# L-moments of the Rice Distribution This function estimates the L-moments of the Rice distribution given the parameters ($\nu$ and $\alpha$) from `parrice`. The L-moments in terms of the parameters are complex. They are computed here by the system of maximum order statistic expectations from `theoLmoms.max.ostat`, which uses `expect.max.ostat`. The connection between $\tau_2$ and $\nu/\alpha$ and a special function (the Laguerre polynomial, `LaguerreHalf`) of $\nu^2/\alpha^2$ and additional algebraic terms is tabulated in the `data.frame` located within `.lmomcohash$RiceTable`. The file SysDataBuilder01.R provides additional details. ```r lmomrice(para, ...) ``` ## Arguments - `para`: The parameters of the distribution. - `...`: Additional arguments passed to `theoLmoms.max.ostat`. ## Returns An `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: lmomrice , but the exact contents of the remainder of the string might vary as limiting distributions of Normal and Rayleigh can be involved for $\nu/\alpha > 52$ (super high SNR, Normal) or $24 < \nu/\alpha \le 52$ (high SNR, Normal) or $\nu/\alpha < 0.08$ (very low SNR, Rayleigh). ## References Asquith, W.H., 2011, Distributional analysis with L-moment statistics using the R environment for statistical computing: Createspace Independent Publishing Platform, ISBN 978--146350841--8. ## Author(s) W.H. Asquith ## See Also `parrice`, `cdfrice`, `cdfrice`, `quarice` ## Examples ```r ## Not run: lmomrice(vec2par(c(65,34), type="rice")) # Use the additional arguments to show how to avoid unnecessary overhead # when using the Rice, which only has two parameters. rice <- vec2par(c(15,14), type="rice") system.time(lmomrice(rice, nmom=2)); system.time(lmomrice(rice, nmom=6)) lcvs <- vector(mode="numeric"); i <- 0 SNR <- c(seq(7,0.25, by=-0.25), 0.1) for(snr in SNR) { i <- i + 1 rice <- vec2par(c(10,10/snr), type="rice") lcvs[i] <- lmomrice(rice, nmom=2)$ratios[2] } plot(lcvs, SNR, xlab="COEFFICIENT OF L-VARIATION", ylab="LOCAL SIGNAL TO NOISE RATIO (NU/ALPHA)") lines(.lmomcohash$RiceTable$LCV, .lmomcohash$RiceTable$SNR) abline(1,0, lty=2) mtext("Rice Distribution") text(0.15,0.5, "More noise than signal") text(0.15,1.5, "More signal than noise") ## End(Not run) ## Not run: # A polynomial expression for the relation between L-skew and # L-kurtosis for the Rice distribution can be readily constructed. T3 <- .lmomcohash$RiceTable$TAU3 T4 <- .lmomcohash$RiceTable$TAU4 LM <- lm(T4~T3+I(T3^2)+I(T3^3)+I(T3^4)+ I(T3^5)+I(T3^6)+I(T3^7)+I(T3^8)) summary(LM) # note shown ## End(Not run) ```

lmomrice function