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%\VignetteIndexEntry{Figure 1 in Elamir and Seheult (2004)}


\title{Figure 1 in Elamir and Seheult (2004)}
\author{Alberto Viglione}
\date{}




\begin{document}
\maketitle




First of all load the library:
<<>>=
library(nsRFA)
@
and generate the samples from the Normal distribution:
<<>>=
Nsim=1000
n=60
@
<<eval=TRUE>>=
campsimulati <- rnorm(n*Nsim)
@
<<a, eval=TRUE>>=
campsimulati <- matrix(campsimulati, ncol=n)
@
Then calculate $l_3$ and $SE(l_3)$:
<<b, eval=TRUE>>=
lmom <- t(apply(campsimulati, 1, Lmoments))
vlmom <- t(apply(campsimulati, 1, varLmoments, matrix=FALSE))

l3 <- lmom[,"lca"]*lmom[,"l2"]
sl3 <- sqrt(vlmom[,"var.l3"])
@
<<eval=TRUE>>=
l3gaussian <- l3/sl3
@
and plot the results:
<<fig=FALSE, eval=TRUE>>=
qqnorm(l3gaussian, main="Normal Q-Q Plot for Gaussian samples")
 qqline(l3gaussian)
@

Repeat the same procedure for the Student distribution:
<<eval=TRUE>>=
campsimulati <- rt(n*Nsim, df=5)
@
<<eval=TRUE>>=
<<a>>
<<b>>
@
<<eval=TRUE>>=
l3student <- l3/sl3
@
the Cauchy distribution:
<<eval=TRUE>>=
campsimulati <- rcauchy(n*Nsim)
@
<<eval=TRUE>>=
<<a>>
<<b>>
@
<<eval=TRUE>>=
l3cauchy <- l3/sl3
@
and the Uniform distribution:
<<eval=TRUE>>=
campsimulati <- runif(n*Nsim)
@
<<eval=TRUE>>=
<<a>>
<<b>>
@
<<eval=TRUE>>=
l3unif <- l3/sl3
@

Plot the result:
<<echo=FALSE, eval=TRUE>>=
#bitmap(file="Fig1.png", type="png256", height=10, width=8, res=144, pointsize=16)
png(filename="Fig1.png", height=720, width=600, res=72, pointsize=16)
@
<<fig=FALSE, eval=TRUE, width=7, height=9>>=
layout(matrix(c(1,2,3,4), 2, 2, byrow = TRUE))
qqnorm(l3gaussian, main="Normal Plot: Gaussian samples")
 qqline(l3gaussian)
qqnorm(l3student, main="Normal Plot: Student (df=5) samples")
 qqline(l3student)
qqnorm(l3cauchy, main="Normal Plot: Cauchy samples")
 qqline(l3cauchy)
qqnorm(l3unif, main="Normal Plot: Uniform samples")
 qqline(l3unif)
@
<<echo=FALSE, eval=TRUE, results=hide>>=
dev.off()
@

\begin{center}
 \includegraphics[width=.9\textwidth]{Fig1.png}
\end{center}
Normal quantile plots and added line for $N=1000$ simulated values of $l_3/SE(l_3)$ from Gaussian, Student(5), Cauchy and Uniform samples of size $n=60$.



\begin{thebibliography}{}

\bibitem[Elamir and Seheult, 2004]{ElamirSeheult2004}
Elamir, E.A.H., and Seheult, A.H. (2004).
\newblock Exact variance structure of sample L-moments.
\newblock {\em Journal of Statistical Planning and Inference}, 124:337--359.

\end{thebibliography}


\end{document}

