# cdfemu: Cumulative Distribution Function of the Eta-Mu Distribution In lmomco: L-Moments, Censored L-Moments, Trimmed L-Moments, L-Comoments, and Many Distributions

 cdfemu R Documentation

## Cumulative Distribution Function of the Eta-Mu Distribution

### Description

This function computes the cumulative probability or nonexceedance probability of the Eta-Mu (\eta:\mu) distribution given parameters (\eta and \mu) computed by parkmu. The cumulative distribution function is complex and numerical integration of the probability density function pdfemu is used or the Yacoub (2007) Y_\nu(a,b) integral. The cumulative distribution function in terms of this integral is

 F(x) = 1- Y_\nu\biggl( \frac{H}{h},\, x\sqrt{2h\mu} \biggr)\mbox{,} 

where

 Y_\nu(a,b) = \frac{2^{3/2 - \nu}\sqrt{\pi}(1-a^2)^\nu}{a^{\nu - 1/2} \Gamma(\nu)} \int_b^\infty x^{2\nu}\,\mathrm{exp}(-x^2)\,I_{\nu-1/2}(ax^2) \; \mathrm{d}x\mbox{,} 

where I_{\nu}(a) is the “\nuth-order modified Bessel function of the first kind.”

### Usage

cdfemu(x, para, paracheck=TRUE, yacoubsintegral=TRUE)


### Arguments

 x A real value vector. para The parameters from paremu or vec2par. paracheck A logical controlling whether the parameters and checked for validity. yacoubsintegral A logical controlling whether the integral by Yacoub (2007) is used instead of numerical integration of pdfemu.

### Value

Nonexceedance probability (F) for x.

W.H. Asquith

### References

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

pdfemu, quaemu, lmomemu, paremu

### Examples

para <- vec2par(c(0.5, 1.4), type="emu")
cdfemu(1.2, para, yacoubsintegral=TRUE)
cdfemu(1.2, para, yacoubsintegral=FALSE)
## Not run:
delx <- 0.01; x <- seq(0,3, by=delx)
nx <- 20*log10(x)
plot(c(-30,10), 10^c(-3,0), log="y", xaxs="i", yaxs="i",
xlab="RHO", ylab="cdfemu(RHO)", type="n")
m <- 0.75
mus <- c(0.7425, 0.7125, 0.675, 0.6, 0.5, 0.45)
for(mu in mus) {
eta <- sqrt((m / (2*mu))^-1 - 1)
lines(nx, cdfemu(x, vec2par(c(eta, mu), type="emu")))
}
mtext("Yacoub (2007, figure 8)")

# Now add some last boundary lines
mu <- m; eta <- sqrt((m / (2*mu))^-1 - 1)
lines(nx, cdfemu(x, vec2par(c(eta, mu), type="emu")),  col=8, lwd=4)
mu <- m/2; eta <- sqrt((m / (2*mu))^-1 - 1)
lines(nx, cdfemu(x, vec2par(c(eta, mu), type="emu")), col=4, lwd=2, lty=2)

delx <- 0.01; x <- seq(0,3, by=delx)
nx <- 20*log10(x)
m <- 0.75; col <- 4; lty <- 2
plot(c(-30,10), 10^c(-3,0), log="y", xaxs="i", yaxs="i",
xlab="RHO", ylab="cdfemu(RHO)", type="n")
for(mu in c(m/2,seq(m/2+0.01,m,by=0.01), m-0.001, m)) {
if(mu > 0.67) { col <- 2; lty <- 1 }
eta <- sqrt((m / (2*mu))^-1 - 1)
lines(nx, cdfemu(x, vec2par(c(eta, mu), type="emu")),
col=col, lwd=.75, lty=lty)
}
## End(Not run)


lmomco documentation built on May 29, 2024, 10:06 a.m.