sc}^{2}(\tilde{\mu}'_{2}-\tilde{\mu}'^{2}_{1})

Parametric skewness

Skewness(X)=μ~3′−3μ~2′μ~1′+2μ~1′3(μ~2′−μ~1′2)1.5\mathrm{Skewness}(X)=\frac{\tilde{\mu}'_{3}-3\tilde{\mu}'_{2}\tilde{\mu}'_{1}+2\tilde{\mu}'^{3}_{1}}{(\tilde{\mu}'_{2}-\tilde{\mu}'^{2}_{1})^{1.5}}

Parametric kurtosis

Kurtosis(X)=μ~4′−4μ~1′μ~3′+6μ~1′2μ~2′−3μ~1′4(μ~2′−μ~1′2)2\mathrm{Kurtosis}(X)=\frac{\tilde{\mu}'_{4}-4\tilde{\mu}'_{1}\tilde{\mu}'_{3}+6\tilde{\mu}'^{2}_{1}\tilde{\mu}'_{2}-3\tilde{\mu}'^{4}_{1}}{(\tilde{\mu}'_{2}-\tilde{\mu}'^{2}_{1})^{2}}

Parametric median

Median(X)=FX−1(12)\mathrm{Median}(X)=F^{-1}_{X}\left(\frac{1}{2}\right)

Parametric mode

Mode(X)=arg⁡max⁡xfX(x)\mathrm{Mode}(X)=\arg\max_{x}f_{X}\left(x\right)

Additional information and definitions

Computing an analytic expression for the inverse of the cumulative distribution function is not feasible. Nonetheless, it is possible to generate a random sample from the distribution.
X~∼NonCentralTStudent(λ,n,0,1)\tilde{X}\sim\mathrm{NonCentralTStudent}\left(\lambda,n,0,1\right)
Loc:Location parameter\text{Loc}:\text{Location parameter}
Sc:Scale parameter\text{Sc}:\text{Scale parameter}
z(x)=(x−Loc)/Scz\left(x\right)=\left(x-\text{Loc}\right)/\text{Sc}
u:Uniform[0,1] random varibleu:\text{Uniform[0,1] random varible}
P−1(a,u):Inverse of regularized lower incomplete gamma function\text{P}^{-1}\left(a,u\right):\text{Inverse of regularized lower incomplete gamma function}
Iα(x):Modified Bessel function of the first kind of order α∈NI_{\alpha}\left(x\right):\text{Modified Bessel function of the first kind of order }\alpha\in\mathbb{N}
1F1(a,b,z):Kummer’s confluent hypergeometric function_{1}F_{1}(a,b,z):\text{Kummer's confluent hypergeometric function}