# L-moments of the Eta-Mu Distribution

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### Description

This function estimates the L-moments of the Eta-Mu (η:μ) distribution given the parameters (η and μ) from paremu. The L-moments in terms of the parameters are complex. They are computed here by the α_r probability-weighted moments in terms of the Yacoub integral (see cdfemu). The linear combination relating the L-moments to the conventional β_r probability-weighted moments is

λ_{r+1} = ∑_{k=0}^{r} (-1)^{r-k} {r \choose k} { r + k \choose k } β_k\mbox{,}

for r ≥ 0 and the linear combination relating the less common α_r to β_r is

α_r = ∑_{k=0}^r (-1)^k { r \choose k } β_k\mbox{,}

and by definition the α_r are the expectations

α_r \equiv E\{ X\,[1-F(X)]^r\}\mbox{,}

and thus

α_r = \int_{-∞}^{∞} 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 α_r for the Eta-Mu distribution with substitution of the Yacoub integral are

α_r = \int_{-∞}^{∞} Y_μ\biggl( η,\; x√{2hμ} \biggr)^r\,x\, f(x)\; \mathrm{d}x\mbox{.}

Yacoub (2007, eq. 21) provides an expectation for the jth moment of the distribution as given by

\mathrm{E}(x^j) = \frac{Γ(2μ+j/2)}{h^{μ+j/2}(2μ)^{j/2}Γ(2μ)}\times {}_2F_1(μ+j/4+1/2, μ+j/4; μ+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-η^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{Γ(c)}{Γ(a)Γ{(b)}} ∑_{i=0}^∞ \frac{Γ(a+i)Γ{(b+i)}}{Γ{(c+i)}}\frac{z^i}{n!}\mbox{.}

Yacoub (2007, eq. 21) is used to compute the mean.

### Usage

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

### 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.

### Value

An R list is returned.

 lambdas Vector of the L-moments. First element is λ_1, second element is λ_2, and so on. ratios Vector of the L-moment ratios. Second element is τ, third element is τ_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.

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

paremu, cdfemu, pdfemu, quaemu
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 ## Not run: 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) ## End(Not run)