Sample L-moments¶

Estimates the generalized trimmed L-moment \(\lambda^{(s, t)}_r\) from the samples along the specified axis. By default, this will be the regular L-moment, \(\lambda_r = \lambda^{(0, 0)}_r\).

Parameters:

Returns:

Examples:

Calculate the L-location and L-scale from student-T(2) samples, for different (symmetric) trim-lengths.

>>> import lmo
>>> import numpy as np
>>> rng = np.random.default_rng(12345)
>>> x = rng.standard_t(2, 99)

>>> lmo.l_moment(x, [1, 2])
array([-0.01412282,  0.94063132])
>>> lmo.l_moment(x, [1, 2], trim=1)
array([-0.0124483 ,  0.40120115])
>>> lmo.l_moment(x, [1, 2], trim=(1, 1))
array([-0.0124483 ,  0.40120115])

The theoretical L- and TL-location is 0, the L-scale is 1.1107, and the TL-scale is 0.4165, respectively.

Estimates the generalized L-moment ratio.

Equivalent to lmo.l_moment(a, r, *, **) / lmo.l_moment(a, s, *, **).

The L-moment with r=0 is 1, so the l_ratio(a, r, 0, *, **) is equivalent to l_moment(a, r, *, **).

Examples:

Estimate the L-location, L-scale, L-skewness and L-kurtosis simultaneously:

>>> import lmo
>>> import numpy as np
>>> rng = np.random.default_rng(12345)
>>> x = rng.lognormal(size=99)
>>> lmo.l_ratio(x, [1, 2, 3, 4], [0, 0, 2, 2])
array([1.53196368, 0.77549561, 0.4463163 , 0.29752178])
>>> lmo.l_ratio(x, [1, 2, 3, 4], [0, 0, 2, 2], trim=(0, 1))
array([0.75646807, 0.32203446, 0.23887609, 0.07917904])

Calculates the L-loc(ation), L-scale, L-skew(ness) and L-kurtosis.

Equivalent to lmo.l_ratio(a, [1, 2, 3, 4], [0, 0, 2, 2], *, **) by default.

Examples:

>>> import lmo, scipy.stats
>>> x = scipy.stats.gumbel_r.rvs(size=99, random_state=12345)
>>> lmo.l_stats(x)
array([0.79014773, 0.68346357, 0.12207413, 0.12829047])

The theoretical L-stats of the standard Gumbel distribution are [0.577, 0.693, 0.170, 0.150].

L-location (or L-loc): unbiased estimator of the first L-moment, \(\lambda^{(s, t)}_1\).

Alias for lmo.l_moment(a, 1, *, **).

Examples:

The first moment (i.e. the mean) of the Cauchy distribution does not exist. This means that estimating the location of a Cauchy distribution from its samples, cannot be done using the traditional average (i.e. the arithmetic mean). Instead, a robust location measure should be used, e.g. the median, or the TL-location.

To illustrate, let’s start by drawing some samples from the standard Cauchy distribution, which is centered around the origin.

>>> import lmo
>>> import numpy as np
>>> rng = np.random.default_rng(12345)
>>> x = rng.standard_cauchy(200)

The mean and the untrimmed L-location (which are equivalent) give wrong results, so don’t do this:

>>> np.mean(x)
-3.6805
>>> lmo.l_loc(x)
-3.6805

Usually, the answer to this problem is to use the median. However, the median only considers one or two samples (depending on whether the amount of samples is odd or even, respectively). So the median ignores most of the available information.

>>> np.median(x)
0.096825
>>> lmo.l_loc(x, trim=(len(x) - 1) // 2)
0.096825

Luckily for us, Lmo knows how to deal with longs tails, as well – trimming them (specifically, by skipping the first \(s\) and last \(t\) expected order statistics).

Let’s try the TL-location (which is equivalent to the median)

>>> lmo.l_loc(x, trim=1)  # equivalent to `trim=(1, 1)`
0.06522

L-scale unbiased estimator for the second L-moment, \(\lambda^{(s, t)}_2\).

Alias for lmo.l_moment(a, 2, *, **).

Examples:

>>> import lmo, numpy as np
>>> x = np.random.default_rng(12345).standard_cauchy(99)
>>> x.std()
72.87715244
>>> lmo.l_scale(x)
9.501123995
>>> lmo.l_scale(x, trim=(1, 1))
0.658993279

The coefficient of L-variation (or L-CV) unbiased sample estimator:

Alias for lmo.l_ratio(a, 2, 1, *, **).

Examples:

>>> import lmo, numpy as np
>>> x = np.random.default_rng(12345).pareto(4.2, 99)
>>> x.std() / x.mean()
1.32161112
>>> lmo.l_variation(x)
0.59073639
>>> lmo.l_variation(x, trim=(0, 1))
0.55395044

Unbiased sample estimator for the L-skewness coefficient.

Alias for lmo.l_ratio(a, 3, 2, *, **).

Examples:

>>> import lmo, numpy as np
>>> x = np.random.default_rng(12345).standard_exponential(99)
>>> lmo.l_skew(x)
0.38524343
>>> lmo.l_skew(x, trim=(0, 1))
0.27116139

L-kurtosis coefficient; the 4th sample L-moment ratio.

Alias for lmo.l_ratio(a, 4, 2, *, **).

Examples:

>>> import lmo, numpy as np
>>> x = np.random.default_rng(12345).standard_t(2, 99)
>>> lmo.l_kurtosis(x)
0.28912787
>>> lmo.l_kurtosis(x, trim=(1, 1))
0.19928182

Non-parmateric auto-covariance matrix of the generalized trimmed L-moment point estimates with orders r = 1, ..., r_max.

Returns:

Examples:

Fitting of the cauchy distribution with TL-moments. The location is equal to the TL-location, and scale should be \(0.698\) times the TL(1)-scale, see Elamir & Seheult (2003).

>>> import lmo, numpy as np
>>> rng = np.random.default_rng(12345)
>>> x = rng.standard_cauchy(1337)
>>> lmo.l_moment(x, [1, 2], trim=(1, 1))
array([0.08142405, 0.68884917])

The L-moment estimates seem to make sense. Let’s check their standard errors, by taking the square root of the variances (the diagonal of the covariance matrix):

>>> lmo.l_moment_cov(x, 2, trim=(1, 1))
array([[ 4.89407076e-03, -4.26419310e-05],
       [-4.26419310e-05,  1.30898414e-03]])
>>> np.sqrt(_.diagonal())
array([0.06995764, 0.03617989])

Non-parametric estimates of the Standard Error (SE) in the L-ratio estimates from lmo.l_ratio.

Examples:

Estimate the values and errors of the TL-loc, scale, skew and kurtosis for Cauchy-distributed samples. The theoretical values are [0.0, 0.698, 0.0, 0.343] (Elamir & Seheult, 2003), respectively.

>>> import lmo, numpy as np
>>> rng = np.random.default_rng(12345)
>>> x = rng.standard_cauchy(42)
>>> lmo.l_ratio(x, [1, 2, 3, 4], [0, 0, 2, 2], trim=(1, 1))
array([-0.25830513,  0.61738638, -0.03069701,  0.25550176])
>>> lmo.l_ratio_se(x, [1, 2, 3, 4], [0, 0, 2, 2], trim=(1, 1))
array([0.32857302, 0.12896501, 0.13835403, 0.07188138])

Calculates the standard errors (SE’s) of the L-stats.

Equivalent to lmo.l_ratio_se(a, [1, 2, 3, 4], [0, 0, 2, 2], *, **) by default.

Examples:

>>> import lmo, scipy.stats
>>> x = scipy.stats.gumbel_r.rvs(size=99, random_state=12345)
>>> lmo.l_stats(x)
array([0.79014773, 0.68346357, 0.12207413, 0.12829047])
>>> lmo.l_stats_se(x)
array([0.12305147, 0.05348839, 0.04472984, 0.03408495])

The theoretical L-stats of the standard Gumbel distribution are [0.577, 0.693, 0.170, 0.150]. The corresponding relative z-scores are [-1.730, 0.181, 1.070, 0.648].

Calculate the Empirical Influence Function (EIF) for a sample L-moment estimate.

Parameters:

Other Parameters:

Returns:

Raises:

Calculate the Empirical Influence Function (EIF) for a sample L-moment ratio estimate.

Parameters:

Other Parameters:

Returns:

Projection matrix of the first \(r\) (T)L-moments for \(n\) samples.

For integer trim is the matrix is a linear combination of the Power Weighted Moment (PWM) weights (the sample estimator of \(\beta_{r_1}\)), and the shifted Legendre polynomials.

If the trimmings are nonzero and integers, a linearized (and corrected) adaptation of the recurrence relations from Hosking (2007) are applied, as well.

for \(s > 0\), and

for \(t > 0\).

If the trim values are floats instead, the weights are calculated directly from the (generalized) order statistics. At the time of writing (07-2023), these “generalized trimmed L-moments” have not been discussed in the literature or the R-packages. It’s probably a good idea to publish this…

Parameters:

Returns:

Examples:

>>> import lmo
>>> lmo.l_weights(3, 4)
array([[ 0.25      ,  0.25      ,  0.25      ,  0.25      ],
       [-0.25      , -0.08333333,  0.08333333,  0.25      ],
       [ 0.25      , -0.25      , -0.25      ,  0.25      ]])
>>> _ @ [-1, 0, 1 / 2, 3 / 2]
array([0.25      , 0.66666667, 0.        ])