lmomtri: L-moments of the Asymmetric Triangular Distribution
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lmomtri R Documentation
L-moments of the Asymmetric Triangular Distribution
Description
This function estimates the L-moments of the Asymmetric Triangular distribution given the parameters (\nu, \omega, and \psi) from partri. The first three L-moments in terms of the parameters are
\lambda_1 = \frac{(\nu+\omega+\psi)}{3}\mbox{,}
\lambda_2 = \frac{1}{15}\biggl[\frac{(\nu-\omega)^2}{(\psi-\nu)^{\phantom{1}}} - (\nu+\omega) + 2\psi\biggr] \mbox{, and}
\lambda_3 = G + H_1 + H_2 + J \mbox{,}
where G is dependent on the integral definining the L-moments in terms of the quantile function (Asquith, 2011, p. 92) with limits of integration of [0,P], H_1 and H_2 are dependent on the integral defining the L-moment in terms of the quantile function with limits of integration of [P,1], and J is dependent on the \lambda_2 and \lambda_1. Finally, the variables G, H_1, H_2, and J are
G = \frac{2}{7}\frac{(\nu+6\omega)(\omega-\nu)^3}{(\psi-\nu)^3}\mbox{,}
H_1 = \frac{12}{7}\frac{(\omega-\psi)^4}{(\nu-\psi)^3} - 2\psi\frac{(\nu-\omega)^3}{(\nu-\psi)^3} + 2\psi\mbox{,}
H_2 = \frac{4}{5}\frac{(5\nu-6\omega+\psi)(\omega-\psi)^2}{(\nu-\psi)^2}\mbox{, and}
J = -\frac{1}{15}\biggl[\frac{3(\nu-\omega)^2}{(\psi-\nu)} + 7(\nu+\omega) + 16\psi\biggl]\mbox{.}
The higher L-moments are even more ponderous and simple expressions for the L-moment ratios appear elusive.
The L-moments of the Symmetrical Triangular distribution for \tau_3 = 0 are considered by Nagaraja (2013) and therein for such a distribution having \lambda_1 = 0.5 then \lambda_4 = 0.0105 and \tau_4 = 0.09. These L-kurtosis values agree with results of this function that are based on the theoLmoms.max.ostat function. Continuing, for the symmetrical case of \omega = (\psi + \nu)/2, then
\lambda_1 = \frac{(\nu+\psi)}{2}\mbox{\ and\ } \lambda_2 = \frac{7}{60}\biggl[\psi - \nu\biggr]\mbox{,}
which might be useful for initial parameter estimation through
\psi = \lambda_1 + \frac{30}{7}\lambda_2 \mbox{\ and\ } \nu = \lambda_1 - \frac{30}{7}\lambda_2 \mbox{.}
Usage
lmomtri(para, paracheck=TRUE)
Arguments
para
The parameters of the distribution.
paracheck
A logical controlling whether the parameters and checked for validity. Overriding of this check might help in numerical optimization of parameters for modes near either the minimum or maximum. The argument here makes code base within partri a little shorter.
Value
An R list is returned.
lambdas
Vector of the L-moments. First element is
\lambda_1, second element is \lambda_2, and so on.
ratios
Vector of the L-moment ratios. Second element is
\tau, third element is \tau_3 and so on.
trim
Level of symmetrical trimming used in the computation, which is 0.
leftrim
Level of left-tail trimming used in the computation, which is NULL.
rightrim
Level of right-tail trimming used in the computation, which is NULL.
tau3poly
A cross check on the polynomial approximation scheme, and tau3poly is \tau_3 from the polynomial and not the direct solution show above, and tau3poly is not used to populate the ratios[3].
source
An attribute identifying the computational
source of the L-moments: “lmomtri”.
Note
Study of some simulation output of inst/newcoes/makelmomtri.R, bounds for \tau_3 and \tau_4 are |\tau_3| \le 0.14285717 or \tau_3 = 1/7 (numerically at tolerance 1E-7) and 1/21 < \tau_4 < 233 / 2585 (numerically at tolerances 1E-7). An approximation for \tau_4 as a function of \tau_3 is
\tau_4 = 0.090128216 - 1.7893336\tau_3^2 - 14.486666\tau_3^4 + 619.08286\tau_3^6 - 29658.04718201972\tau_3^8 \mbox{,}
where the residual standard error is {<}1.750\times 10^{-5} and the absolute value of the maximum residual is <9.338\times 10^{-5}. Lastly, an expression for \tau_4 in terms of \tau_3 by theoLmoms numerical integration is
"tau4tri" <- function(t3) {
t3[t3 < -0.1428571650715543 | t3 > 0.1428571650715543] <- NA
b <- 9.012821605998006e-02
a <- c(0, -1.789333565397665e+00, 0,
-1.448666575025029e+01, 0,
66.190828635274557e+02, 0, -2.965804718201972e+04)
t4 <- b + a[2]*t3^2 + a[4]*t3^4 + a[6]*t3^6 + a[8]*t3^8
return(t4)
}
Deprecated May 2026 in favor of built-in polynomial approximations—The 4th and 5th L-moments \lambda_4 and \lambda_5, respectively, are computed using expectations of order statistic maxima
(expect.max.ostat) and are defined (Asquith, 2011, p. 95) as
\lambda_4 = 5\mathrm{E}[X_{4:4}] - 10\mathrm{E}[X_{3:3}] + 6\mathrm{E}[X_{2:2}] - \mathrm{E}[X_{1:1}]
and
\lambda_5 = 14\mathrm{E}[X_{5:5}] - 35\mathrm{E}[X_{4:4}] + 30\mathrm{E}[X_{3:3}] - 10\mathrm{E}[X_{2:2}] + \mathrm{E}[X_{1:1}]\mbox{.}
These expressions are solved using the expect.max.ostat function to compute the \mathrm{E}[X_{r:r}]. The historical logic is retained in lmomtri but is below the return() in the source code for lmomtri().
Author(s)
W.H. Asquith
References
Asquith, W.H., 2011, Distributional analysis with L-moment statistics using the R environment for statistical computing: Createspace Independent Publishing Platform, ISBN 978–146350841–8.
Nagaraja, H.N., 2013, Moments of order statistics and L-moments for the symmetric triangular distribution: Statistics and Probability Letters, v. 83, no. 10, pp. 2357–2363.
See Also
partri, cdftri, pdftri, quatri
Examples
lmr <- lmoms(c(46, 70, 59, 36, 48, 46, 63, 35, 52))
print( lmr)
print( lmomtri(partri(lmr)) )
## Not run:
par <- vec2par(c(-405, -390, -102), type="tri")
lmomtri(par)$lambdas
# -299.0000000 39.4495050 5.5670228 1.9317517 0.8007754 0.5057529
theoLmoms.max.ostat(para=par, qua=quatri, nmom=6)$lambdas
# -299.0000126 39.4494885 5.5670486 1.9318732 0.8002989 0.5070410
# The -299 is the correct by exact solution as are 39.4495050 and 5.5670228, the
# 4-6th L-moments slightly diverge from theoLmoms.max.ostat() and its numerical
# integration relative the theoLmoms() used to form basis on which the built-in
# polynomial expressions for the 3-6 Lmoment ratios were made.
## End(Not run)
## Not run:
# At extreme limit of Tau3 for the triangular distribution, L-moment ratio diagram
# shows convergence to the trajectory of the Generalized Pareto distribution.
FF <- seq(0, 1, by=0.001)
lmr <- vec2lmom( c(10, 9, 1/7, 233 / 2585) )
parA <- partri(lmr); parB <- pargpa(lmr)
xA <- qlmomco(FF, parA); xB <- qlmomco(FF, parB); x <- sort(unique(c(xA, xB)))
plot(x, pdftri(x, parA), type="l", col=8, lwd=4) # Compare Asym. Tri. to
lines(x, pdfgpa(x, parB), col=2) # Gen. Pareto
## End(Not run)
## Not run:
# Alternative view of the higher L-moment ratios for the triangular distribution.
df <- NULL
for(i in seq_len(2000)) {
l <- -runif(1, min=0, max=1000); r <- runif(1, min=0, max=1000)
lmr <- theoLmoms(vec2par(c(l,0, r), type="tri"), nmom=6, subdivisions=200)
t3 <- lmr$ratios[3]; t4 <- lmr$ratios[4]; t5 <- lmr$ratios[5]; t6 <- lmr$ratios[6]
df <- rbind(df, data.frame(l=l, r=r, rho=abs(l/r), t3=t3, t4=t4, t5=t5, t6=t6))
}
df <- df[order(df$t3),]
rho <- 10^seq(-3, 3, by=0.01)
t3x <- approx(log10(df$rho), df$t3, xout=log10(rho), rule=2)$y
plot(rho, t3x, log="x", type="l")
t4x <- approx(log10(df$rho), df$t4, xout=log10(rho), rule=2)$y
plot(rho, t4x, log="x", type="l")
t5x <- approx(log10(df$rho), df$t5, xout=log10(rho), rule=2)$y
plot(rho, t5x, log="x", type="l")
t6x <- approx(log10(df$rho), df$t6, xout=log10(rho), rule=2)$y
plot(rho, t6x, log="x", type="l")
dfo <- data.frame(rho, t3=t3x, t4=t4x, t5=t5x, t6=t6x)
t3s <- unique(c(seq(min(dfo$t3), max(dfo$t3), by=0.001), max(dfo$t3)))
suppressWarnings( t4s <- approx(dfo$t3, dfo$t4, xout=t3s, rule=2)$y )
# In regularize.values(x, y, ties, missing(ties), na.rm = na.rm) :
# collapsing to unique 'x' values
plotlmrdia(lmrdia(), xlim=c(-0.2, .2), ylim=c(0,0.2))
lines(t3s, t4s) #
## End(Not run)
lmomco documentation built on Oct. 18, 2026, 5:07 p.m.
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| lmomtri | R Documentation |
L-moments of the Asymmetric Triangular Distribution
Description
This function estimates the L-moments of the Asymmetric Triangular distribution given the parameters (\nu, \omega, and \psi) from partri. The first three L-moments in terms of the parameters are
\lambda_1 = \frac{(\nu+\omega+\psi)}{3}\mbox{,}
\lambda_2 = \frac{1}{15}\biggl[\frac{(\nu-\omega)^2}{(\psi-\nu)^{\phantom{1}}} - (\nu+\omega) + 2\psi\biggr] \mbox{, and}
\lambda_3 = G + H_1 + H_2 + J \mbox{,}
where G is dependent on the integral definining the L-moments in terms of the quantile function (Asquith, 2011, p. 92) with limits of integration of [0,P], H_1 and H_2 are dependent on the integral defining the L-moment in terms of the quantile function with limits of integration of [P,1], and J is dependent on the \lambda_2 and \lambda_1. Finally, the variables G, H_1, H_2, and J are
G = \frac{2}{7}\frac{(\nu+6\omega)(\omega-\nu)^3}{(\psi-\nu)^3}\mbox{,}
H_1 = \frac{12}{7}\frac{(\omega-\psi)^4}{(\nu-\psi)^3} - 2\psi\frac{(\nu-\omega)^3}{(\nu-\psi)^3} + 2\psi\mbox{,}
H_2 = \frac{4}{5}\frac{(5\nu-6\omega+\psi)(\omega-\psi)^2}{(\nu-\psi)^2}\mbox{, and}
J = -\frac{1}{15}\biggl[\frac{3(\nu-\omega)^2}{(\psi-\nu)} + 7(\nu+\omega) + 16\psi\biggl]\mbox{.}
The higher L-moments are even more ponderous and simple expressions for the L-moment ratios appear elusive.
The L-moments of the Symmetrical Triangular distribution for \tau_3 = 0 are considered by Nagaraja (2013) and therein for such a distribution having \lambda_1 = 0.5 then \lambda_4 = 0.0105 and \tau_4 = 0.09. These L-kurtosis values agree with results of this function that are based on the theoLmoms.max.ostat function. Continuing, for the symmetrical case of \omega = (\psi + \nu)/2, then
\lambda_1 = \frac{(\nu+\psi)}{2}\mbox{\ and\ } \lambda_2 = \frac{7}{60}\biggl[\psi - \nu\biggr]\mbox{,}
which might be useful for initial parameter estimation through
\psi = \lambda_1 + \frac{30}{7}\lambda_2 \mbox{\ and\ } \nu = \lambda_1 - \frac{30}{7}\lambda_2 \mbox{.}
Usage
lmomtri(para, paracheck=TRUE)
Arguments
para |
The parameters of the distribution. |
paracheck |
A logical controlling whether the parameters and checked for validity. Overriding of this check might help in numerical optimization of parameters for modes near either the minimum or maximum. The argument here makes code base within |
Value
An R list is returned.
lambdas |
Vector of the L-moments. First element is
|
ratios |
Vector of the L-moment ratios. Second element is
|
trim |
Level of symmetrical trimming used in the computation, which is |
leftrim |
Level of left-tail trimming used in the computation, which is |
rightrim |
Level of right-tail trimming used in the computation, which is |
tau3poly |
A cross check on the polynomial approximation scheme, and |
source |
An attribute identifying the computational source of the L-moments: “lmomtri”. |
Note
Study of some simulation output of inst/newcoes/makelmomtri.R, bounds for \tau_3 and \tau_4 are |\tau_3| \le 0.14285717 or \tau_3 = 1/7 (numerically at tolerance 1E-7) and 1/21 < \tau_4 < 233 / 2585 (numerically at tolerances 1E-7). An approximation for \tau_4 as a function of \tau_3 is
\tau_4 = 0.090128216 - 1.7893336\tau_3^2 - 14.486666\tau_3^4 + 619.08286\tau_3^6 - 29658.04718201972\tau_3^8 \mbox{,}
where the residual standard error is {<}1.750\times 10^{-5} and the absolute value of the maximum residual is <9.338\times 10^{-5}. Lastly, an expression for \tau_4 in terms of \tau_3 by theoLmoms numerical integration is
"tau4tri" <- function(t3) {
t3[t3 < -0.1428571650715543 | t3 > 0.1428571650715543] <- NA
b <- 9.012821605998006e-02
a <- c(0, -1.789333565397665e+00, 0,
-1.448666575025029e+01, 0,
66.190828635274557e+02, 0, -2.965804718201972e+04)
t4 <- b + a[2]*t3^2 + a[4]*t3^4 + a[6]*t3^6 + a[8]*t3^8
return(t4)
}
Deprecated May 2026 in favor of built-in polynomial approximations—The 4th and 5th L-moments \lambda_4 and \lambda_5, respectively, are computed using expectations of order statistic maxima
(expect.max.ostat) and are defined (Asquith, 2011, p. 95) as
\lambda_4 = 5\mathrm{E}[X_{4:4}] - 10\mathrm{E}[X_{3:3}] + 6\mathrm{E}[X_{2:2}] - \mathrm{E}[X_{1:1}]
and
\lambda_5 = 14\mathrm{E}[X_{5:5}] - 35\mathrm{E}[X_{4:4}] + 30\mathrm{E}[X_{3:3}] - 10\mathrm{E}[X_{2:2}] + \mathrm{E}[X_{1:1}]\mbox{.}
These expressions are solved using the expect.max.ostat function to compute the \mathrm{E}[X_{r:r}]. The historical logic is retained in lmomtri but is below the return() in the source code for lmomtri().
Author(s)
W.H. Asquith
References
Asquith, W.H., 2011, Distributional analysis with L-moment statistics using the R environment for statistical computing: Createspace Independent Publishing Platform, ISBN 978–146350841–8.
Nagaraja, H.N., 2013, Moments of order statistics and L-moments for the symmetric triangular distribution: Statistics and Probability Letters, v. 83, no. 10, pp. 2357–2363.
See Also
partri, cdftri, pdftri, quatri
Examples
lmr <- lmoms(c(46, 70, 59, 36, 48, 46, 63, 35, 52))
print( lmr)
print( lmomtri(partri(lmr)) )
## Not run:
par <- vec2par(c(-405, -390, -102), type="tri")
lmomtri(par)$lambdas
# -299.0000000 39.4495050 5.5670228 1.9317517 0.8007754 0.5057529
theoLmoms.max.ostat(para=par, qua=quatri, nmom=6)$lambdas
# -299.0000126 39.4494885 5.5670486 1.9318732 0.8002989 0.5070410
# The -299 is the correct by exact solution as are 39.4495050 and 5.5670228, the
# 4-6th L-moments slightly diverge from theoLmoms.max.ostat() and its numerical
# integration relative the theoLmoms() used to form basis on which the built-in
# polynomial expressions for the 3-6 Lmoment ratios were made.
## End(Not run)
## Not run:
# At extreme limit of Tau3 for the triangular distribution, L-moment ratio diagram
# shows convergence to the trajectory of the Generalized Pareto distribution.
FF <- seq(0, 1, by=0.001)
lmr <- vec2lmom( c(10, 9, 1/7, 233 / 2585) )
parA <- partri(lmr); parB <- pargpa(lmr)
xA <- qlmomco(FF, parA); xB <- qlmomco(FF, parB); x <- sort(unique(c(xA, xB)))
plot(x, pdftri(x, parA), type="l", col=8, lwd=4) # Compare Asym. Tri. to
lines(x, pdfgpa(x, parB), col=2) # Gen. Pareto
## End(Not run)
## Not run:
# Alternative view of the higher L-moment ratios for the triangular distribution.
df <- NULL
for(i in seq_len(2000)) {
l <- -runif(1, min=0, max=1000); r <- runif(1, min=0, max=1000)
lmr <- theoLmoms(vec2par(c(l,0, r), type="tri"), nmom=6, subdivisions=200)
t3 <- lmr$ratios[3]; t4 <- lmr$ratios[4]; t5 <- lmr$ratios[5]; t6 <- lmr$ratios[6]
df <- rbind(df, data.frame(l=l, r=r, rho=abs(l/r), t3=t3, t4=t4, t5=t5, t6=t6))
}
df <- df[order(df$t3),]
rho <- 10^seq(-3, 3, by=0.01)
t3x <- approx(log10(df$rho), df$t3, xout=log10(rho), rule=2)$y
plot(rho, t3x, log="x", type="l")
t4x <- approx(log10(df$rho), df$t4, xout=log10(rho), rule=2)$y
plot(rho, t4x, log="x", type="l")
t5x <- approx(log10(df$rho), df$t5, xout=log10(rho), rule=2)$y
plot(rho, t5x, log="x", type="l")
t6x <- approx(log10(df$rho), df$t6, xout=log10(rho), rule=2)$y
plot(rho, t6x, log="x", type="l")
dfo <- data.frame(rho, t3=t3x, t4=t4x, t5=t5x, t6=t6x)
t3s <- unique(c(seq(min(dfo$t3), max(dfo$t3), by=0.001), max(dfo$t3)))
suppressWarnings( t4s <- approx(dfo$t3, dfo$t4, xout=t3s, rule=2)$y )
# In regularize.values(x, y, ties, missing(ties), na.rm = na.rm) :
# collapsing to unique 'x' values
plotlmrdia(lmrdia(), xlim=c(-0.2, .2), ylim=c(0,0.2))
lines(t3s, t4s) #
## End(Not run)
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