# Numerically Integrated L-moments of Smoothed Quantiles from Bernstein or Kantorovich Polynomials Compute the L-moment by numerical integration of the smoothed quantiles from Bernstein or Kantorovich polynomials (see `dat2bernqua`). Letting $\tilde{X}_n(F)$ be the smoothed quantile function for nonexceedance probability $F$ for a sample of size $n$, from Asquith (2011) the first five L-moments in terms of quantile function integration are [REMOVE_ME]$$ \lambda_1 = \int_0^1 \tilde{X}_n(F)\;\mathrm{d}F \mbox{,} [REMOVE_ME_2]$$ [REMOVE_ME]$$ \lambda_2 = \int_0^1 \tilde{X}_n(F)\times(2F - 1)\;\mathrm{d}F\mbox{,} [REMOVE_ME_2]$$ [REMOVE_ME]$$ \lambda_3 = \int_0^1 \tilde{X}_n(F)\times(6F^2 - 6F + 1)\;\mathrm{d}F\mbox{,} [REMOVE_ME_2]$$ [REMOVE_ME]$$ \lambda_4 = \int_0^1 \tilde{X}_n(F)\times(20F^3 - 30F^2 + 12F - 1)\;\mathrm{d}F\mbox{, and} [REMOVE_ME_2]$$ [REMOVE_ME]$$ \lambda_5 = \int_0^1 \tilde{X}_n(F)\times(70F^4 - 140F^3 + 90F^2 - 20F + 1)\;\mathrm{d}F\mbox{.} [REMOVE_ME_2]$$ utf8 ## Description Compute the L-moment by numerical integration of the smoothed quantiles from Bernstein or Kantorovich polynomials (see `dat2bernqua`). Letting $\tilde{X}_n(F)$ be the smoothed quantile function for nonexceedance probability $F$ for a sample of size $n$, from Asquith (2011) the first five L-moments in terms of quantile function integration are $$ \lambda_1 = \int_0^1 \tilde{X}_n(F)\;\mathrm{d}F \mbox{,} $$ $$ \lambda_2 = \int_0^1 \tilde{X}_n(F)\times(2F - 1)\;\mathrm{d}F\mbox{,} $$ $$ \lambda_3 = \int_0^1 \tilde{X}_n(F)\times(6F^2 - 6F + 1)\;\mathrm{d}F\mbox{,} $$ $$ \lambda_4 = \int_0^1 \tilde{X}_n(F)\times(20F^3 - 30F^2 + 12F - 1)\;\mathrm{d}F\mbox{, and} $$ $$ \lambda_5 = \int_0^1 \tilde{X}_n(F)\times(70F^4 - 140F^3 + 90F^2 - 20F + 1)\;\mathrm{d}F\mbox{.} $$ ```r lmoms.bernstein(x, bern.control=NULL, poly.type=c("Bernstein", "Kantorovich", "Cheng"), bound.type=c("none", "sd", "Carv", "either"), fix.lower=NULL, fix.upper=NULL, p=0.05) ``` ## Arguments - `x`: A vector of data values. - `bern.control`: A `list` that holds `poly.type`, `bound.type`, `fix.lower`, and `fix.upper`. And this list will supersede the respective values provided as separate arguments. - `poly.type`: Same argument as for `dat2bernqua`. - `bound.type`: Same argument as for `dat2bernqua`. - `fix.lower`: Same argument as for `dat2bernqua`. - `fix.upper`: Same argument as for `dat2bernqua`. - `p`: The p-factor is the same argument as for `dat2bernqua`. ## Returns An `vector` is returned. ## Author(s) 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. ## See Also `dat2bernqua`, `pfactor.bernstein`, `lmoms` ## Examples ```r ## Not run: X <- exp(rnorm(100)) lmoms.bernstein(X)$ratios lmoms.bernstein(X, fix.lower=0)$ratios lmoms.bernstein(X, fix.lower=0, bound.type="sd")$ratios lmoms.bernstein(X, fix.lower=0, bound.type="Carv")$ratios lmoms(X)$ratios lmoms.bernstein(X, poly.type="Kantorovich")$ratios lmoms.bernstein(X, fix.lower=0, poly.type="Kantorovich")$ratios lmoms.bernstein(X, fix.lower=0, bound.type="sd", poly.type="Kantorovich")$ratios lmoms.bernstein(X, fix.lower=0, bound.type="Carv", poly.type="Kantorovich")$ratios lmoms(X)$ratios ## End(Not run) ## Not run: lmr <- vec2lmom(c(1,.2,.3)) par <- lmom2par(lmr, type="gev") lmr <- lmorph(par2lmom(par)) lmT <- c(lmr$lambdas[1:2], lmr$ratios[3:5]) ns <- 200; nsim <- 1000; empty <- rep(NA, nsim) sink("ChengLmomentTest.txt") cat(c("N errmeanA errlscaleA errtau3A errtau4A errtau5A", "errmeanB errlscaleB errtau3B errtau4B errtau5B\n")) for(n in 1:ns) { message(n); SIM <- data.frame(errmeanA=empty, errlscaleA=empty, errtau3A=empty, errtau4A=empty, errtau5A=empty, errmeanB=empty, errlscaleB=empty, errtau3B=empty, errtau4B=empty, errtau5B=empty) for(i in 1:nsim) { X <- rlmomco(30, par) lmrA <- lmoms(X) lmA <- c(lmrA$lambdas[1:2], lmrA$ratios[3:5]) lmrB <- lmoms.bernstein(X, poly.type="Cheng") lmB <- c(lmrB$lambdas[1:2], lmrB$ratios[3:5]) EA <- lmA - lmT; EB <- lmB - lmT SIM[i,] <- c(EA,EB) } MeanErr <- sapply(1:length(SIM[1,]), function(x) { return(mean(SIM[,x])) }) line <- paste(c(n, round(MeanErr, digits=6), "\n"), sep=" ") cat(line) } sink() ## End(Not run) ```

lmoms.bernstein function