lmomemu: L-moments of the Eta-Mu Distribution

Description

This function estimates the L-moments of the Eta-Mu ($\eta:\mu$) distribution given the parameters ($\eta$ and $\mu$) from paremu. The L-moments in terms of the parameters are complex. They are computed here by the $\alpha_r$ probability-weighted moments in terms of the Yacoub integral (see cdfemu). The linear combination relating the L-moments to the conventional $\beta_r$ probability-weighted moments is $$ \lambda_{r+1} = \sum_{k=0}^{r} (-1)^{r-k} {r \choose k} { r + k \choose k } \beta_k\mbox{,} $$ for $r \ge 0$ and the linear combination relating the less common $\alpha_r$ to $\beta_r$ is $$ \alpha_r = \sum_{k=0}^r (-1)^k { r \choose k } \beta_k\mbox{,} $$ and by definition the $\alpha_r$ are the expectations $$ \alpha_r \equiv E\{ X\,[1-F(X)]^r\}\mbox{,} $$ and thus $$ \alpha_r = \int_{-\infty}^{\infty} x\, [1 - F(x)]^r f(x)\; \mathrm{d}x\mbox{,} $$ in terms of $x$, the PDF $f(x)$, and the CDF $F(x)$. Lastly, the $\alpha_r$ for the Eta-Mu distribution with substitution of the Yacoub integral are $$ \alpha_r = \int_{-\infty}^{\infty} Y_\mu\biggl( \eta,\; x\sqrt{2h\mu} \biggr)^r\,x\, f(x)\; \mathrm{d}x\mbox{.} $$

Yacoub (2007, eq. 21) provides an expectation for the $j$th moment of the distribution as given by $$ \mathrm{E}(x^j) = \frac{\Gamma(2\mu+j/2)}{h^{\mu+j/2}(2\mu)^{j/2}\Gamma(2\mu)}\times {}_2F_1(\mu+j/4+1/2, \mu+j/4; \mu+1/2; (H/h)^2)\mbox{,} $$ where ${}_2F_1(a,b;c;z)$ is the Gauss hypergeometric function of Abramowitz and Stegun (1972, eq. 15.1.1) and $h = 1/(1-\eta^2)$ (format 2 of Yacoub's paper and the format exclusively used by lmomco). The lmomemu function optionally solves for the mean ($j=1$) using the above equation in conjunction with the mean as computed by the order statistic minimums. The ${}_2F_1(a,b;c;z)$ is defined as $$ {}_2F_1(a,b;c;z) = \frac{\Gamma(c)}{\Gamma(a)\Gamma{(b)}} \sum_{i=0}^\infty \frac{\Gamma(a+i)\Gamma{(b+i)}}{\Gamma{(c+i)}}\frac{z^i}{n!}\mbox{.} $$ Yacoub (2007, eq. 21) is used to compute the mean.

Usage

lmomemu(para, nmom=5, paracheck=TRUE, tol=1E-6, maxn=100)

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.
source: An attribute identifying the computational source of the L-moments: “lmomemu”.
yacoubsmean: A list containing the mean, convergence error, and number of iterations in the series until convergence.

Arguments

para: The parameters of the distribution.
nmom: The number of L-moments to compute.
paracheck: A logical controlling whether the parameters and checked for validity.
tol: An absolute tolerance term for series convergence of the Gauss hypergeometric function when the Yacoub (2007) mean is to be computed.
maxn: The maximum number of interations in the series of the Gauss hypergeometric function when the Yacoub (2007) mean is to be computed.

Author

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.

Yacoub, M.D., 2007, The kappa-mu distribution and the eta-mu distribution: IEEE Antennas and Propagation Magazine, v. 49, no. 1, pp. 68--81

Examples

Run this code

if (FALSE) {
emu <- vec2par(c(.19,2.3), type="emu")
lmomemu(emu)

par <- vec2par(c(.67, .5), type="emu")
lmomemu(par)$lambdas
cdf2lmoms(par, nmom=4)$lambdas
system.time(lmomemu(par))
system.time(cdf2lmoms(par, nmom=4))

# This extensive sequence of operations provides very important
# perspective on the L-moment ratio diagram of L-skew and L-kurtosis.
# But more importantly this example demonstrates the L-moment
# domain of the Kappa-Mu and Eta-Mu distributions and their boundaries.
#
t3 <- seq(-1,1,by=.0001)
plotlmrdia(lmrdia(), xlim=c(-0.05,0.5), ylim=c(-0.05,.2))
# The following polynomials are used to define the boundaries of
# both distributions. The applicable inequalities for these
# are not provided for these polynomials as would be in deeper
# implementation---so don't worry about wild looking trajectories.
"KMUup" <- function(t3) {
             return(0.1227 - 0.004433*t3 - 2.845*t3^2 +
                    + 18.41*t3^3 - 50.08*t3^4 + 83.14*t3^5 +
                    - 81.38*t3^6 + 43.24*t3^7 - 9.600*t3^8)}

"KMUdnA" <- function(t3) {
              return(0.1226 - 0.3206*t3 - 102.4*t3^2 - 4.753E4*t3^3 +
                     - 7.605E6*t3^4 - 5.244E8*t3^5 - 1.336E10*t3^6)}

"KMUdnB" <- function(t3) {
              return(0.09328 - 1.488*t3 + 16.29*t3^2 - 205.4*t3^3 +
                     + 1545*t3^4 - 5595*t3^5 + 7726*t3^6)}

"KMUdnC" <- function(t3) {
              return(0.07245 - 0.8631*t3 + 2.031*t3^2 - 0.01952*t3^3 +
                     - 0.7532*t3^4 + 0.7093*t3^5 - 0.2156*t3^6)}

"EMUup" <- function(t3) {
              return(0.1229 - 0.03548*t3 - 0.1835*t3^2 + 2.524*t3^3 +
                     - 2.954*t3^4 + 2.001*t3^5 - 0.4746*t3^6)}

# Here, we are drawing the trajectories of the tabulated parameters
# and L-moments within the internal storage of lmomco.
lines(.lmomcohash$EMU_lmompara_byeta$T3,
      .lmomcohash$EMU_lmompara_byeta$T4,   col=7, lwd=0.5)
lines(.lmomcohash$KMU_lmompara_bykappa$T3,
      .lmomcohash$KMU_lmompara_bykappa$T4, col=8, lwd=0.5)

# Draw the polynomials
lines(t3, KMUdnA(t3), lwd=4, col=2, lty=4)
lines(t3, KMUdnB(t3), lwd=4, col=3, lty=4)
lines(t3, KMUdnC(t3), lwd=4, col=4, lty=4)
lines(t3, EMUup(t3),  lwd=4, col=5, lty=4)
lines(t3, KMUup(t3),  lwd=4, col=6, lty=4)
}

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