L-moments of common probability distributions¶

This page lists theoretical L-moments 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¶

An overview of the L-location, L-scale, L-skewness and L-kurtosis, of a bunch of popular univariate probability distributions, for which they exist (in closed form).

L-stats¶

TL-stats¶

Symmetrically trimmed TL-stats of some symmetric distributions.

LL-stats¶

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

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 ¹, can’t be expressed as easily as the distribution itself:

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:

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

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 ² with shape parameter \( \alpha > 0 \) and \( x \ge 0 \), has the following CDF and PPF:

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

GEV¶

The GEV distribution ³ unifies the Gumbel, Fréchet, and Weibull distributions. It has one shape parameter \( \alpha \in \mathbb{R} \), and the following distribution functions:

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

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

GLO¶

The GLO ⁴, also known as the shifted log-logistic distribution , with shape parameter \( \alpha \in \mathbb{R} \), is characterized by the following distribution functions:

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

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 ⁵, with shape parameter \( \alpha \in \mathbb{R} \), has for \( x \ge 0 \) the distribution functions:

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

where \( H_n \) is a harmonic number.

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

Burr III / Dagum¶

The Burr III distribution ⁶, 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:

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

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.

Burr XII / Pareto IV¶

The Burr XII distribution ⁶ 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:

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

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.

Kumaraswamy¶

For Kumaraswamy’s distribution ⁷ 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 ⁸. Lmo has extended these results for the general trimmed L-moments.

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

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

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

Wakeby¶

The Wakeby distribution ⁹ 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

The PPF is defined to be

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

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

where \( H_n \) is a harmonic number.

GLD¶

The GLD ¹⁰ 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:

The domain is

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

and the arbitrarily-trimmed L-location is

where

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:

Constants and special functions¶

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