scipy.stats.Mixture.

lmoment#

Mixture.lmoment(order=1, *, standardize=False, method=None)[source]#

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

The L-moment of order \(n\) of a continuous random variable \(X\) is:

\[\lambda_n(X) = \frac{1}{n} \sum_{k=0}^{n-1} (-1)^{k} {{n-1}\choose k} E[X_{(n-k)}]\]

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 \(\widetilde{P}_{n-1}\):

\[\lambda_n(X) = \int_0^1 F^{-1}(p) \widetilde{P}_{n-1}(p) dp\]

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:

\[\tau_n(X) = \frac{\lambda_n(X)} {\lambda_2(X)}\]
Parameters:
orderint

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

standardizebool, 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:
outarray

The L-moment of the random variable of the specified order.

See also

moment
order_statistic

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].

References

[1]

L-moment, Wikipedia, https://en.wikipedia.org/wiki/L-moment

[2]

@jorenham, Lmo, jorenham/Lmo

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)
1.0
>>> X.lmoment(order=1) == X.mean() == X.mu
True

Evaluate the second L-moment:

>>> X.lmoment(order=2)
np.float64(1.1283791670955123)
>>> 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)
np.float64(0.12260171954089069)
>>> X.lmoment(order=4) == X.lmoment(order=4, standardize=False) / X.lmoment(order=2)
True
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