`scipy.stats._probability_distribution:_ProbabilityDistribution.lmoment`

source: build-install/usr/lib/python3.14/site-packages/scipy/stats/_probability_distribution.py:343

Signature

def lmoment(self, order, *, standardize, method)

Summary

L-moment or L-moment ratio of positive integer order.

Extended Summary

The L-moment of order $n$ of a continuous random variable $X$ is:

λ_{n} (X) = \frac{1}{n} k = 0 \sum n - 1 (- 1)^{k} (k n - 1) E [X_{(n - k)}]

where $X_{(1)}, \dots, X_{(r)}, \dots, X_{(n)}$ are the order statistics of an independent sample of size $n$ .

The L-moment can also be expressed in terms of the random variable's inverse cumulative distribution function $F^{- 1}$ and the shifted Legendre polynomial $P_{n - 1}$ :

λ_{n} (X) = \int_{0}^{1} F^{- 1} (p) P_{n - 1} (p) d p

For order $n \geq 3$ , the "standardized" L-moment, known as the L-moment ratio, is the L-moment normalized by the L-moment of order 2, resulting in a scale invariant quantity:

τ_{n} (X) = \frac{λ _{n} ( X )}{λ _{2} ( X )}

Parameters

order : int

The positive integer order of the L-moment; i.e. $n$ in the formulae above.

standardize : bool, default: True

Whether to return L-moment ratios for orders 3 and higher. L-moment ratios are analogous to standardized conventional moments: they are the non-standardized L-moments divided by the L-moment of order 2.

method : {None, 'formula', 'general', 'order_statistics', 'quadrature_icdf', 'cache'}

The strategy used to evaluate the L-moment. By default (None), the infrastructure chooses between the following options, listed in order of precedence.

'cache': use the value of the L-moment most recently calculated via another method
'formula': use a formula specific to the distribution.
'general': use a general result that is true for all distributions with finite L-moments; for instance, the first L-moment is identically equal to the mean.
'quadrature_icdf': numerically integrate according to the definition in terms of the inverse cumulative distribution function.
'order_statistics': compute according to the definition in terms of order statistics.

Not all method options are available for all orders and distributions. If the selected method is not available, a NotImplementedError will be raised.

Returns

out : array: The L-moment of the random variable of the specified order.

Notes

L-moments are only defined for distributions with finite mean. If a formula for the L-moment is not specifically implemented for the chosen distribution, SciPy will attempt to compute the moment via a generic method, which may yield a finite result where none exists. This is not a critical bug, but an opportunity for an enhancement.

SciPy offers only basic capabilities for working with L-moments. For more advanced features, consider the lmo package ^[2].

Examples

Instantiate a distribution with the desired parameters:

import numpy as np
from scipy import stats
X = stats.Normal(mu=1., sigma=2.)

✓

Evaluate the first L-moment:

X.lmoment(order=1)
X.lmoment(order=1) == X.mean() == X.mu

✗

np.True_

Evaluate the second L-moment:

X.lmoment(order=2)

✗

np.float64(1.1283791670955126)

np.allclose(X.lmoment(order=2), X.sigma / np.sqrt(np.pi))

✓

True

Evaluate the fourth L-moment ratio, that is, the L-kurtosis:

X.lmoment(order=4)
X.lmoment(order=4) == X.lmoment(order=4, standardize=False) / X.lmoment(order=2)

✗

np.True_

Aliases

scipy.stats._distribution_infrastructure._ProbabilityDistribution.lmoment