facmodCS (version 1.0)

dCornishFisher: Cornish-Fisher expansion

Description

Density, distribution function, quantile function and random generation using Cornish-Fisher approximation.

Usage

dCornishFisher(x, n, skew, ekurt)
pCornishFisher(q, n, skew, ekurt)
qCornishFisher(p, n, skew, ekurt)
rCornishFisher(n, sigma, skew, ekurt, dp = NULL, seed = NULL)

Value

dCornishFisher gives the density, pCornishFisher gives the distribution function, qCornishFisher gives the quantile function, and rCornishFisher generates n random simulations.

Arguments

x, q

vector of standardized quantiles.

n

scalar; number of simulated values in random simulation, sample length in density, distribution and quantile functions.

skew

scalar; skewness.

ekurt

scalar; excess kurtosis.

p

vector of probabilities.

sigma

scalar standard deviation.

dp

a vector of length 3, whose elements represent sigma, skew and ekurt, respectively. If dp is specified, the individual parameters cannot be set. Default is NULL.

seed

scalar; set seed. Default is NULL.

Author

Eric Zivot and Yi-An Chen.

Details

CDF(q) = Pr(sqrt(n)*(x_bar-mu)/sigma < q) dCornishFisher Computes Cornish-Fisher density from two term Edgeworth expansion given mean, standard deviation, skewness and excess kurtosis. pCornishFisher Computes Cornish-Fisher CDF from two term Edgeworth expansion given mean, standard deviation, skewness and excess kurtosis. qCornishFisher Computes Cornish-Fisher quantiles from two term Edgeworth expansion given mean, standard deviation, skewness and excess kurtosis. rCornishFisher simulates observations based on Cornish-Fisher quantile expansion given mean, standard deviation, skewness and excess kurtosis.

References

DasGupta, A. (2008). Asymptotic theory of statistics and probability. Springer. Severini, T. A., (2000). Likelihood Methods in Statistics. Oxford University Press.

Examples

# generate 1000 observation from Cornish-Fisher distribution
rc <- rCornishFisher(1000,1,0,5)
hist(rc, breaks=100, freq=FALSE, 
     main="simulation of Cornish Fisher Distribution", xlim=c(-10,10))
lines(seq(-10,10,0.1), dnorm(seq(-10,10,0.1), mean=0, sd=1), col=2)
# compare with standard normal curve

# exponential example from A.dasGupta p.188
# x is iid exp(1) distribution, sample size = 5
# then x_bar is Gamma(shape=5, scale=1/5) distribution
q <- c(0,0.4,1,2)
# exact cdf
pgamma(q/sqrt(5)+1, shape=5, scale=1/5)
# use CLT
pnorm(q)
# use edgeworth expansion
pCornishFisher(q, n=5, skew=2, ekurt=6)