| lmomkmu {lmomco} | R Documentation |
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 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
\mbox{,}
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
\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 Kappa-Mu distribution with substitutions of the Marcum Q-function are
\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\mbox{.}
Although multiple methods for Marcum Q-function computation are in cdfkmu and discussed in that documentation, the lmomkmu presenting is built only using the “chisq” approach.
Yacoub (2007, eq. 5) provides an expectation for the jth 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)
\mbox{,}
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!}
\mbox{,}
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.
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 |
source |
An attribute identifying the computational source of the L-moments: “lmomkmu”. |
yacoubsmean |
A list containing the mean, convergence error, and number of iterations in the series until convergence. |
Author(s)
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.
See Also
parkmu, cdfkmu, pdfkmu, quakmu
Examples
kmu <- vec2par(c(1.19,2.3), type="kmu")
lmomkmu(kmu)
## Not run:
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))
## End(Not run)
# See the examples under lmomemu() so visualize L-moment
# relations on the L-skew and L-kurtosis diagram