pdfemu: Probability Density Function of the Eta-Mu Distribution

Description

This function computes the probability density of the Eta-Mu ($\eta:\mu$) distribution given parameters ($\eta$ and $\mu$) computed by paremu. The probability density function is $$ f(x) = \frac{4\sqrt{\pi}\mu^{\mu - 1/2}h^\mu}{\gamma(\mu)H^{\mu - 1/2}}\,x^{2\mu}\,\exp(-2\mu h x^2)\,I_{\mu-1/2}(2\mu H x^2)\mbox{,} $$ where $f(x)$ is the nonexceedance probability for quantile $x$, and the modified Bessel function of the first kind is $I_k(x)$, and the $h$ and $H$ are $$ h = \frac{1}{1-\eta^2}\mbox{,} $$ and $$ H = \frac{\eta}{1-\eta^2}\mbox{,} $$ for “Format 2” as described by Yacoub (2007). This format is exclusively used in the algorithms of the lmomco package.

If $\mu=1$, then the Rice distribution results, although pdfrice is not used. If $\kappa \rightarrow 0$, then the exact Nakagami-m density function results with a close relation to the Rayleigh distribution.

Define $m$ as $$m = 2\mu\biggl[1 + {\biggr(\frac{H}{h}\biggl)}^2 \biggr]\mbox{,}$$ where for a given $m$, the parameter $\mu$ must lie in the range $$m/2 \le \mu \le m\mbox{.}$$

The $I_k(x)$ for real $x > 0$ and noninteger $k$ is $$I_k(x) = \frac{1}{\pi} \int_0^\pi \exp(x\cos(\theta)) \cos(k \theta)\; \mathrm{d}\theta - \frac{\sin(k\pi)}{\pi}\int_0^\infty \exp(-x \mathrm{cosh}(t) - kt)\; \mathrm{d}t\mbox{.}$$

Usage

pdfemu(x, para, paracheck=TRUE)

Value

Probability density ($f$) for $x$.

Arguments

x: A real value vector.
para: The parameters from paremu or vec2par.
paracheck: A logical controlling whether the parameters and checked for validity.

Author

W.H. Asquith

References