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 \(j\)th moment of the distribution as given by
$$
\mathrm{E}(x^j) = \frac{\Gamma(\mu+j/2)\mathrm{exp}(-\kappa\mu)}{\Gamma(\mu)[(1+\kappa)\mu]^{j/2}} \times {}_1F_1(\mu+j/2; \mu; \kappa\mu)
\mbox{,}$$
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.
lmomkmu(para, nmom=5, paracheck=TRUE, tol=1E-6, maxn=100)An R
list is returned.
Vector of the L-moments. First element is \(\lambda_1\), second element is \(\lambda_2\), and so on.
Vector of the L-moment ratios. Second element is \(\tau\), third element is \(\tau_3\) and so on.
Level of symmetrical trimming used in the computation, which is 0.
Level of left-tail trimming used in the computation, which is NULL.
Level of right-tail trimming used in the computation, which is NULL.
An attribute identifying the computational source of the L-moments: “lmomkmu”.
A list containing the mean, convergence error, and number of iterations in the series until convergence.
The parameters of the distribution.
The number of moments to compute.
A logical controlling whether the parameters and checked for validity.
An absolute tolerance term for series convergence of the confluent hypergeometric function when the Yacoub (2007) mean is to be computed.
The maximum number of interations in the series of the confluent hypergeometric function when the Yacoub (2007) mean is to be computed.
W.H. Asquith
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.
parkmu, cdfkmu, pdfkmu, quakmu
kmu <- vec2par(c(1.19,2.3), type="kmu")
lmomkmu(kmu)
if (FALSE) {
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
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