PLAYGROUND

DISTRIBUTIONS / CONTINUOUS / ALPHA

Alpha distribution

What it describes

The Alpha distribution models a positive variable obtained by inverting a left-truncated normal variable. In the parameterization used here, alpha controls the shape while loc and scale translate and stretch the support to the right.

Its support starts at loc and extends to infinity. The right tail can be noticeable, so a mean or a fit driven by extreme observations should be checked against the plots.

History and names

It should not be confused with the α-stable family or with the alpha parameter used by other distributions. Johnson, Kotz, and Balakrishnan describe it as a transformation of a truncated normal; reliability literature also uses the name Alpha distribution.

How it connects to other distributions

Its construction links it to lifetime models derived from normal transformations and to the logic of the truncated normal. It is a positive, skewed alternative when lognormal or Weibull shapes do not reproduce the mass near a threshold.

Where it appears

  • component lifetimes and reliability analysis
  • accelerated life testing, where times are positive and skewed

Fitting cautions

The inverse transformation makes it especially important to enforce scale > 0 and check that the observed minimum is compatible with loc.

A Normal law seen through its reciprocal

The Alpha distribution arises by truncating a Normal variable and transforming its distance through a reciprocal. The result is positive and can produce the pronounced skewness seen in lifetime data. In 1985 Salvia proposed reliability applications and showed that its hazard could express patterns unavailable to an Exponential law.

The Alpha name is easily confused with a generic shape parameter or unrelated namesakes. Here it denotes one specific distribution. Its tail and hazard depend strongly on alpha, while loc and scale change origin and unit. The transformation does not provide a physical story by itself: a defensible application should connect the failure mechanism or empirical evidence with the reciprocal behaviour.

Decision guide

A good candidate when: the data are positive and strongly skewed, and a reciprocal-normal construction is physically meaningful, especially near a lower threshold.

Compare it with: Lognormal and Weibull. Check which model reproduces both the mass near the minimum and high quantiles; the generating story should decide between similar fits.

References

  • SciPy reference: scipy.stats.alpha — definition and parameterization
  • Johnson, N. L., Kotz, S. & Balakrishnan, N. (1994). Continuous Univariate Distributions, 2nd ed., Vol. 1. Wiley.
  • Evans, M., Hastings, N. & Peacock, B. (2000). Statistical Distributions, 3rd ed. Wiley.
  • Salvia, A. A. (1985). Reliability applications of the Alpha Distribution. IEEE Transactions on Reliability, 34(3), 251–252.

Alpha Distribution: equations and calculator

Distribution defintion

X∼Alpha(α,Loc,Sc)X\sim\mathrm{Alpha}\left(\alpha,\text{Loc},\text{Sc}\right)

Distribution domain

x∈(Loc,∞)x\in\left(\text{Loc},\infty\right)

Parameters domain and parameters constraints

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

Cumulative distribution function

FX(x)=Φ(α−1z(x))Φ(α)F_{X}\left(x\right)=\frac{\Phi\left(\alpha-\frac{1}{z(x)}\right)}{\Phi\left(\alpha\right)}

Probability density function

fX(x)=1Sc⋅z(x)2⋅Φ(α)⋅2πexp⁡(−12(α−1z(x))2)f_{X}\left(x\right)=\frac{1}{\text{Sc}\cdot z(x)^{2}\cdot \Phi\left(\alpha\right)\cdot\sqrt{2\pi}}\exp\left(-\frac{1}{2}\left(\alpha-\frac{1}{z(x)}\right)^{2}\right)

Percent point function/Sample

FX−1(u)=Loc+Sc×1α−Φ−1(uΦ(α))F_{X}^{-1}\left(u\right)=\text{Loc}+\text{Sc}\times \frac{1}{\alpha-\Phi^{-1}\left(u\Phi\left(\alpha\right)\right)}

Non-central parametric moments

μ~k′=E[X~k]=∫0∞xkfX~(x)dx\tilde{\mu}'_{k}=E[\tilde{X}^k]=\int_{0}^{\infty}x^{k}f_{\tilde{X}}\left(x\right)dx

Parametric mean

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

Parametric variance

Variance(X)=Sc2⋅(μ~2′−μ~1′2)\mathrm{Variance}(X)=\text{Sc}^{2}\cdot(\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)=Loc+Scα−Φ−1(12Φ(α))\mathrm{Median}(X)=\text{Loc}+\frac{\text{Sc}}{\alpha-\Phi^{-1}\left(\frac{1}{2}\Phi\left(\alpha\right)\right)}

Parametric mode

Mode(X)=Loc+Sc(α2+8−α)4\mathrm{Mode}(X)=\text{Loc}+\text{Sc}\frac{(\sqrt{\alpha^{2}+8}-\alpha)}{4}

Additional information and definitions

X~∼Alpha(α,0,1)\tilde{X}\sim \mathrm{Alpha}\left(\alpha,0,1\right)
Loc:Location parameter\text{Loc}:\text{Location parameter}
Sc:Scale parameter\text{Sc}:\text{Scale parameter}
z(x)=(x−Loc)/Scz(x)=\left(x-\text{Loc}\right)/\text{Sc}
u:Uniform[0,1] random varibleu:\text{Uniform[0,1] random varible}
Φ(x):CDF normal standard distribution\Phi\left(x\right):\text{CDF normal standard distribution}
Φ−1(x):PPF normal standard distribution\Phi^{-1}\left(x\right):\text{PPF normal standard distribution}