NON CENTRAL F DISTRIBUTION

Phitter implementation

Distribution Definition

import phitter

distribution = phitter.continuous.NonCentralF({"lambda": *, "n1": *, "n2": *})

💡 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

$$X\sim \mathrm{NonCentralF}(\lambda ,n_1,n_2)$$

Distribution Domain

$$x\in [0,\mathrm{\infty })$$

Parameters Domain and Constraints

$$\lambda \in {\mathbb{R}}^{+},n_1\in {\mathbb{R}}^{+},n_2\in {\mathbb{R}}^{+}$$

Cumulative Distribution Function

$$F_X\left(x\right)=\sum _{j=0}^{\mathrm{\infty }}\left(\frac{{\left(\frac{1}{2}\lambda \right)}^j}{j!}{e}^{-\lambda /2}\right)I_{n_{1}x/(n_2+n_{1}x)}(\frac{n_1}{2}+j,\frac{n_2}{2})$$

Probability Density Function

$$f_X\left(x\right)=\sum _{k=0}^{\mathrm{\infty }}\frac{e^{-\lambda /2}(\lambda /2{)}^k}{\text{Beta}(\frac{n_2}{2},\frac{n_1}{2}+k)k!}{\left(\frac{n_1}{n_2}\right)}^{\frac{n_1}{2}+k}{\left(\frac{n_2}{n_2+n_{1}x}\right)}^{\frac{n_1+n_2}{2}+k}{x}^{n_1/2-1+k}$$

Percent Point Function / Sample

$${\text{Sample}}_X=\frac{\left(\sum _{i=1}^{n_1}{(\sqrt{\frac{\lambda }{n_1}}+{\mathrm{\Phi }}^{-1}\left(u_i\right))}^2\right)/n_1}{\left(2{\text{P}}^{-1}(\frac{n_2}{2},u)\right)/n_2}$$

Parametric Centered Moments

$${\mu }_k^{\prime }=E[X^k]={\int }_0^{\mathrm{\infty }}{x}^k{f}_X\left(x\right)dx=e^{-\lambda /2}{\left(\frac{n1}{n2}\right)}^k\frac{\mathrm{\Gamma }(n_1/2-k)}{\mathrm{\Gamma }\left(n_1/2\right)}\sum _{r=0}^{\mathrm{\infty }}\left(\frac{1}{r!}\right){\left(\frac{\lambda }{2}\right)}^r\frac{\mathrm{\Gamma }(\frac{n_1}{2}+r+k)}{\mathrm{\Gamma }(\frac{n_1}{2}+r)}$$

Parametric Mean

$$\mathrm{Mean}(X)={\mu }_1^{\prime }$$

Parametric Variance

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

Parametric Skewness

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

Parametric Kurtosis

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

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

• $$\begin{gathered} \text{Computing an analytic expression for the inverse of the cumulative distribution function is not} \\ \text{feasible. Nonetheless, it is possible to generate a random sample from the distribution.} \end{gathered}$$
• $u:\text{Uniform[0,1] random varible}$
• $u_i:\text{Uniform[0,1] random varible}$
• ${\mathrm{\Phi }}^{-1}\left(x\right):\text{PPF normal standard distribution}$
• ${\text{P}}^{-1}(a,u):\text{Inverse of regularized lower incomplete gamma function}$
• $I(x,a,b):\text{Regularized incomplete beta function}$
• $\text{Beta}(x,y):\text{Beta function}$

Spreadsheet Documents

• Phitter playground
• Download Excel Spreadsheet
• Excel file from GitHub repository
• Google spreadsheet document