# Estimate the Parameters of the Gamma Distribution This function estimates the parameters of the Gamma distribution given the L-moments of the data in an L-moment object such as that returned by `lmoms`. Both the two-parameter Gamma and three-parameter Generalized Gamma distributions are supported based on the desired choice of the user, and numerical-hybrid methods are required. The `pdfgam` documentation provides further details. ```r pargam(lmom, p=c("2", "3"), checklmom=TRUE, ...) ``` ## Arguments - `lmom`: A L-moment object created by `lmoms` or `vec2lmom`. - `p`: The number of parameters to estimate for the 2-p Gamma or 3-p Generalized Gamma. - `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). However, for some circumstances or large simulation exercises then one might want to bypass this check. - `...`: Other arguments to pass. ## Returns An `list` is returned. - **type**: The type of distribution: `gam`. - **para**: The parameters of the distribution. - **source**: The source of the parameters: pargam . ## Note The two-parameter Gamma is supported by Hosking's code-based approximations to avoid direct numerical techniques. The three-parameter version is based on a dual approach to parameter optimization. The $\log(\sigma)$ and $\sqrt{\log(\lambda_1/\lambda_2)}$ conveniently has a relatively narrow range of variation. A polynomial approximation to provide a first estimate of $\sigma$ (named $\sigma'$) is used through the `optim()` function to isolated the best estimates of $\mu'$ and $\nu'$ of the distribution holding $\sigma$ constant at $\sigma = \sigma'$---a 2D approach is thus involved. Then, the initial parameter for a second three-dimensional optimization is made using the initial parameter estimates as the tuple $\mu', \sigma', \nu'$. This 2D approach seems more robust and effectively canvases more of the Generalized Gamma parameter domain, though a doubled-optimization is not quite as fast as a direct 3D optimization. The following code was used to derive the polynomial coefficients used for the first approximation of $sigma'$: ``` nsim <- 10000; mu <- sig <- nu <- l1 <- l2 <- t3 <- t4 <- rep(NA, nsim) for(i in 1:nsim) { m <- exp(runif(1, min=-4, max=4)); s <- exp(runif(1, min=-8, max=8)) n <- runif(1, min=-14, max=14); mu[i] <- m; sig[i] <- s; nu[i] <- n para <- vec2par(c(m,s,n), type="gam"); lmr <- lmomgam(para) if(is.null(lmr)) next lam <- lmr$lambdas[1:2]; rat <- lmr$ratios[3:4] l1[i]<-lam[1]; l2[i]<-lam[2];t3[i]<-rat[1]; t4[i]<-rat[2] } ZZ <- data.frame(mu=mu, sig=sig, nu=nu, l1=l1, l2=l2, t3=t3, t4=t4) ZZ$ETA <- sqrt(log(ZZ$l1/ZZ$l2)); ZZ <- ZZ[complete.cases(ZZ), ] ix <- 1:length(ZZ$ETA); ix <- ix[(ZZ$ETA < 0.025 & log(ZZ$sig) < 1)] ZZ <- ZZ[-ix,] with(ZZ, plot(ETA, log(sig), xlim=c(0,4), ylim=c(-8,8))) LM <- lm(log(sig)~ I(1/ETA^1)+I(1/ETA^2)+I(1/ETA^3)+I(1/ETA^4)+I(1/ETA^5)+ ETA +I( ETA^2)+I( ETA^3)+I( ETA^4)+I( ETA^5), data=ZZ) ETA <- seq(0,4,by=0.002) # so the line of fit can be seen lines(ETA, predict(LM, newdata=list(ETA=ETA)), col=2) The.Coefficients.In.pargam.Function <- LM$coefficients ``` ## References Hosking, J.R.M., 1990, L-moments---Analysis and estimation of distributions using linear combinations of order statistics: Journal of the Royal Statistical Society, Series B, v. 52, pp. 105--124. Hosking, J.R.M., 1996, FORTRAN routines for use with the method of L-moments: Version 3, IBM Research Report RC20525, T.J. Watson Research Center, Yorktown Heights, New York. Hosking, J.R.M., and Wallis, J.R., 1997, Regional frequency analysis---An approach based on L-moments: Cambridge University Press. ## Author(s) W.H. Asquith ## See Also `lmomgam`, `cdfgam`, `pdfgam`, `quagam` ## Examples ```r pargam(lmoms(abs(rnorm(20, mean=10)))) ## Not run: pargam(lmomgam(vec2par(c(0.3,0.4,+1.2), type="gam")), p=3)$para pargam(lmomgam(vec2par(c(0.3,0.4,-1.2), type="gam")), p=3)$para # mu sigma nu # 0.2999994 0.3999990 1.1999696 # 0.2999994 0.4000020 -1.2000567 ## End(Not run) ```

pargam function