# Estimate the Parameters of the Kappa-Mu Distribution This function estimates the parameters ($\nu$ and $\alpha$) of the Kappa-Mu ($\kappa:\mu$) distribution given the L-moments of the data in an L-moment object such as that returned by `lmoms`. The relations between distribution parameters and L-moments are seen under `lmomkmu`. The basic approach for parameter optimization is to extract initial guesses for the parameters from the table `KMU_lmompara_bykappa` in the `.lmomcohash` environment. The parameters having a minimum Euclidean error as controlled by three arguments are used for initial guesses in a Nelder-Mead simplex multidimensional optimization using the function `optim` and default arguments. Limited testing indicates that of the error term controlling options that the default values as shown in the Usage section seem to provide superior performance in terms of recovering the **a priori known** parameters in experiments. It seems that only Euclidean optimization using L-skew and L-kurtosis is preferable, but experiments show the general algorithm to be slow. ```r parkmu(lmom, checklmom=TRUE, checkbounds=TRUE, alsofitT3=FALSE, alsofitT3T4=FALSE, alsofitT3T4T5=FALSE, justfitT3T4=TRUE, boundary.tolerance=0.001, verbose=FALSE, trackoptim=TRUE, ...) ``` ## Arguments - `lmom`: An L-moment object created by `lmoms` or `pwm2lmom`. - `checklmom`: Should the `lmom` be checked for validity using the `are.lmom.valid` function. Normally this should be left as the default and it is very unlikely that the L-moments will not be viable (particularly in the $\tau_4$ and $\tau_3$ inequality). - `checkbounds`: Should the L-skew and L-kurtosis boundaries of the distribution be checked. - `alsofitT3`: Logical when true will add the error term $(\hat\tau_3 - \tau_3)^2$ to the sum of square errors for the mean and L-CV. - `alsofitT3T4`: Logical when true will add the error term $(\hat\tau_3 - \tau_3)^2 + (\hat\tau_4 - \tau_4)^2$ to the sum of square errors for the mean and L-CV. - `alsofitT3T4T5`: Logical when true will add the error term $(\hat\tau_3 - \tau_3)^2 + (\hat\tau_4 - \tau_4)^2 + (\hat\tau_5 - \tau_5)^2$ to the sum of square errors for the mean and L-CV. - `justfitT3T4`: Logical when true will only consider the sum of squares errors for L-skew and L-kurtosis as mathematically shown for `alsofitT3T4`. - `boundary.tolerance`: A fudge number to help guide how close to the boundaries an arbitrary list of $\tau_3$ and $\tau_4$ can be to consider them formally in or out of the attainable $\{\tau_3, \tau_4\}$ domain. - `verbose`: A logical to control a level of diagnostic output. - `trackoptim`: A logical to control specific messaging through each iteration of the objective function. - `...`: Other arguments to pass. ## Returns An `list` is returned. - **type**: The type of distribution: `kmu`. - **para**: The parameters of the distribution. - **source**: The source of the parameters: parkmu . ## 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 ## Author(s) W.H. Asquith ## See Also `lmomkmu`, `cdfkmu`, `pdfkmu`, `quakmu` ## Examples ```r ## Not run: par1 <- vec2par(c(0.7, 0.2), type="kmu") lmr1 <- lmomkmu(par1, nmom=4) par2.1 <- parkmu(lmr1, alsofitT3=TRUE, verbose=TRUE, trackoptim=TRUE) par2.1$para par2.2 <- parkmu(lmr1, alsofitT3T4=TRUE, verbose=TRUE, trackoptim=TRUE) par2.2$para par2.3 <- parkmu(lmr1, alsofitT3=FALSE, verbose=TRUE, trackoptim=TRUE) par2.3$para par2.4 <- parkmu(lmr1, justfitT3T4=TRUE, verbose=TRUE, trackoptim=TRUE) par2.4$para x <- seq(0,3,by=.01) plot(x, pdfkmu(x, par1), type="l", lwd=6, col=8, ylim=c(0,5)) lines(x, pdfkmu(x, par2.1), col=2, lwd=2, lty=2) lines(x, pdfkmu(x, par2.2), col=4) lines(x, pdfkmu(x, par2.3), col=3, lty=3, lwd=2) lines(x, pdfkmu(x, par2.4), col=5, lty=2, lwd=2) ## End(Not run) ## Not run: par1 <- vec2par(c(1, 0.65), type="kmu") lmr1 <- lmomkmu(par1, nmom=4) par2.1 <- parkmu(lmr1, alsofitT3=TRUE, verbose=TRUE, trackoptim=TRUE) par2.1$para # eta=1.0 mu=0.65 par2.2 <- parkmu(lmr1, alsofitT3T4=TRUE, verbose=TRUE, trackoptim=TRUE) par2.2$para # eta=1.0 mu=0.65 par2.3 <- parkmu(lmr1, alsofitT3=FALSE, verbose=TRUE, trackoptim=TRUE) par2.3$para # eta=8.5779 mu=0.2060 par2.4 <- parkmu(lmr1, justfitT3T4=TRUE, verbose=TRUE, trackoptim=TRUE) par2.4$para # eta=1.0 mu=0.65 x <- seq(0,3,by=.01) plot(x, pdfkmu(x, par1), type="l", lwd=6, col=8, ylim=c(0,1)) lines(x, pdfkmu(x, par2.1), col=2, lwd=2, lty=2) lines(x, pdfkmu(x, par2.2), col=4) lines(x, pdfkmu(x, par2.3), col=3, lty=3, lwd=2) lines(x, pdfkmu(x, par2.4), col=5, lty=2, lwd=2) lines(x, dlmomco(x, lmom2par(lmr1, type="gam")), lwd=2, col=2) lines(x, dlmomco(x, lmom2par(lmr1, type="ray")), lwd=2, col=2, lty=2) lines(x, dlmomco(x, lmom2par(lmr1, type="rice")), lwd=2, col=4, lty=2) ## End(Not run) ```

parkmu function