Estimate L-Moments

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Description

Estimate the r'th L-moment from a random sample.

Usage

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  lMoment(x, r = 1, method = "unbiased", 
    plot.pos.cons = c(a = 0.35, b = 0), na.rm = FALSE)

Arguments

x

numeric vector of observations.

r

positive integer specifying the order of the moment.

method

character string specifying what method to use to compute the L-moment. The possible values are "unbiased" (method based on the U-statistic; the default), or "plotting.position" (method based on the plotting position formula). See the DETAILS section for more information.

plot.pos.cons

numeric vector of length 2 specifying the constants used in the formula for the plotting positions when method="plotting.position". The default value is plot.pos.cons=c(a=0.35, b=0). If this vector has a names attribute with the value c("a","b") or c("b","a"), then the elements will be matched by name in the formula for computing the plotting positions. Otherwise, the first element is mapped to the name "a" and the second element to the name "b". See the DETAILS section for more information. This argument is ignored if method="ubiased".

na.rm

logical scalar indicating whether to remove missing values from x. If na.rm=FALSE (the default) and x contains missing values, then a missing value (NA) is returned. If na.rm=TRUE, missing values are removed from x prior to computing the L-moment.

Details

Definitions: L-Moments and L-Moment Ratios
The definition of an L-moment given by Hosking (1990) is as follows. Let X denote a random variable with cdf F, and let x(p) denote the p'th quantile of the distribution. Furthermore, let

x_{1:n} ≤ x_{2:n} ≤ … ≤ x_{n:n}

denote the order statistics of a random sample of size n drawn from the distribution of X. Then the r'th L-moment is given by:

λ_r = \frac{1}{r} ∑^{r-1}_{k=0} (-1)^k {r-1 \choose k} E[X_{r-k:r}]

for r = 1, 2, ….

Hosking (1990) shows that the above equation can be rewritten as:

λ_r = \int^1_0 x(u) P^*_{r-1}(u) du

where

P^*_r(u) = ∑^r_{k=0} p^*_{r,k} u^k

p^*_{r,k} = (-1)^{r-k} {r \choose k} {r+k \choose k} = \frac{(-1)^{r-k} (r+k)!}{(k!)^2 (r-k)!}

The first four L-moments are given by:

λ_1 = E[X]

λ_2 = \frac{1}{2} E[X_{2:2} - X_{1:2}]

λ_3 = \frac{1}{3} E[X_{3:3} - 2X_{2:3} + X_{1:3}]

λ_4 = \frac{1}{4} E[X_{4:4} - 3X_{3:4} + 3X_{2:4} - X_{1:4}]

Thus, the first L-moment is a measure of location, and the second L-moment is a measure of scale.

Hosking (1990) defines the L-moment ratios of X to be:

τ_r = \frac{λ_r}{λ_2}

for r = 2, 3, …. He shows that for a non-degenerate random variable with a finite mean, these quantities lie in the interval (-1, 1). The quantity

τ_3 = \frac{λ_3}{λ_2}

is the L-moment analog of the coefficient of skewness, and the quantity

τ_4 = \frac{λ_4}{λ_2}

is the L-moment analog of the coefficient of kurtosis. Hosking (1990) also defines an L-moment analog of the coefficient of variation (denoted the L-CV) as:

λ = \frac{λ_2}{λ_1}

He shows that for a positive-valued random variable, the L-CV lies in the interval (0, 1).

Relationship Between L-Moments and Probability-Weighted Moments
Hosking (1990) and Hosking and Wallis (1995) show that L-moments can be written as linear combinations of probability-weighted moments:

λ_r = (-1)^{r-1} ∑^{r-1}_{k=0} p^*_{r-1,k} α_k = ∑^{r-1}_{j=0} p^*_{r-1,j} β_j

where

α_k = M(1, 0, k) = \frac{1}{k+1} E[X_{1:k+1}]

β_j = M(1, j, 0) = \frac{1}{j+1} E[X_{j+1:j+1}]

See the help file for pwMoment for more information on probability-weighted moments.

Estimating L-Moments
The two commonly used methods for estimating L-moments are the “unbiased” method based on U-statistics (Hoeffding, 1948; Lehmann, 1975, pp. 362-371), and the “plotting-position” method. Hosking and Wallis (1995) recommend using the unbiased method for almost all applications.

Unbiased Estimators (method="unbiased")
Using the relationship between L-moments and probability-weighted moments explained above, the unbiased estimator of the r'th L-moment is based on unbiased estimators of probability-weighted moments and is given by:

l_r = (-1)^{r-1} ∑^{r-1}_{k=0} p^*_{r-1,k} a_k = ∑^{r-1}_{j=0} p^*_{r-1,j} b_j

where

a_k = \frac{1}{n} ∑^{n-k}_{i=1} x_{i:n} \frac{{n-i \choose k}}{{n-1 \choose k}}

b_j = \frac{1}{n} ∑^{n}_{i=j+1} x_{i:n} \frac{{i-1 \choose j}}{{n-1 \choose j}}

Plotting-Position Estimators (method="plotting.position")
Using the relationship between L-moments and probability-weighted moments explained above, the plotting-position estimator of the r'th L-moment is based on the plotting-position estimators of probability-weighted moments and is given by:

\tilde{λ}_r = (-1)^{r-1} ∑^{r-1}_{k=0} p^*_{r-1,k} \tilde{α}_k = ∑^{r-1}_{j=0} p^*_{r-1,j} \tilde{β}_j

where

\tilde{α}_k = \frac{1}{n} ∑^n_{i=1} (1 - p_{i:n})^k x_{i:n}

\tilde{β}_j = \frac{1}{n} ∑^{n}_{i=1} p^j_{i:n} x_{i:n}

and

p_{i:n} = \hat{F}(x_{i:n})

denotes the plotting position of the i'th order statistic in the random sample of size n, that is, a distribution-free estimate of the cdf of X evaluated at the i'th order statistic. Typically, plotting positions have the form:

p_{i:n} = \frac{i-a}{n+b}

where b > -a > -1. For this form of plotting position, the plotting-position estimators are asymptotically equivalent to their unbiased estimator counterparts.

Estimating L-Moment Ratios
L-moment ratios are estimated by simply replacing the population L-moments with the estimated L-moments. The estimated ratios based on the unbiased estimators are given by:

t_r = \frac{l_r}{l_2}

and the estimated ratios based on the plotting-position estimators are given by:

\tilde{τ}_r = \frac{\tilde{λ}_r}{\tilde{λ}_2}

In particular, the L-moment skew is estimated by:

t_3 = \frac{l_3}{l_2}

or

\tilde{τ}_3 = \frac{\tilde{λ}_3}{\tilde{λ}_2}

and the L-moment kurtosis is estimated by:

t_4 = \frac{l_4}{l_2}

or

\tilde{τ}_4 = \frac{\tilde{λ}_4}{\tilde{λ}_2}

Similarly, the L-moment coefficient of variation can be estimated using the unbiased L-moment estimators:

l = \frac{l_2}{l_1}

or using the plotting-position L-moment estimators:

\tilde{λ} = \frac{\tilde{λ}_2}{\tilde{λ}_1}

Value

A numeric scalar–the value of the r'th L-moment as defined by Hosking (1990).

Note

Hosking (1990) introduced the idea of L-moments, which are expectations of certain linear combinations of order statistics, as the basis of a general theory of describing theoretical probability distributions, computing summary statistics from observed data, estimating distribution parameters and quantiles, and performing hypothesis tests. The theory of L-moments parallels the theory of conventional moments. L-moments have several advantages over conventional moments, including:

  • L-moments can characterize a wider range of distributions because they always exist as long as the distribution has a finite mean.

  • L-moments are estimated by linear combinations of order statistics, so estimators based on L-moments are more robust to the presence of outliers than estimators based on conventional moments.

  • Based on the author's and others' experience, L-moment estimators are less biased and approximate their asymptotic distribution more closely in finite samples than estimators based on conventional moments.

  • L-moment estimators are sometimes more efficient (smaller RMSE) than even maximum likelihood estimators for small samples.

Hosking (1990) presents a table with formulas for the L-moments of common probability distributions. Articles that illustrate the use of L-moments include Fill and Stedinger (1995), Hosking and Wallis (1995), and Vogel and Fennessey (1993).

Hosking (1990) and Hosking and Wallis (1995) show the relationship between probabiity-weighted moments and L-moments.

Author(s)

Steven P. Millard (EnvStats@ProbStatInfo.com)

References

Fill, H.D., and J.R. Stedinger. (1995). L Moment and Probability Plot Correlation Coefficient Goodness-of-Fit Tests for the Gumbel Distribution and Impact of Autocorrelation. Water Resources Research 31(1), 225–229.

Hosking, J.R.M. (1990). L-Moments: Analysis and Estimation of Distributions Using Linear Combinations of Order Statistics. Journal of the Royal Statistical Society, Series B 52(1), 105–124.

Hosking, J.R.M., and J.R. Wallis (1995). A Comparison of Unbiased and Plotting-Position Estimators of L Moments. Water Resources Research 31(8), 2019–2025.

Vogel, R.M., and N.M. Fennessey. (1993). L Moment Diagrams Should Replace Product Moment Diagrams. Water Resources Research 29(6), 1745–1752.

See Also

cv, skewness, kurtosis, pwMoment.

Examples

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  # Generate 20 observations from a generalized extreme value distribution 
  # with parameters location=10, scale=2, and shape=.25, then compute the 
  # first four L-moments. 
  # (Note: the call to set.seed simply allows you to reproduce this example.)

  set.seed(250) 
  dat <- rgevd(20, location = 10, scale = 2, shape = 0.25) 

  lMoment(dat) 
  #[1] 10.59556 

  lMoment(dat, 2) 
  #[1] 1.0014 

  lMoment(dat, 3) 
  #[1] 0.1681165
 
  lMoment(dat, 4) 
  #[1] 0.08732692

  #----------

  # Now compute some L-moments based on the plotting-position estimators:

  lMoment(dat, method = "plotting.position") 
  #[1] 10.59556

  lMoment(dat, 2, method = "plotting.position") 
  #[1] 1.110264 

  lMoment(dat, 3, method="plotting.position", plot.pos.cons = c(.325,1)) 
  #[1] -0.4430792
 
  #----------

  # Clean up
  #---------
  rm(dat)

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