Exact L-moments of probability distributions¶

This page lists theoretical L-moments ¹ of popular probability distributions.

All distributions are in the “standardized” form, similar to the convention used in the scipy.stats distribution documentation. Shifting a distribution only affects the L-location \( \tlmoment{s,t}{1} \), just like the expectation and the median. Scaling a distribution simply scales all L-moments \( \tlmoment{s,t}{r}, \; r \ge 1 \) analogous to e.g. the standard deviation or MAD. Note that neither shifting nor scaling affects the L-moment ratio’s \( \tlratio{s,t}{r} \).

Each of the listed expressions have been validated, both numerically and symbolically (with either Wolfram Alpha, SymPy, or pen and paper).

Most of the closed-form expressions that are listed here, have been previously reported in the literature. But for the sake of interpretability, several have been algebraically rearranged.

Due to the exploratory use of symbolic computation software, this listing is likely to include some novel solutions. This is also the reason for the lack of references. But this should pose no problems in practise, since Lmo makes it trivial to check if they aren’t incorrect.

Simple distributions¶

Tip

Numerical calculation of these L-statistics using scipy.stats distributions, refer to rv_continuous.l_stats.

For direct calculation of the L-stats from a CDF or PPF function, see

An overview of the exact L-location, L-scale, L-skewness and L-kurtosis of some well-known (univariate) probability distributions.

L-stats¶

Distribution	Shape	\( \lmoment{1} \)	\( \lmoment{2} \)	\( \lratio{3} = \lmoment{3}/\lmoment{2} \)	\( \lratio{4} = \lmoment{4}/\lmoment{2} \)
Uniform `uniform()`		\( \dfrac 1 2 \\ = 0.5 \)	\( \dfrac 1 6 \\ = 0.16\overline{6} \dots \)	\( 0 \)	\( 0 \)
Normal `norm()`		\( 0 \)	\( \dfrac{1}{\sqrt \pi} \\ \approx 0.5642 \)	\( 0 \)	\( 30 \dfrac{\href{#const-theta_m}{\theta_m}}{\pi} - 9 \\ \approx 0.1226 \)
Logistic `logistic()`		\( 0 \)	\( 1 \)	\( 0 \)	\( \dfrac 1 6 \\ = 0.16\overline{6} \dots \)
Laplace `laplace()`		\( 0 \)	\( \dfrac 3 4 \\ = 0.75 \)	\( 0 \)	\( \dfrac{17}{72} \\ \approx 0.2361 \)
Student’s t `t(2)`	\( \nu = 2 \)	\( 0 \)	\( \dfrac{\pi}{2 \sqrt 2} \\ \approx 1.1107 \)	\( 0 \)	\( \dfrac 3 8 \\ = 0.375 \)
Student’s t `t(3)`	\( \nu = 3 \)	\( 0 \)	\( \dfrac{3 \sqrt 3}{2 \pi} \\ \approx 0.8270 \)	\( 0 \)	\( 1 - \dfrac{175}{24 \pi^2} \\ \approx 0.2612 \)
Student’s t `t(4)`	\( \nu = 4 \)	\( 0 \)	\( \dfrac{15}{64} \pi \\ \approx 0.7363 \)	\( 0 \)	\( \dfrac{111}{512} \\ \approx 0.2168 \)
Exponential `expon()`		\( 1 \)	\( \dfrac 1 2 \\ = 0.5 \)	\( \dfrac 1 3 \\ = 0.3\overline{3} \dots \)	\( \dfrac 1 6 \\ = 0.16\overline{6} \dots \)
Half-normal `halfnorm()`		\( 1 \)	\( \sqrt 2 - 1 \\ \approx 0.4142 \)	\( 7 + 4 \sqrt 2 - 12 (2 - \sqrt 2) \dfrac{\href{#const-theta_m}{\theta_m}}{\pi} \\ \approx 0.1983 \)	\( \approx 0.09732 \)
Half-logistic `halflogistic()`		\( 2 \ln 2 \\ \approx 1.386 \)	\( 1 \)	\( \dfrac{1}{1 - \ln 2} - 3 \\ \approx 0.2589 \)	\( 11 - \dfrac{10}{3 - \ln 8} \\ \approx 0.1370 \)
Rayleigh `rayleigh()`		\( \sqrt{\pi / 2} \\ \approx 1.253 \)	\( \sqrt{\pi / 2} - \sqrt{\pi / 4} \\ \approx 0.3671 \)	\( \dfrac{2 + \sqrt 2}{\sqrt 3} - \dfrac{4 + \sqrt 2}{\sqrt 2} \\ \approx 0.1140 \)	\( 10 \dfrac{2 + \sqrt 2}{\sqrt 3} - 3 \dfrac{5 + 3 \sqrt 2}{\sqrt 2} \\ \approx 0.1054 \)
Gumbel `gumbel_r()`		\( \href{#const-euler}{\gamma_e} \\ \approx 0.5772 \)	\( \ln{2} \\ \approx 0.6931 \)	\( 2 \log_2(3) - 3 \\ \approx 0.1699 \)	\( 16 - 10 \log_2(3) \\ \approx 0.1504 \)
\( \chi^2 \) `chi2(2)`	\( k = 2 \)	\( 2 \)	\( 1 \)	\( \dfrac 1 3 \\ = 0.3\overline{3} \dots \)	\( \dfrac 1 6 \\ = 0.16\overline{6} \dots \)
\( \chi^2 \) `chi2(4)`	\( k = 4 \)	\( 4 \)	\( \dfrac 3 2 \\ = 1.5 \)	\( \dfrac{19}{81} \\ \approx 0.2346 \)	\( \dfrac{367}{2^5 \cdot 3^4} \\ \approx 0.1416 \)
\( \chi^2 \) `chi2(6)`	\( k = 6 \)	\( 6 \)	\( \dfrac{15}{8} \\ = 1.875 \)	\( \dfrac{139}{729} \\ \approx 0.1907 \)	\( \dfrac{200\ 827}{2^{11} \cdot 3^6} \\ \approx 0.1345 \)

TL-stats¶

Symmetrically trimmed TL-stats of some symmetric distributions.

Distribution	Shape	\( \tlmoment{1}{1} \)	\( \tlmoment{1}{2} \)	\( \tlratio{1}{3} \)	\( \tlratio{1}{4} \)
Uniform `uniform()`		\( \dfrac 1 2 \\ = 0.5 \)	\( \dfrac{1}{10} \\ = 0.1 \)	\( 0 \)	\( 0 \)
Normal `norm()`		\( 0 \)	\( \dfrac{6}{\sqrt \pi} \left( 1 - 3 \dfrac{\href{#const-theta_m}{\theta_m}}{\pi}\right ) \\ \approx 0.2970 \)	\( 0 \)	\( \approx 0.06248 \)
Logistic `logistic()`		\( 0 \)	\( \dfrac 1 2 \\ = 0.5 \)	\( 0 \)	\( \dfrac{1}{12} \\ = 0.083\overline{3} \dots \)
Laplace `laplace()`		\( 0 \)	\( \dfrac{11}{32} \\ = 0.34375 \)	\( 0 \)	\( \dfrac{3}{22} \\ = 0.136\overline{36} \dots \)
Cauchy `cauchy()`		\( 0 \)	\( \dfrac{18}{\pi^3} \ \zeta(3) \\ \approx 0.6978 \)	\( 0 \)	\( \dfrac{25}{6} - \dfrac{175}{4 \pi^2} \frac \zeta(5) \zeta(3) \\ \approx 0.3428 \)
Student’s t `t(2)`	\( \nu = 2 \)	\( 0 \)	\( \dfrac{3 \pi}{16 \sqrt 2} \\ \approx 0.4165 \)	\( 0 \)	\( \dfrac{5}{32} \\ = 0.15625 \)
Student’s t `t(3)`	\( \nu = 3 \)	\( 0 \)	\( \dfrac{105 \sqrt 3}{16 \pi^3} \\ \approx 0.3666 \)	\( 0 \)	\( \dfrac{25}{6} - \frac{23 \ 023}{(24 \pi)^2} \\ \approx 0.1168 \)
Student’s t `t(4)`	\( \nu = 4 \)	\( 0 \)	\( \dfrac{3\ 609\ \pi}{32\ 768} \\ \approx 0.3460 \)	\( 0 \)	\( \dfrac{164 \ 975}{1 \ 642 \ 496} \\ \approx 0.1004 \)
Gumbel `gumbel_r()`		\( \href{#const-euler}{\gamma_e} - 2 \ln 3 + 3 \ln 2 \\ \approx 0.4594 \)	\( 6 \ln 3 - 9 \ln 2 \\ \approx 0.3533 \)	\( - \dfrac{10}{9} \dfrac{5 - 2 \log_2(5)}{3 - 2 \log_2(3)} - \frac{20}{9} \\ \approx 0.1065 \)	\( \dfrac{35}{6} \dfrac{7 - 3 \log_2(5)}{3 - 2 \log_2(3)} + \dfrac{5}{4} \\ \approx 0.07541 \)

LL-stats¶

Right-trimmed LL-stats of some simple left-bound distributions.

Distribution	Shape	\( \tlmoment{0, 1}{1} \)	\( \tlmoment{0, 1}{2} \)	\( \tlratio{0, 1}{3} \)	\( \tlratio{0, 1}{4} \)
Exponential `expon()`		\( \dfrac 1 2 \\ = 0.5 \)	\( \dfrac 1 4 \\ = 0.25 \)	\( \dfrac 2 9 \\ = 0.2\overline{2} \dots \)	\( \dfrac{1}{12} \\ = 0.083\overline{3} \dots \)
Half-normal `halfnorm()`		\( 2 - \sqrt 2 \\ \approx 0.5858 \)	\( 9 \sqrt 2 \ \dfrac{\href{#const-theta_m}{\theta_m}}{\pi} - 3 \dfrac{1 + \sqrt 2}{2} \\ \approx 0.2491 \)	\( \approx 0.1119 \)	\( \approx 0.04489 \)
Half-logistic `halflogistic()`		\( 4 \ln 2 - 2 \\ \approx 0.7726 \)	\( 1 \)	\( \dfrac{4}{27} \left( \dfrac{5}{3 - 4 \ln 2} - 21 \right) \\ \approx 0.1462 \)	\( \dfrac{5}{36} \left( \dfrac{-19}{3 - 4 \ln 2} + 84 \right) \\ \approx 0.06263 \)
Half-Cauchy `halfcauchy()`		\( \dfrac{4 \ln 2}{\pi} \\ \approx 0.8825 \)	\( \dfrac{63 \href{#const-zeta}{\mathop{\zeta}}(3)}{2 \pi^3} - \dfrac{3 \ln 2}{\pi} \\ \approx 0.5593 \)	\( \dfrac {40 \mathop{\zeta}(3)} {63 \mathop{\zeta}(3) - 9 \pi^2 \ln 2} - \frac 8 9 \\ \approx 0.4974 \)	\( \dfrac{155}{2} \dfrac {3 \mathop{\zeta}(3) - 35 \pi^{-2} \mathop{\zeta}(5)} {21 \mathop{\zeta}(3) - 2 \pi^2 \ln 2} + \dfrac 5 6 \\ \approx 0.3572 \)
Rayleigh `rayleigh()`		\( \dfrac{\sqrt \pi}{2} \\ \approx 0.8862 \)	\( \dfrac{3 - \sqrt 6}{4} \sqrt \pi \\ \approx 0.2439 \)	\( \dfrac{10}{9} \left( 3 \sqrt 2 + 2 \sqrt 3 - 2 \sqrt 6 \right) - \frac{28}{9} \\ \approx 0.008\ 625 \)	\( \dfrac 1 4 \dfrac{80 - 75 \sqrt 2 + 14 \sqrt{10}}{\sqrt 6 - 3} + \dfrac{25}{3} \\ \approx 0.06561 \)
Gumbel `gumbel_r()`		\( \href{#const-euler}{\gamma_e} - \ln 2 \\ \approx -0.1159 \)	\( 3 \ln 2 - \dfrac 3 2 \ln 3 \\ \approx 0.4315 \)	\( \dfrac 4 9 \left( \dfrac{5}{2 - \log_2(3)} - 12 \right) \\ \approx 0.02094 \)	\( \dfrac 5 6 \left( \dfrac{8 - 7 \log_2(5)}{2 - \log_2(3)} + 20 \right) \\ \approx 0.09488 \)
\( \chi^2 \) `chi2(2)`	\( k = 2 \)	\( 1 \)	\( \dfrac 1 2 \\ = 0.5 \)	\( \dfrac 2 9 \\ = 0.2\overline{2}\ldots \)	\( \dfrac{1}{12} \\ = 0.083\overline{3}\ldots \)
\( \chi^2 \) `chi2(4)`	\( k = 4 \)	\( \dfrac 5 2 \\ = 2.5 \)	\( \dfrac{31}{36} \\ = 0.861\overline{1}\dots \)	\( \dfrac{241}{2\ 232} \\ \approx 0.1080 \)	\( \dfrac{282\ 127}{372 \cdot 10^4} \\ \approx 0.07584 \)
\( \chi^2 \) `chi2(6)`	\( k = 6 \)	\( \dfrac{33}{8} \\ = 4.125 \)	\( \dfrac{1\ 475}{1\ 296} \\ \approx 1.138 \)	\( \dfrac{16\ 769}{271\ 872} \\ \approx 0.06168 \)	\( \dfrac{550\ 465\ 668\ 887}{708 \cdot 10^{10}} \\ \approx 0.07775 \)

General L-moments¶

Lmo derived a bunch of closed-form solutions for L-moments of several distributions. The proofs are not published, but it isn’t difficult to validate their correctness, e.g. numerically, or symbolically with sympy or wolfram alpha / mathematica.

Bernoulli¶

Surprisingly, the L-moments of the discrete Bernoulli distribution ², can’t be expressed as easily as the distribution itself:

\[ \tlmoment{s, t}{r} = \frac{(-1)^r}{r} (1 - p)^{s + 1} \jacobi{r + t - 1}{s + 1}{-t - 1}{2p - 1} + \ffact{1}{r} \]

Here, \( \jacobi{n}{\alpha}{\beta}{x} \) is a Jacobi polynomial (although it’s not orthogonal for \( t >= 0 \), since \( \beta > -1 \) does not hold).

PFD¶

With support on the interval \( (0, 1] \) and shape parameter \( \alpha > 0 \), the power function distribution has the following CDF and PPF:

\[ \begin{align*} F(x) &= x^\alpha \\ x(F) &= F^{\frac 1 \alpha} \end{align*} \]

The trimmed L-moments of order \( r \in \naturals \setminus \{ 0 \} \) and trim \( (s, t) \in \naturals^2 \) are

\[ \tlmoment{s, t}{r} = \frac{\ffact{\frac 1 \alpha}{r - 1}}{r} \frac{\rfact{r + s}{1 + t}}{\rfact{\frac 1 \alpha + s}{r + t}} , \]

where \( \ffact{x}{n} \) and \( \rfact{x}{n} \) are pochhammer symbols for the falling and rising factorials, respectively.

See scipy.stats.powerlaw for an Lmo-compatible implementation.

Gompertz¶

The Gompertz distribution ³ with shape parameter \( \alpha > 0 \) and \( x \ge 0 \), has the following CDF and PPF:

\[ \begin{align*} F(x) &= 1 - e^{\alpha (1 - e^x)} \\ x(F) &= \ln\left( 1 - \frac{\ln(1-F)}{\alpha} \right) \end{align*} \]

The general trimmed L-moments of the Gompertz distribution are:

\[ \tlmoment{s, t}{r} = \frac{1}{r} \sum_{k = t + 1}^{r + s + t} (-1)^{k - t - 1} \binom{r + k - 2}{r + t - 1} \binom{r + s + t}{k} e^{\alpha k} \ \Gamma_{\alpha k}(0) \]

GEV¶

The GEV distribution ⁴ unifies the Gumbel, Fréchet, and Weibull distributions. It has one shape parameter \( \alpha \in \mathbb{R} \), and the following distribution functions:

\[ \begin{align*} F(x) &= e^{-\qexp{1 - \alpha}{-x}} \\ x(F) &= -\qlog{1 - \alpha}{-\ln(F)} \end{align*} \]

Here, \( \qexp{q}{y} \) and \( \qlog{q}{y} \) are the Tsallis \( q \)-exponential and the \( q \)-logarithm, respectively.

An alternative parametrization is sometimes used, e.g. on Wikipedia, where \( \xi = -\alpha \). The convention that is used here, is the same as in scipy.stats.genextreme, where c corresponds to \( \alpha \).

The trimmed L-moments of the GEV are

\[ \tlmoment{s, t}{r} = \frac{(-1)^{r}}{r} \sum_{k = s + 1}^{r + s + t} (-1)^{k - s} \binom{r + k - 2}{r + s - 1} \binom{r + s + t}{k} \left( \begin{cases} \gamma_e + \ln(k) & \text{if } \alpha = 0 \\ 1 / \alpha - \Gamma(\alpha) \ k^{-\alpha} & \text{if } \alpha \neq 0 \end{cases} \right) \]

Note that the GEV is effectively a reparametrized \( q \)-Gumbel Tsallis distribution, with \( q = 1 - \alpha \).

GLO¶

The GLO ⁵, also known as the shifted log-logistic distribution , with shape parameter \( \alpha \in \mathbb{R} \), is characterized by the following distribution functions:

\[ \begin{align*} F(x) &= \frac{1}{1 + \qexp{1 - \alpha}{x}} \\ x(F) &= -\qlog{1 - \alpha}{\frac{1 - F}{F}} \end{align*} \]

For \( -1 < \alpha < 1 \), the general trimmed L-moments of the GLO are:

\[ \tlmoment{s, t}{r} = \begin{cases} \displaystyle \digamma(s + 1) - \digamma(t + 1) & \text{if } \alpha = 0 \wedge r = 1 \\ \displaystyle \frac{(-1)^r}{r} \B(r - 1,\ s + 1) + \frac 1 r \B(r - 1,\ t + 1) & \text{if } \alpha = 0 \\ \displaystyle \frac{\ffact{1}{r}}{\alpha} + \sum_{k = s + 1}^{r + s + t} (-1)^{r + s - k } \binom{r + k - 2}{r + s - 1} \binom{r + s + t}{k} \B(\alpha,\ k - \alpha) & \text{if } -1 < \alpha < 1 \end{cases} \]

Where \( \digamma(z) \) is the digamma function.

The corresponding scipy.stats implementation is kappa4, with h = -1 and k set to \( \alpha \); not genlogistic.

Note that the GLO is effectively a reparametrized \( q \)-logistic Tsallis distribution, with \( q = 1 - \alpha \).

GPD¶

The GPD ⁶, with shape parameter \( \alpha \in \mathbb{R} \), has for \( x \ge 0 \) the distribution functions:

\[ \begin{align*} F(x) &= 1 - \qexp{1 + \alpha}{-x} \\ x(F) &= -\qlog{1 + \alpha}{1 - F} \end{align*} \]

The L-moments of the GPD exist when \( \alpha < 1 + t \), and can be compactly expressed as

\[ \tlmoment{s,t}{r} = \begin{cases} \displaystyle H_{s + t + 1} - H_t & \text{if } \alpha = 0 \wedge r = 1 \\ \displaystyle \frac{1}{\alpha r} \left[ \frac {\B(r + \alpha - 1,\ t - \alpha + 1)} {\B(\alpha,\ r + s + t - \alpha + 1)} - \ffact{1}{r} \right] & \text{otherwise,} \end{cases} \]

where \( H_n \) is a harmonic number.

See scipy.stats.genpareto for an Lmo-compatible implementation.

Special cases

There are several notable special cases of the GPD:

\( q \)-Exponential: When \( \alpha > -1 \), GPD is \( q \)-exponential with shape \( q = 2 - 1 / (1 + \alpha) \) and rate (inverse scale) \( \lambda = \alpha + 1 \).
Exponential: When \( \alpha = 0 \), GPD is standard exponential.
Uniform: When \( \alpha = 1 \) GPD is uniform on \( [0, 1] \).

Generalizations

Wakeby’s distribution: Implemented as lmo.distributions.wakeby. See below for details, including the general L-moments in closed-form.
Kappa distribution: Implemented in as scipy.stats.kappa4.

Burr III / Dagum¶

The Burr III distribution ⁷, also known as the Dagum distribution, has two shape parameters \( \alpha \) and \( \beta \), both restricted to the positive reals

For \( x > 0 \), the distribution functions are:

\[ \begin{align*} F(x) &= (1 + x^{-\alpha})^{-\beta} \\ x(F) &= (F^{-1 / \beta} - 1)^{-1 / \alpha} \end{align*} \]

For \( \alpha > 1 \), the general L-moments are:

\[ \tlmoment{s,t}{r} = (-1)^{t - 1 / \alpha} \ \beta \ \frac{r + s + t}{r} \sum_{k = s}^{r + s + t - 1} (-1)^{k} \binom{k + r - 1}{k - s} \binom{r + s + t - 1}{k} \B(1 - 1 / \alpha, -\beta - k \beta) \]

The Burr III distribution is implemented in scipy.stats.burr, where the shape parameters c and d correspond to \( \alpha \) and \( \beta \), respectively. Equivalently, scipy.stats.mielke can be used, by setting k and s to \( \alpha \beta \) and \( \alpha \), respectively.

Special cases

Log-logistic / Fisk: With \( \beta = 1 \) Burr III is Log-logistic with shape \( \gamma \equiv \alpha \) and scale \( \sigma = 1 \).

Burr XII / Pareto IV¶

The Burr XII distribution ⁷ has two shape parameters \( \alpha \) and \( \beta \), both restricted to the positive reals. It is also known as the Singh-Maddala distribution. The alternative parametrization \( \alpha \mapsto 1 / \gamma \), where \( \gamma > 0 \), is known as the (standard) type IV Pareto distribution

The distribution functions for \( x > 0 \) are defined as:

\[ \begin{align*} F(x) &= 1 - (1 + x^\alpha)^{-\beta} \\ x(F) &= \bigl((1 - F)^{-1/\beta} - 1 \bigr)^{1/\alpha} \end{align*} \]

Burr XII distribution PDF

When \( \beta > 1 / \alpha \), the general \( r \)-th trimmed L-moment is:

\[ \tlmoment{s,t}{r} = \beta \ \frac{r + s + t}{r} \sum_{k = t}^{r + s + t - 1} (-1)^k \binom{k + r - 1}{k - t} \binom{r + s + t - 1}{k} \B\bigl(1 + 1 / \alpha,\ \beta + k \beta - 1 / \alpha \bigr) \]

This distribution is implemented in scipy.stats.burr12, where the shape parameters c and d correspond to \( \alpha \) and \( \beta \), respectively.

The Burr XII and Burr III distributions are related as \( Y = 1 / X \), where \( X \) and \( Y \) are RV’s with Burr XII \( (\alpha, \beta) \) and Burr III \( (1 / \alpha, \beta) \) distributions (or vice-versa), respectively.

Special cases

There are several notable special cases of the Burr XII distribution:

Pareto IV: Burr XII is a reparametrized standard Pareto type IV distribution with shapes \( \gamma \equiv 1 / \alpha \) and \( \alpha^{\prime} \equiv \beta \), location \( \mu = 0 \), and scale \( \sigma = 1 \).
Lomax: With \( \alpha = 1 \) Burr XII is Lomax with shape \( \alpha^{\prime} \equiv \beta \) and scale \( \sigma = 1 \).
Log-logistic / Fisk: With \( \beta = 1 \) Burr XII is Log-logistic with shape \( \gamma \equiv \alpha \) and scale \( \sigma = 1 \).

Kumaraswamy¶

For Kumaraswamy’s distribution ⁸ with parameters \( \alpha \in \mathbb{R}_{>0} \) and \( \beta \in \mathbb{R}_{>0} \), the general solution for the \( r \)th untrimmed L-moment has been derived by M.C. Jones in 2009 ⁹. Lmo has extended these results for the general trimmed L-moments.

The distribution functions are for \( 0 \le x \le 1 \) defined as:

\[ \begin{align*} F(x) &= 1 - (1 - x^\alpha)^\beta \\ x(F) &= \bigl(1 - (1 - F)^{1/\beta} \bigr)^{1/\alpha} \end{align*} \]

Kumaraswamy distribution PDF

Its general \( r \)-th trimmed L-moment are:

\[ \tlmoment{s,t}{r} = \frac{1}{r} \sum_{k = t + 1}^{r + s + t} (-1)^{k - t - 1} \binom{r + k - 2}{r + t - 1} \binom{r + s + t}{k} \frac{\B\bigl(1 / \alpha,\ 1 + k \beta \bigr)}{\alpha} \]

The Kumaraswamy distribution is implemented in lmo.distributions.kumaraswamy.

Special cases

There are several notable special cases of the Kumaraswamy distribution:

Beta: With \( \alpha = 1 \) or \( \beta = 1 \) Kumaraswamy is \( \text{Beta}(1, \beta) \) or \( \text{Beta}(\alpha, 1) \), respectively.
Uniform: With \( \alpha = \beta = 1 \) Kumaraswamy is uniform on \( [0, 1] \).

Wakeby¶

The Wakeby distribution ¹⁰ is quantile-based – the CDF and PDF are not analytically expressible for the general case. Without loss of generality, Lmo uses a 3-parameter “standardized” paremetrization, with shape parameters \( \beta,\ \delta,\ \phi \).

See lmo.distributions.wakeby for the implementation.

Each of the following restrictions apply:

\( \beta + \delta \ge 0 \)
\( 0 \le \phi \le 1 \)
if \( \beta + \delta = 0 \), then \( \phi = 1 \)
if \( \phi = 0 \), then \( \beta = 0 \)
if \( \phi = 1 \), then \( \delta = 0 \)

The domain of the distribution is

\[ 0 \le x \le \begin{cases} \displaystyle \frac \phi \beta - \frac{1 - \phi}{\delta} & \text{if } \delta < 0 \\ \displaystyle \frac 1 \beta & \text{if } \beta > 0 \wedge \phi = 1 \\ \displaystyle \infty & \text{otherwise} \end{cases} \]

The PPF is defined to be

\[ x(F) = -\phi \qlog{1 - \beta}{1 - F} - (1 - \phi) \qlog{1 + \delta}{1 - F} \]

or, if \( \beta \neq 0 \) and \( \delta \neq 0 \), this is equivalent to

\[ x(F) = \frac{\phi}{\beta} (1 - (1 - F)^\beta) - \frac{1 - \phi}{\delta} (1 - (1 - F)^{-\delta}) \]

Wakeby distribution PDF

Alternative parametrization

This 3-parameter Wakeby distribution is equivalent to the 5-parameter variant that is generally used, after scaling by \( \sigma \) and shifting by \( \xi \). The shape parameters \( \beta \) and \( \delta \) are (intentionally) equivalent, the scale parameters are related by \( \alpha \equiv \sigma \phi \) and \( gamma \equiv \sigma (1 - \phi) \), and the location parameter is precisely \( \xi \).

Conversely, Lmo’s “standard” Wakeby distribution can by obtained from 5-Wakeby, by shifting and scaling s.t. \( \xi = 0 \) and \( \alpha + \gamma = 1 \). Finally, \( \phi \equiv \alpha = 1 - \gamma \) effectively combines the two scale parameters.

Lmo figured out that when \( \delta < t + 1 \), all of Wakeby’s (trimmed) L-moments can be expressed as

\[ \begin{align*} \tlmoment{s,t}{1} &= \phi \left( \begin{cases} \displaystyle H_{s + t + 1} - H_t & \text{if } \beta = 0 \\ \displaystyle \frac{1}{\beta} - \frac {\rfact{t + 1}{s + 1}} {\beta \ \rfact{t + 1 + \beta }{s + 1}} & \text{if } \beta \neq 0 \end{cases} \right) + (1 - \phi) \left( \begin{cases} \displaystyle H_{s + t + 1} - H_t & \text{if } \delta = 0 \\ \displaystyle \frac {\rfact{t + 1}{s + 1}} {\delta \rfact{t + 1 - \delta}{s + 1}} - \frac{1}{\delta} & \text{if } \delta \neq 0 \end{cases} \right) \\ \tlmoment{s,t}{r} &= \frac{\rfact{r + t}{s + 1}}{r} \left( \phi \frac {\rfact{1 - \beta}{r - 2}} { \rfact{1 + \beta + t}{r + s}} + (1 - \phi) \frac {\rfact{1 + \delta}{r - 2}} {\rfact{1 - \delta + t}{r + s}} \right) \quad \text{for } r > 1 \end{align*} \]

where \( H_n \) is a harmonic number.

Special cases

There are several notable special cases of the Wakeby distribution:

GPD – Generalized Pareto

With \( \phi = 0 \), Wakeby is the standard GPD, and \( \delta \) its shape parameter.

Conversely, \( \phi = 1 \) yields a bounded GPD variant, with shape parameter \( -\beta \), and \( 1 / \beta \) the upper bound.

Exponential

With \( \beta = \delta = 0 \) and \( \phi = 1 \), Wakeby is standard exponential.

Uniform

With \( \beta = \phi = 1 \) (and therefore \( \delta = 0 \)) Wakeby is uniform on \( [0, 1] \).

GLD¶

The GLD ¹¹ is a flexible generalization of the Tukey lambda distribution. Lmo uses an unconventional “standardized” paremetrization, with shape parameters \( \beta,\ \delta,\ \phi \), where \( \phi \in [-1, 1] \) replaces the more commonly used shape parameters \( \alpha \mapsto 1 + \phi \) and \( \gamma \mapsto 1 - \phi \).

The GLD is implemented as lmo.distributions.genlamda.

As with the Wakeby distribution, the PDF and CDF of the GLD are not analytically expressible. Instead, the GLD is defined through its PPF:

\[ x(F) = (1 + \phi) \qlog{1 - \beta}{F} - (1 - \phi) \qlog{1 - \delta}{1 - F} \]

The domain is

\[ \left. \begin{array}{ll} \text{if } \beta \le 0: & \displaystyle -\infty \\ \text{if } \beta > 0: & \displaystyle -\frac{1 + \phi}{\beta} \end{array} \right\} \le x \le \left\{ \begin{array}{ll} \displaystyle \infty \ , & \text{if } \delta \le 0 \\ \displaystyle \frac{1 - \phi}{\delta} \ , & \text{if } \delta > 0 \end{array} \right. \]

Generalized Tukey-Lambda distribution PDF

Unlike GLD’s central product-moments, which have no general closed-form expression, its trimmed L-moments can be compactly expressed. When \( \beta > -s - 1 \) and \( \delta > -t - 1 \), the L-moments are defined for \( r = 2, 3, \ldots \) and \( s, t \ge 0 \) as

\[ r \tlmoment{s, t}{r} = (-1)^r \frac{1 + \phi}{\beta + r + s + t} \frac {\B(\beta + s + 1 ,\ r - 1 - \beta)} {\B(\beta + s + t + r,\ 1 - \beta)} + \frac{1 - \phi}{\delta + r + s + t} \frac {\B(\delta + 1 + t ,\ r - 1 - \delta)} {\B(\delta + s + t + r,\ 1 - \delta)} \ , \]

and the arbitrarily-trimmed L-location is

\[ \tlmoment{s, t}{1} = -(1 + \phi) \mathfrak{L}_{1}^{(s)}(\beta) + (1 - \phi) \mathfrak{L}_{1}^{(t)}(\delta) \, \]

where

\[ \mathfrak{L}_{1}^{(k)}(\theta) = \begin{cases} \displaystyle H_{s + t + 1} - H_k & \text{if } \theta = 0 \\ \displaystyle \frac{1}{\theta}\left( 1 - \frac {\B(1 + k + \theta,\ 2 + s + t)} {\B(1 + k,\ \theta + 2 + s + t)} \right) & \text{otherwise.} \end{cases} \]

These equations look scarier that they actually are. To see why, take a look at the first 4 L-moment, with 4 styles of trimming:

L-momentsLL-momentsLH-momentsTL-moments

If \( \beta > -1 \) and \( \delta > -1 \):

\[ \begin{align*} \lmoment{1} &= -(1 + \phi) \frac {1} {1 + \beta} &&+ (1 - \phi) \frac {1} {1 + \delta} \\ \lmoment{2} &= \hphantom{-}(1 + \phi) \frac {1} {(1 + \beta)(2 + \beta)} &&+ (1 - \phi) \frac {1} {(1 + \delta)(2 + \delta)} \\ \lmoment{3} &= -(1 + \phi) \frac {1 - \beta} {(1 + \beta)(2 + \beta)(3 + \beta)} &&+ (1 - \phi) \frac {1 - \delta} {(1 + \delta)(2 + \delta)(3 + \delta)} \\ \lmoment{4} &= \hphantom{-}(1 + \phi) \frac {(1 - \beta)(2 - \beta)} {(1 + \beta)(2 + \beta)(3 + \beta)(4 + \beta)} &&+ (1 - \phi) \frac {(1 - \delta)(2 - \delta)} {(1 + \delta)(2 + \delta)(3 + \delta)(4 + \delta)} \end{align*} \]

If \( \beta > -1 \) and \( \delta > -2 \):

\[ \begin{align*} \tlmoment{0, 1}{1} &= -(1 + \phi) \frac {3 + \beta} {(1 + \beta)(2 + \beta)} &&+ (1 - \phi) \frac {1} {2 + \delta} \\ \frac{1}{3}\tlmoment{0, 1}{2} &= \hphantom{-} (1 + \phi) \frac {1} {(1 + \beta)(2 + \beta)(3 + \beta)} &&+ \frac{1 - \phi}{2} \frac {1} {(2 + \delta)(3 + \delta)} \\ \frac{1}{4} \tlmoment{0, 1}{3} &= -(1 + \phi) \frac {1 - \beta} {(1 + \beta)(2 + \beta)(3 + \beta)(4 + \beta)} &&+ \frac{1 - \phi}{3} \frac {(1 - \delta)} {(2 + \delta)(3 + \delta)(4 + \delta)} \\ \frac{1}{5} \tlmoment{0, 1}{4} &= \hphantom{-} (1 + \phi) \frac {(1 - \beta)(2 - \beta)} {(1 + \beta)(2 + \beta)(3 + \beta)(4 + \beta)(5 + \beta)} &&+ \frac{1 - \phi}{4} \frac {(1 - \delta)(2 - \delta)} {(2 + \delta)(3 + \delta)(4 + \delta)(5 + \delta)} \end{align*} \]

If \( \beta > -2 \) and \( \delta > -1 \):

\[ \begin{align*} \tlmoment{1, 0}{1} &= -(1 + \phi) \frac {1} {(2 + \beta)} &&+ (1 - \phi) \frac {3 + \beta} {(1 + \delta)(2 + \delta)} \\ \frac{1}{3}\tlmoment{1, 0}{2} &= \hphantom{-} \frac{1 + \phi}{2} \frac {1} {(2 + \beta)(3 + \beta)} &&+ (1 - \phi) \frac {1} {(1 + \delta)(2 + \delta)(3 + \delta)} \\ \frac{1}{4} \tlmoment{1, 0}{3} &= -\frac{1 + \phi}{3} \frac {1 - \beta} {(2 + \beta)(3 + \beta)(4 + \beta)} &&+ (1 - \phi) \frac {(1 - \delta)} {(1 + \delta)(2 + \delta)(3 + \delta)(4 + \delta)} \\ \frac{1}{5} \tlmoment{1, 0}{4} &= \hphantom{-} \frac{1 + \phi}{4} \frac {(1 - \beta)(2 - \beta)} {(2 + \beta)(3 + \beta)(4 + \beta)(5 + \beta)} &&+ (1 - \phi) \frac {(1 - \delta)(2 - \delta)} {(1 + \delta)(2 + \delta)(3 + \delta)(4 + \delta)(5 + \delta)} \end{align*} \]

If \( \beta > -2 \) and \( \delta > -2 \):

\[ \begin{align*} \tlmoment{1}{1} &= -(1 + \phi) \frac {5 + \beta} {(2 + \beta)(3 + \beta)} &&+ (1 - \phi) \frac {5 + \delta} {(2 + \delta)(3 + \delta)} \\ \frac{2}{3 \cdot 4} \tlmoment{1}{2} &= \hphantom{-} (1 + \phi) \frac {1} {(2 + \beta)(3 + \beta)(4 + \beta)} &&+ (1 - \phi) \frac {1} {(2 + \delta)(3 + \delta)(4 + \delta)} \\ \frac{3}{4 \cdot 5} \tlmoment{1}{3} &= -(1 + \phi) \frac {(1 - \beta)} {(2 + \beta)(3 + \beta)(4 + \beta)(5 + \beta)} &&+ (1 - \phi) \frac {(1 - \delta)} {(2 + \delta)(3 + \delta)(4 + \delta)(5 + \delta)} \\ \frac{4}{5 \cdot 6} \tlmoment{1}{4} &= \hphantom{-} (1 + \phi) \frac {(1 - \beta)(2 - \beta)} {(2 + \beta)(3 + \beta)(4 + \beta)(5 + \beta)(6 + \beta)} &&+ (1 - \phi) \frac {(1 - \delta)(2 - \delta)} {(2 + \delta)(3 + \delta)(4 + \delta)(5 + \delta)(6 + \delta)} \end{align*} \]

Special cases

There are several notable special cases of the GLD:

GPD

With \( \phi = -1 \), GLD is GPD with shape \( \alpha \equiv -\delta \) and scale \( \sigma = 2 \).

Lomax

With \( \phi = -1 \) and \( \delta < 0 \), GLD is the Lomax distribution with shape \( \alpha = -1 / \delta \) and scale \( \sigma = -2 / \delta \).

Exponential

With \( \beta = \delta = 0 \) and \( \phi = -1 \), GLD is exponential with rate \( \lambda = \frac 1 2 \), or scale \( \sigma = 2 \).

Tukey-Lambda

With \( \lambda \equiv \beta = \delta \) and \( \phi = 0 \), GLD is the standard Tukey-lambda distribution, and \( \lambda \) its shape parameter.

Logistic

With \( \beta = \delta = 0 \) and \( \phi = 0 \), GLD is standard logistic.

Uniform

With \( \beta = \delta = 1 \), GLD is uniform on \( [-1 - \phi,\ 1 - \phi] \).

With \( \beta = \delta = 2 \) and \( \phi = 0 \) GLD is uniform on \( \left[-\frac 1 2,\ \frac 1 2\right] \).

With \( \delta = 1 \) and \( \phi = -1 \), GLD is uniform on \( [0,\ 2] \)

Constants and special functions¶

An overview of the (non-obvious) mathematical notation of special functions and constants.

Name	Notation	Definition	Python
Euler–Mascheroni constant	\( \gamma_e \)	\( \displaystyle = \lim_{x \to 0} \left( {1 \over x} - \Gamma(x) \right) \\ = \int_1^\infty \left( {1 \over \lfloor x \rfloor} - {1 \over x} \right) \dd{x} \\ \approx 0.5772 \vphantom{\frac 1 1} \)	`numpy.euler_gamma`
Magic angle	\( \theta_m \)	\( = \arctan \sqrt 2 \\ = \arccos \dfrac{1}{\sqrt 3} \\ \approx 0.9553 \)	`lmo.constants.theta_m`
Incomplete Gamma function	\( \Gamma_a(z) \)	\( \displaystyle = \int_a^\infty t^{z - 1} \ e^{-t} \dd{t} \)	`lmo.special.gamma2`
Gamma function	\( \Gamma(z) \)	\( = \Gamma_0(z) \)	`math.gamma` `scipy.special.gamma`
Digamma function (a.k.a. \( \psi(z) \ \), yet “psi” \( \neq \) “digamma”)	\( \digamma(z) \)	\( \displaystyle = \frac {\Gamma^{\prime}(z)} {\Gamma(z)} \\ = \int_{\lbrack 0, 1 \rbrack} \frac {1 - t^z} {1 - t^{\hphantom{1}}} \dd{t} - \gamma_e \)	`scipy.special.digamma`
Beta function	\( \B(x,\ y) \)	\( \displaystyle = \frac {\Gamma(x) \ \Gamma(y)} {\Gamma(x + y)} \\ = \int_{\lbrack 0, 1 \rbrack} t^{x - 1} \ (1 - t)^{y - 1} \dd{t} \)	`scipy.special.beta`
Riemann zeta function	\( \zeta(z) \)	\( = \displaystyle \sum_{n = 1}^{\infty} n^{-z} \)	`scipy.special.zeta`
Factorial	\( n! \vphantom{\prod_{k=1}^n k} \)	\( = \displaystyle \prod_{k = 1}^n n \\ = \Gamma(n - 1) \)	`math.factorial` `scipy.special.factorial`
Falling factorial (a.k.a. the falling power)	\( \ffact{x}{n} \)	\( \displaystyle = \frac{x!}{(x - n)!} \\ = \frac{\Gamma(x + 1)}{\Gamma(x - n + 1)} \\ = \rfact{x - n + 1}{n} \\ \)	`lmo.special.fpow`
Rising factorial (a.k.a. the pochhammer symbol)	\( \rfact{x}{n} \)	\( \displaystyle = \frac{\Gamma(x + n)}{\Gamma(x)} \\ = \frac{(x + n - 1)!}{(x - 1)!} \\ = \ffact{x + n - 1}{n} \)	`scipy.special.poch`
Binomial coefficient	\( \displaystyle \binom n k \)	\( \displaystyle = \frac{n!}{k! \ (n - k)!} \\ = \frac{1}{k \ \B(k,\ n - k + 1)} \)	`math.comb` `scipy.special.comb`
Harmonic number	\( H_n \)	\( = \begin{cases} \displaystyle \sum_{k=1}^n \frac 1 k & \text{if } n \in \naturals \\ \digamma(n + 1) + \gamma_e & \text{otherwise} \end{cases} \)	`lmo.special.harmonic`
Jacobi polynomial	\( \jacobi{n}{\alpha}{\beta}{x} \)	\( = \displaystyle \frac{1}{2^n} \sum_{k=0}^n \binom{n + \alpha}{k} \binom{n + \beta}{n - k} (x + 1)^{n + k} (x - 1)^{n - k} \)	`scipy.special.eval_jacobi`
q-exponential	\( \qexp{1 - q}{x} \)	\( = \begin{cases} e^x & \text{if } q = 0 \\ (1 + q x)^{\frac{1}{q}} & \text{otherwise} \end{cases} \)	`scipy.special.inv_boxcox`
q-logarithm (a.k.a. the Box-Cox transform)	\( \qlog{1 - q}{y} \)	\( = \begin{cases} \ln y & \text{if } q = 0 \\ (y^q - 1) / q & \text{otherwise} \end{cases} \)	`scipy.special.boxcox`

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Arthur F Jenkinson. The frequency distribution of the annual maximum (or minimum) values of meteorological elements. Quarterly Journal of the Royal meteorological society, 81(348):158–171, 1955. ↩
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