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T Student 3P distribution

Portrait

The three-parameter Student t shifts and rescales Student’s t. It can describe heavy-tailed errors with a calibrated centre, but classical inferential meaning belongs to untransformed t.

Historical trail

The 3P extension is a later modelling convention following Gosset’s work. Software libraries use it to separate shape, location, and unit.

Two useful connections

  • With loc=0 and scale=1 it becomes standard t. The F relationship holds in standardized form, while noncentral t changes the generation mechanism.
  • For df ≤ 1 the standardized mean does not exist and for df ≤ 2 variance does not exist; location and scale do not change that condition.

A use case

It is a descriptive choice for real observations with heavier tails than normal. If used for a t statistic, document the transform and do not report it as standard t.

Diagnostic advice

Do not read loc as a mean when the moment is undefined or use scale as standard deviation without checking df.

t tails for errors, with no hidden t test

Student t 3P uses the heavy-tailed shape as a location-scale family. loc sets the centre, scale the units, and degrees of freedom the tail thickness. It is a common choice for robust errors.

In this role, degrees need not come from n-1 or a sample variance. They are an estimated shape parameter. Quantiles should therefore not be read as critical values for a classical t test without reconstructing its statistic. As degrees increase, the family approaches a Normal law with the same location and scale. With few degrees, mean or variance may fail to exist even though centre and median remain defined.

Decision guide

A good candidate when: a symmetric heavy-tailed t model is needed with centre and scale estimated in the original units.

Compare it with: Normal and Generalized Normal with matching location and scale. Check that degrees of freedom are not determined by one or two outliers.

References

  • SciPy reference: scipy.stats.t — definition and parameterization
  • Johnson, N. L., Kotz, S. & Balakrishnan, N. (1994). Continuous Univariate Distributions, 2nd ed., Vol. 1. Wiley.
  • Student (W. S. Gosset) (1908). The probable error of a mean. Biometrika, 6(1), 1–25.
  • Evans, M., Hastings, N. & Peacock, B. (2000). Statistical Distributions, 3rd ed. Wiley.

T Student 3P Distribution: equations and calculator

Distribution defintion

X∼TStudent3P(df,Loc,Sc)X\sim\mathrm{TStudent_{3P}}\left(\text{df},\text{Loc},\text{Sc}\right)\\

Distribution domain

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

Parameters domain and parameters constraints

df∈R+,Loc∈R,Sc∈R+\text{df}\in\mathbb{R}^{+},\text{Loc}\in\mathbb{R},\text{Sc}\in\mathbb{R}^{+}\\

Cumulative distribution function

FX(x)=I(z(x)+z(x)2+df2z(x)2+df,df2,df2)F_{X}\left(x\right)=I\left(\frac{z(x)+\sqrt{z(x)^{2}+\text{df}}}{2\sqrt{z(x)^{2}+\text{df}}},\frac{\text{df}}{2},\frac{\text{df}}{2}\right)\\

Probability density function

fX(x)=(1+z(x)2/df)−(1+df)/2df×Beta(12,df2)f_{X}\left(x\right)=\frac{\left(1+z(x)^{2}/\text{df}\right)^{-(1+\text{df})/2}}{\sqrt{\text{df}}\times \text{Beta}\left(\frac{1}{2},\frac{\text{df}}{2}\right)}\\

Percent point function/Sample

FX−1(u)={Loc+Sc df(1−I−1(u,df/2,df/2))I−1(u,df/2,df/2)if  u≥12Loc−Sc df(1−I−1(u,df/2,df/2))I−1(u,df/2,df/2)if  u<12F^{-1}_{X}\left(u\right)=\left\{\begin{array}{cl} \text{Loc}+\text{Sc} \ \sqrt{\frac{\text{df}(1-I^{-1}\left(u,\text{df}/2,\text{df}/2\right))}{I^{-1}\left(u,\text{df}/2,\text{df}/2\right)}} & \text{if } \ u \geq \frac{1}{2} \\ \text{Loc}-\text{Sc} \ \sqrt{\frac{\text{df}(1-I^{-1}\left(u,\text{df}/2,\text{df}/2\right))}{I^{-1}\left(u,\text{df}/2,\text{df}/2\right)}} & \text{if } \ u < \frac{1}{2} \end{array} \right.\\

Non-central parametric moments

μ~k′=E[X~k]=∫0∞xkfX~(x)dx={0if  k odd ∧ 0<k<dfdfk2 ∏i=1k/22i−1df−2iif  k even ∧ 0<k<df\tilde{\mu}'_{k}=E[\tilde{X}^k]=\int_{0}^{\infty}x^{k}f_{\tilde{X}}\left(x\right)dx=\left\{\begin{array}{cl} 0 & \text{if } \ k\text{ odd} \ \wedge \ 0 < k < \text{df} \\ \text{df}^{\frac{k}{2}} \,\prod_{i=1}^{k/2}\frac{2i-1}{\text{df}-2i} & \text{if } \ k\text{ even} \ \wedge \ 0 < k < \text{df} \end{array} \right.\\

Parametric mean

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

Parametric variance

Variance(X)=Sc2×(μ~2′−μ~1′2)={Sc2 df/(df+2)if  df>2undefinedif  df≤2\mathrm{Variance}(X)=\text{Sc}^{2}\times (\tilde{\mu}'_{2}-\tilde{\mu}'^{2}_{1})=\left\{\begin{array}{cl} \text{Sc}^{2} \ \text{df}/(\text{df}+2) & \text{if } \ \text{df} > 2 \\ \text{undefined} & \text{if } \ \text{df} \leq 2 \end{array} \right.\\

Parametric skewness

Skewness(X)=μ~3′−3μ~2′μ~1′+2μ~1′3(μ~2′−μ~1′2)1.5={0if  df>3undefinedif  df≤3\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}}=\left\{\begin{array}{cl} 0 & \text{if } \ \text{df} > 3 \\ \text{undefined} & \text{if } \ \text{df} \leq 3 \end{array} \right.\\

Parametric kurtosis

Kurtosis(X)=μ~4′−4μ~1′μ~3′+6μ~1′2μ~2′−3μ~1′4(μ~2′−μ~1′2)2={3+6/(df−4)if  df>4undefinedif  df≤4\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}}=\left\{\begin{array}{cl} 3+6/(\text{df}-4) & \text{if } \ \text{df} > 4 \\ \text{undefined} & \text{if } \ \text{df} \leq 4 \end{array} \right.\\

Parametric median

Median(X)=Loc\mathrm{Median}(X)=\text{Loc}\\

Parametric mode

Mode(X)=Loc\mathrm{Mode}(X)=\text{Loc}

Additional information and definitions

X~∼TStudent(df)\tilde{X}\sim\mathrm{TStudent}\left(\text{df}\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}
I(x,a,b):Regularized incomplete beta functionI\left(x,a,b\right):\text{Regularized incomplete beta function}
I−1(x,a,b):Inverse of regularized incomplete beta functionI^{-1}\left(x,a,b\right):\text{Inverse of regularized incomplete beta function}
Beta(x,y):Beta function\text{Beta}\left(x,y\right):\text{Beta function}