# lmomrice: L-moments of the Rice Distribution In lmomco: L-Moments, Censored L-Moments, Trimmed L-Moments, L-Comoments, and Many Distributions

## Description

This function estimates the L-moments of the Rice distribution given the parameters (ν and α) 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 τ_2 and ν/α and a special function (the Laguerre polynomial, `LaguerreHalf`) of ν^2/α^2 and additional algebraic terms is tabulated in the R `data.frame` located within .lmomcohash\$RiceTable. The file ‘SysDataBuilder.R’ provides additional details.

## Usage

 `1` ```lmomrice(para, ...) ```

## Arguments

 `para` The parameters of the distribution. `...` Additional arguments passed to `theoLmoms.max.ostat`.

## Value

An R `list` is returned.

 `lambdas` Vector of the L-moments. First element is λ_1, second element is λ_2, and so on. `ratios` Vector of the L-moment ratios. Second element is τ, third element is τ_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 ν/α > 52 (super high SNR, Normal) or 24 < ν/α ≤ 52 (high SNR, Normal) or ν/α < 0.08 (very low SNR, Rayleigh).

W.H. Asquith

## 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.

`parrice`, `cdfrice`, `cdfrice`, `quarice`
 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35``` ```## 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) ```