Computes the Edgeworth expansion of either the
standardized mean, the mean or the sum of i.i.d. random variables.
Usage
edgeworth(x, n, rho3, rho4, mu, sigma2, deg=3,
type = c("standardized", "mean", "sum"))
Arguments
x
a numeric vector or array giving the values at which the
approximation should be evaluated.
n
a positive integer giving the number of i.i.d. random
variables in the sum.
rho3
a numeric value giving the standardized 3rd cumulant. May
be missing if deg <= 1.
rho4
a numeric value giving the standardized 4th cumulant. May
be missing if deg <= 2.
mu
a numeric value giving the mean.
May be missing if type = "standardized", since it is
only needed for transformation purposes.
sigma2
a positive numeric value giving the variance.
May be missing if type= "standardized".
deg
an integer value giving the order of the approximation:
deg=1: corresponds to a normal approximation
deg=2: takes 3rd cumulant into account
deg=3: allows for the 4th cumulant as well.
The default value is 3.
type
determines which sum should be approximated. Must be one
of (“standardized”, “mean”, “sum”),
representing the shifted and scaled sum, the weighted sum and the
raw sum. Can be abbreviated.
Details
The Edgeworth approximation (EA) for the density of the
standardized mean Z=(S_n-n*mu)/(n*sigma^2)^(1/2), where
S_n = Y_1 + … + Y_n denotes the sum of i.i.d. random
variables,
mu denotes the expected value of Y_i,
sigma^2 denotes the variance of Y_i
is given by:
f_Z(s) =
phi(z)*[1 + rho3/(6*n^(1/2))*H_3(z) +
rho4/(24*n)*H_4(z) + rho3^2/(72*n)*H_6(z)],
with phi denoting the density of the standard normal
distribution and rho3 and rho4 denoting
the 3rd and the 4th standardized cumulants of Y_i
respectively. H_n(x) denotes the nth Hermite polynomial (see
hermite for details).
The EA for the mean and the sum can be obtained by applying
the transformation theorem for densities. In this case, the expected
value mu and the variance sigma2 must be given to allow
for an appropriate transformation.
Value
edgeworth returns an object of the class
approximation. See approximation for further
details.
Author(s)
Thorn Thaler
References
Reid, N. (1991). Approximations and Asymptotics. Statistical
Theory and Modelling, London: Chapman and Hall.
See Also
approximation,hermite,saddlepoint
Examples
# Approximation of the mean of n iid Chi-squared(2) variables
n <- 10
df <- 2
mu <- df
sigma2 <- 2*df
rho3 <- sqrt(8/df)
rho4 <- 12/df
x <- seq(max(df-3*sqrt(2*df/n),0), df+3*sqrt(2*df/n), length=1000)
ea <- edgeworth(x, n, rho3, rho4, mu, sigma2, type="mean")
plot(ea, lwd=2)
# Mean of n Chi-squared(2) variables is n*Chi-squared(n*2) distributed
lines(x, n*dchisq(n*x, df=n*mu), col=2)
Results
R version 3.3.1 (2016-06-21) -- "Bug in Your Hair"
Copyright (C) 2016 The R Foundation for Statistical Computing
Platform: x86_64-pc-linux-gnu (64-bit)
R is free software and comes with ABSOLUTELY NO WARRANTY.
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Type 'license()' or 'licence()' for distribution details.
R is a collaborative project with many contributors.
Type 'contributors()' for more information and
'citation()' on how to cite R or R packages in publications.
Type 'demo()' for some demos, 'help()' for on-line help, or
'help.start()' for an HTML browser interface to help.
Type 'q()' to quit R.
> library(EQL)
Loading required package: ttutils
> png(filename="/home/ddbj/snapshot/RGM3/R_CC/result/EQL/edgeworth.Rd_%03d_medium.png", width=480, height=480)
> ### Name: edgeworth
> ### Title: Edgeworth Approximation
> ### Aliases: edgeworth
>
> ### ** Examples
>
> # Approximation of the mean of n iid Chi-squared(2) variables
>
> n <- 10
> df <- 2
> mu <- df
> sigma2 <- 2*df
> rho3 <- sqrt(8/df)
> rho4 <- 12/df
> x <- seq(max(df-3*sqrt(2*df/n),0), df+3*sqrt(2*df/n), length=1000)
> ea <- edgeworth(x, n, rho3, rho4, mu, sigma2, type="mean")
> plot(ea, lwd=2)
>
> # Mean of n Chi-squared(2) variables is n*Chi-squared(n*2) distributed
> lines(x, n*dchisq(n*x, df=n*mu), col=2)
>
>
>
>
>
> dev.off()
null device
1
>