# L-moments of the Gamma Difference Distribution This function estimates the L-moments of the Gamma Difference distribution (Klar, 2015) given the parameters ($\alpha_1 > 0$, $\beta_1 > 0$, $\alpha_2 > 0$, $\beta_2 > 0$) from `pargdd`. The L-moments in terms of the parameters higher than the mean are complex and numerical methods are required. The mean is [REMOVE_ME]$$ \lambda_1 = \frac{\alpha_1}{\beta_1} - \frac{\alpha_2}{\beta_2} \mbox{.} [REMOVE_ME_2]$$ The product moments, however, have simple expressions, the variance and skewness, respectively are [REMOVE_ME]$$ \sigma^2 = \frac{\alpha_1}{\beta_2^2} + \frac{\alpha_2}{\beta_2^2}\mbox{,} [REMOVE_ME_2]$$ and [REMOVE_ME]$$ \gamma = \frac{2\bigl(\alpha_1{\beta_2^3} + \alpha_2{\beta_2^2}\bigr)}{\bigl(\alpha_2{\beta_1^2} + \alpha_2{\beta_1^2}\bigr)^{3/2}}\mbox{.} [REMOVE_ME_2]$$ ## Description This function estimates the L-moments of the Gamma Difference distribution (Klar, 2015) given the parameters ($\alpha_1 > 0$, $\beta_1 > 0$, $\alpha_2 > 0$, $\beta_2 > 0$) from `pargdd`. The L-moments in terms of the parameters higher than the mean are complex and numerical methods are required. The mean is $$ \lambda_1 = \frac{\alpha_1}{\beta_1} - \frac{\alpha_2}{\beta_2} \mbox{.} $$ The product moments, however, have simple expressions, the variance and skewness, respectively are $$ \sigma^2 = \frac{\alpha_1}{\beta_2^2} + \frac{\alpha_2}{\beta_2^2}\mbox{,} $$ and $$ \gamma = \frac{2\bigl(\alpha_1{\beta_2^3} + \alpha_2{\beta_2^2}\bigr)}{\bigl(\alpha_2{\beta_1^2} + \alpha_2{\beta_1^2}\bigr)^{3/2}}\mbox{.} $$ ```r lmomgdd(para, nmom=6, paracheck=TRUE, silent=TRUE, ...) ``` ## Arguments - `para`: The parameters of the distribution. - `nmom`: The number of L-moment to numerically compute for the distribution. - `paracheck`: A logical controlling whether the parameters are checked for validity. - `silent`: The argument of `silent` for the `try()` operation wrapped on `integrate()`. - `...`: Additional argument to pass. ## 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: lmomgdd . ## Note Experimental Summer 2024 ---For a symmetrical version of the distribution, the relation between $\tau_4$ and $\tau_6$ and other coupling to $\lambda_2$ can be computed using the following recipe: ``` LMR <- NULL plotlmrdia46(lmrdia46(), autolegend=TRUE, xleg="topleft") for(i in 1:4000) { if(length(grep("00$", as.character(i)))) message(i) para <- 10^runif(2, min=-3, max=3) para <- list(para=c(para[1], para[2], para[1], para[2]), type="gdd") lmr <- lmomgdd(para, nmom=6, subdivisions=200) points(lmr$ratios[4], lmr$ratios[6], pch=1, cex=0.8, lwd=0.8) LMR <- rbind(LMR, data.frame(A12=para$para[1], B12=para$para[2], L1=lmr$lambdas[1], L2=lmr$lambdas[2], T3=lmr$ratios[3], T4=lmr$ratios[ 4], T5=lmr$ratios[ 3], T6=lmr$ratios[6])) } LMR <- LMR[completed.cases(LMR), ] LMR <- LMR[abs( LMR$T3) < 0.01, ] LMR <- LMR[order(LMR$T4), ] ``` We have swept through, hopefully, a sufficiently large span of viable parameter values under a constrain of symmetry. The following recipe continues in post-processing with the goal of producing a polynomial approximation between $\tau_4$ and $\tau_6$ for `lmrdia46`. ``` plotlmrdia46(lmrdia46(), autolegend=TRUE, xleg="topleft") points(LMR$T4, LMR$T6, pch=1, cex=0.8, lwd=0.8) LM <- lm(T6~I(T4 ) + I(T4^2) + I(T4^3) + I(T4^4) + I(T4^5) + I(T4^6) + I(T4^7) + I(T4^8), data=LMR) lines(LMR$T4, fitted.values(LM), col="blue", lwd=3) res <- residuals(LM) plot(fitted.values(LM), res, ylim=c(-0.02, 0.02)) abline(h=c(-0.002, 0.002), col="red") LMRthin <- LMR[abs(res) < 0.002, ] LM <- lm(T6~I(T4 ) + I(T4^2) + I(T4^3) + I(T4^4)+ I(T4^5) + I(T4^6) + I(T4^7) + I(T4^8), data=LMRthin) plot( LMRthin$T4, fitted.values(LM), col="blue", type="l", lwd=3 ) points(LMRthin$T4, LMRthin$T6, col="red", cex=0.4, lwd=0.5) tau4 <- c(lmrdia46()$nor$tau4, 0.1227, 0.123, 0.125, seq(0.13, 1, by=0.01)) tau6 <- predict(LM, newdata=data.frame(T4=tau4)) names(tau6) <- NULL gddsymt46 <- data.frame(tau4=tau4, tau6=tau6) gddsymt46f <- function(t4) { # print(coefficients(LM)) coe <- c( -0.0969112, 2.1743687, -12.8878580, 47.8931168, -108.0871549, 156.9200440, -139.5599813, 69.3492358, -14.7052424) ix <- seq_len(length(coes))-1 sapply(t4, function(t) sum(coes[ix+1]*t^ix)) } # This function is inserted into the lmrdia46() for deployment as symgdd. plotlmrdia46(lmrdia46(), autolegend=TRUE, xleg="topleft") lines( tau4, gddsymt46f(tau4), lwd=3, col="deepskyblue3") lines(gddsymt46$tau4, gddsymt46$tau6, lwd=3, col="deepskyblue3") legend("bottomright", "Symmetrical Gamma Difference distribution", bty="n", cex=0.9, lwd=3, col="deepskyblue3") ``` This is the first known derivation of the relation between these two L-moment ratios for the symmetrical version of this distribution. The quantities recorded in the `LMR` data frame in the recipe can be useful for additional study of the quality of numerical implementation of the distribution by `lmomco`. Next, for purposes of helping parameter estimation for $\alpha_1 = \alpha_2$ and $\beta_1 = \beta_2$ and $\tau_4$, let us build a polynomial for $\alpha$ estimation from $\tau_4$: ``` tlogit <- function(x) log(x/(1-x)) ilogit <- function(x) 1/(1+exp(-x)) A12l <- log(LMRthin$A12) T4l <- tlogit(LMRthin$T4 ) A12p <- exp( approx(T4l, y=A12l, xout=tlogit(tau4))$y ) plot( tlogit(tau4), A12p, log="y", col="blue", type="l", lwd=3 ) points(LMRthin$T4, LMRthin$A12, col="red", cex=0.4, lwd=0.5) ``` ## Author(s) W.H. Asquith ## See Also `pargdd`, `cdfgdd`, `pdfgdd`, `quagdd` ## Examples ```r # ```

lmomgdd function