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Topp-Leone distribution

What it represents

The Topp–Leone distribution is bounded and can concentrate near the lower endpoint or rise to an interior peak. It was designed to provide a flexible density on a unit-like scale.

Historical clue

Topp and Leone introduced the family in 1955 as J-shaped frequency functions. Modern work has revisited it for survival and bounded data.

Relationships that clarify its use

It is related to Beta and transforms of uniform variables. Topp–Leone-G extensions apply its CDF to a base distribution, but the basic form is enough for its bounded shape.

Data examples

  • unit proportions and non-uniform bounded variables
  • survival or reliability models with known limits

Modelling warning

A J-shape does not itself imply early or late failure cause; connect density to the observed process.

The J-shaped family that can also raise a peak

Topp and Leone titled their 1955 paper a family of J-shaped functions. On the unit interval, the CDF combines a variable with its reflection through a symmetric product and then applies a power. This algebraic symmetry creates a simple yet flexible law.

For some parameter values density falls from an endpoint; for others it can rise to an interior peak. A quadratic transformation of its complement helps connect it with a Power Function. Its hazard can provide useful shapes for bounded reliability data. Since the name now appears in many generated families, distinguish the original distribution from a Topp Leone G construction based on another CDF.

Decision guide

A good candidate when: a bounded proportion has a J shape or characteristic endpoint concentration that one parameter can summarize.

Compare it with: Beta and Kumaraswamy. Topp–Leone is parsimonious but less flexible; inspect both endpoints before treating observed accumulation as structural.

References

Topp-Leone Distribution: equations and calculator

Distribution defintion

X∼ToppLeone(α,min,max)X\sim\mathrm{ToppLeone}\left(\alpha,\text{min},\text{max}\right)

Distribution domain

x∈(min,max)x\in\left(\text{min},\text{max}\right)

Parameters domain and parameters constraints

α∈R+,min∈R,max∈R\alpha\in\mathbb{R}^{+},\text{min}\in\mathbb{R},\text{max}\in\mathbb{R}

Cumulative distribution function

FX(x)=[z(x)(2−z(x))]αF_{X}\left(x\right)=\left[z(x)\left(2-z(x)\right)\right]^{\alpha}

Probability density function

fX(x)=2αmax−min(1−z(x))[z(x)(2−z(x))]α−1f_{X}\left(x\right)=\frac{2\alpha}{\text{max}-\text{min}}\left(1-z(x)\right)\left[z(x)\left(2-z(x)\right)\right]^{\alpha-1}

Percent point function/Sample

FX−1(u)=min+(max−min)(1−1−u1/α)F^{-1}_{X}\left(u\right)=\text{min}+\left(\text{max}-\text{min}\right)\left(1-\sqrt{1-u^{1/\alpha}}\right)

Non-central parametric moments

μ~k′=E[X~k]=∑i=0k(ki)(−1)iα Beta ⁣(α,1+i2)\tilde{\mu}'_{k}=E[\tilde{X}^{k}]=\sum_{i=0}^{k}\binom{k}{i}\left(-1\right)^{i}\alpha\,\text{Beta}\!\left(\alpha,1+\tfrac{i}{2}\right)

Parametric mean

Mean(X)=min+(max−min)×μ~1′\mathrm{Mean}(X)=\text{min}+\left(\text{max}-\text{min}\right)\times \tilde{\mu}'_{1}

Parametric variance

Variance(X)=(max−min)2(μ~2′−μ~1′2)\mathrm{Variance}(X)=\left(\text{max}-\text{min}\right)^{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)=min+(max−min)(1−1−0.51/α)\mathrm{Median}(X)=\text{min}+\left(\text{max}-\text{min}\right)\left(1-\sqrt{1-0.5^{1/\alpha}}\right)

Parametric mode

Mode(X)=min+(max−min)(1−α−12α−1)\mathrm{Mode}(X)=\text{min}+\left(\text{max}-\text{min}\right)\left(1-\sqrt{\tfrac{\alpha-1}{2\alpha-1}}\right)

Additional information and definitions

X~∼ToppLeone(α,0,1)\tilde{X}\sim\mathrm{ToppLeone}\left(\alpha,0,1\right)
z(x)=(x−min)/(max−min)z\left(x\right)=\left(x-\text{min}\right)/\left(\text{max}-\text{min}\right)
u:Uniform[0,1] random varibleu:\text{Uniform[0,1] random varible}
Beta(x,y):Beta function\text{Beta}\left(x,y\right):\text{Beta function}
Mode formula valid when α>1, otherwise Mode(X)=min\text{Mode formula valid when } \alpha>1,\text{ otherwise }\mathrm{Mode}(X)=\text{min}