lmomkmu: L-moments of the Kappa-Mu Distribution

In wasquith/lmomco: L-Moments, Censored L-Moments, Trimmed L-Moments, L-Comoments, and Many Distributions

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 \lambda_1, second element is \lambda_2, and so on.
ratios	Vector of the L-moment ratios. Second element is \tau, third element is \tau_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: “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

wasquith/lmomco documentation built on Nov. 13, 2024, 4:53 p.m.