NON CENTRAL T STUDENT DISTRIBUTION

Phitter implementation

Distribution Definition

import phitter

distribution = phitter.continuous.NonCentralTStudent({"lambda": *, "n": *, "loc": *, "scale": *})

💡 The distribution's parameters are defined equation section below

Distribution Methods and Attributes

## CDF, PDF, PPF receive float or numpy.ndarray.
distribution.cdf(float | numpy.ndarray) # -> float | numpy.ndarray
distribution.pdf(float | numpy.ndarray) # -> float | numpy.ndarray
distribution.ppf(float | numpy.ndarray) # -> float | numpy.ndarray
distribution.sample(int) # -> numpy.ndarray

## STATS
distribution.mean # -> float
distribution.variance # -> float
distribution.standard_deviation # -> float
distribution.skewness # -> float
distribution.kurtosis # -> float
distribution.median # -> float
distribution.mode # -> float

Equations

Distribution Definition

$$X\sim \mathrm{NonCentralTStudent}(\lambda ,n,\text{Loc},\text{Sc})$$

Distribution Domain

$$x\in (-\mathrm{\infty },\mathrm{\infty })$$

Parameters Domain and Constraints

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

Cumulative Distribution Function

$$F_X\left(x\right)=\{\begin{array}{cl}\frac{1}{2}\sum _{j=0}^{\mathrm{\infty }}\frac{1}{j!}(-\lambda \sqrt{2}{)}^j{e}^{\frac{-{\lambda }^2}{2}}\frac{\mathrm{\Gamma }(\frac{j+1}{2})}{\sqrt{\pi }}{I}_{n/(n+z(x{)}^2)}(\frac{n}{2},\frac{j+1}{2})& \text{if }z(x)\ge 0\\ 1-\frac{1}{2}\sum _{j=0}^{\mathrm{\infty }}\frac{1}{j!}(-\lambda \sqrt{2}{)}^j{e}^{\frac{-{\lambda }^2}{2}}\frac{\mathrm{\Gamma }(\frac{j+1}{2})}{\sqrt{\pi }}{I}_{n/(n+z(x{)}^2)}(\frac{n}{2},\frac{j+1}{2})& \text{if }z(x)<0\end{array}$$

Probability Density Function

$$\begin{gathered} f_X\left(x\right)=\frac{1}{\text{Sc}}\frac{n^{n/2}\mathrm{\Gamma }(n+1)}{2^n{e}^{{\lambda }^2/2}{(n+z(x{)}^2)}^{n/2}\mathrm{\Gamma }\left(n/2\right)}\times \\ \{\frac{\sqrt{2}\lambda z(x{)}_1{F}_1(\frac{n}{2}+1,\frac{3}{2},\frac{{\lambda }^{2}z(x{)}^2}{2(n+z(x{)}^2)})}{(n+z(x{)}^2)\mathrm{\Gamma }\left(\frac{n+1}{2}\right)}-\frac{{}_1{F}_1(\frac{n+1}{2},\frac{1}{2},\frac{{\lambda }^{2}z(x{)}^2}{2(n+z(x{)}^2)})}{\sqrt{n+z(x{)}^2}\mathrm{\Gamma }(\frac{n}{2}+1)}\} \end{gathered}$$

Percent Point Function / Sample

$${\text{Sample}}_X=\text{Loc}+\text{Sc}\frac{(\lambda +{\mathrm{\Phi }}^{-1}\left(u\right))}{\left(\sqrt{2{\text{P}}^{-1}(\frac{n}{2},u)}\right)/n}$$

Parametric Centered Moments

$${\tilde{\mu }}_k^{\prime }=E[{\tilde{X}}^k]={\int }_0^{\mathrm{\infty }}{x}^k{f}_{\tilde{X}}\left(x\right)dx=\frac{e^{-{\lambda }^2/2}}{\sqrt{n\pi }\mathrm{\Gamma }\left(n/2\right)}\mathrm{\Gamma }\left(\frac{n-k}{2}\right)n^{k/2}\sum _{r=0}^{\mathrm{\infty }}\frac{{\lambda }^r{2}^{r/2}}{r!}\mathrm{\Gamma }\left(\frac{r+k+1}{2}\right)$$

Parametric Mean

$$\mathrm{Mean}(X)=\text{Loc}+\text{Sc}{\tilde{\mu }}_1^{\prime }$$

Parametric Variance

$$\mathrm{Variance}(X)={\text{Sc}}^2({\tilde{\mu }}_2^{\prime }-{\tilde{\mu }}_1^{\prime 2})$$

Parametric Skewness

$$\mathrm{Skewness}(X)=\frac{{\tilde{\mu }}_3^{\prime }-3{\tilde{\mu }}_2^{\prime }{\tilde{\mu }}_1^{\prime }+2{\tilde{\mu }}_1^{\prime 3}}{({\tilde{\mu }}_2^{\prime }-{\tilde{\mu }}_1^{\prime 2}{)}^{1.5}}$$

Parametric Kurtosis

$$\mathrm{Kurtosis}(X)=\frac{{\tilde{\mu }}_4^{\prime }-4{\tilde{\mu }}_1^{\prime }{\tilde{\mu }}_3^{\prime }+6{\tilde{\mu }}_1^{\prime 2}{\tilde{\mu }}_2^{\prime }-3{\tilde{\mu }}_1^{\prime 4}}{({\tilde{\mu }}_2^{\prime }-{\tilde{\mu }}_1^{\prime 2}{)}^2}$$

Parametric Median

$$\mathrm{Median}(X)=F_X^{-1}\left(\frac{1}{2}\right)$$

Parametric Mode

$$\mathrm{Mode}(X)=\arg \underset{x}{max}{f}_X\left(x\right)$$

Additional Information and Definitions

• $\text{Computing an analytic expression for the inverse of the cumulative distribution function is not}$ $\text{feasible. Nonetheless, it is possible to generate a random sample from the distribution.}$
• $\tilde{X}\sim \mathrm{NonCentralTStudent}(\lambda ,n,0,1)$
• $\text{Loc}:\text{Location parameter}$
• $\text{Sc}:\text{Scale parameter}$
• $z\left(x\right)=(x-\text{Loc})/\text{Sc}$
• $u:\text{Uniform[0,1] random varible}$
• ${\text{P}}^{-1}(a,u):\text{Inverse of regularized lower incomplete gamma function}$
• $I_{\alpha }\left(x\right):\text{Modified Bessel function of the first kind of order }\alpha \in \mathbb{N}$
• ${}_1{F}_1(a,b,z):\text{Kummer’s confluent hypergeometric function}$

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