lmomco (version 2.0.1)

lmoms.bernstein: Numerically Integrated L-moments of Smoothed Quantiles from Bernstein or Kantorovich Polynomials

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 $$\lambda_1 = \int_0^1 \tilde{X}_n(F)\;\mathrm{d}F \mbox{,}$$ $$\lambda_2 = \int_0^1 \tilde{X}_n(F)\times(2F - 1)\;\mathrm{d}F\mbox{,}$$ $$\lambda_3 = \int_0^1 \tilde{X}_n(F)\times(6F^2 - 6F + 1)\;\mathrm{d}F\mbox{,}$$ $$\lambda_4 = \int_0^1 \tilde{X}_n(F)\times(20F^3 - 30F^2 + 12F - 1)\;\mathrm{d}F\mbox{, and}$$ $$\lambda_5 = \int_0^1 \tilde{X}_n(F)\times(70F^4 - 140F^3 + 90F^2 - 20F + 1)\;\mathrm{d}F\mbox{.}$$

Usage

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 that is the same argument as for dat2bernqua.

Value

• An R vector is returned.

encoding

utf8

References

Asquith, W.H., 2011, Distributional analysis with L-moment statistics using the R environment for statistical computing: First edition, CreateSpace Independent Publishing Platform, ISBN 978--1463508418, 2nd printing, 348 p.

Asquith, W.H., 201X. Empirical quantile function smoothing and distal-tail estimation by Bernstein and Kantorovich polynomials incorporating regional L-moment information. [in review].

See Also

dat2bernqua, pfactor.bernstein, lmoms

Examples

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

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()