lmomkmu: L-moments of the Kappa-Mu Distribution

Description

This function estimates the L-moments of the Kappa-Mu ($\kappa:\mu$) distribution given the parameters ($\nu$ and $\alpha$) from parkmu. 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 Marcum Q-function (see cdfkmu). The conventional linear combination relating the L-moments to the $\beta_r$ probability-weighted moments is $$\lambda_{r+1} = \sum_{k=0}^{r} (-1)^{r-k} {r \choose k} { r + k \choose k } \beta_k$$ for $r \ge 0$ and the linear combination relating $\alpha_r$ to $\beta_r$ is $$\alpha_r = \sum_{k=0}^r (-1)^k { r \choose k } \beta_k$$ and by definition the $\alpha_r$ are the expectations $$\alpha_r \equiv E{ X\,[1-F(X)]^r}$$ and thus $$\alpha_r = \int_{-\infty}^{\infty} x\, [1 - F(x)]^r f(x)\; \mathrm{d}x$$ in terms of $x$, the PDF $f(x)$, and the CDF $F(x)$. Lastly, the $\alpha_r$ for the Eta-Mu distribution with substitutions of the Marcum Q-function $$\alpha_r = \int_{-\infty}^{\infty} Q_\mu\biggl(\sqrt{2\kappa\mu},\; x\sqrt{2(1+\kappa)\mu}\biggr)^r\,x\, f(x)\; \mathrm{d}x$$

Yacoub (2007, eq. 5) provides an expectation for the $j$th moment of the distribution as given by $$\mathrm{E}(x^j) = \frac{\Gamma(\mu+j/2)\mathrm{exp}(-\kappa\mu)}{\Gamma(\mu)[(1+\kappa)\mu]^{j/2}} \times {}_1F_1(\mu+j/2; \mu; \kappa\mu)$$ where ${}_1F_1(a;b;z)$ is the confluent hypergeometric function of Abramowitz and Stegun (1972, eq. 13.1.2). The lmomkmu 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 ${}_1F_1(a;b;z)$ is defined as $${}_1F_1(a;b;z) = \sum_{i=0}^\infty \frac{a^{(i)}}{b^{(i)}}\frac{z^i}{n!}$$ where the notation $a^{(n)}$ represents rising factorials that are defined as $a^{(0)} = 1$ and $a^{(n)} = a(a+1)(a+2)\ldots(a+n-1)$. The rising factorials are readily computed by $a^{(n)} = \Gamma(n+1)/\Gamma(n)$ without resorting to a series computation. Yacoub (2007, eq. 5) is used to compute the mean.

Usage

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

Arguments

para

The parameters of the distribution.

nmom

The number of moments to compute.

paracheck

A logical controlling whether the parameters and checked for validity.

tol

An absolute tolerance term for series convergence of the confluent hypergeometric function when the Yacoub (2007) mean is to be computed.

maxn

The maximum number of interations in the series of the confluent hypergeometric function when the Yacoub (2007) mean is to be computed.

Value

An R list is returned.
lambdasVector of the L-moments. First element is $\lambda_1$, second element is $\lambda_2$, and so on.
ratiosVector of the L-moment ratios. Second element is $\tau$, third element is $\tau_3$ and so on.
trimLevel of symmetrical trimming used in the computation, which will equal NULL until trimming support is made.
leftrimLevel of left-tail trimming used in the computation, which will equal NULL until trimming support is made.
rightrimLevel of right-tail trimming used in the computation, which will equal NULL until trimming support is made.
sourceAn attribute identifying the computational source of the L-moments: lmomkmu.
yacoubsmeanA list containing the mean, convergence error, and number of iterations in the series until convergence.

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.

Examples

Run this code

kmu <- vec2par(c(1.19,2.3), type="kmu")
lmomkmu(kmu)

par <- vec2par(c(1.67, .5), type="kmu")
lmomkmu(par)$lambdas
cdf2lmoms(par, nmom=4)$lambdas

system.time(lmomkmu(par))
system.time(cdf2lmoms(par, nmom=4))
# See the examples under lmomemu() so visualize L-moment
# relations on the L-skew and L-kurtosis diagram

Run the code above in your browser using DataLab