EQL (version 1.0-1)

saddlepoint: Saddlepoint Approximation

Description

Computes the (normalized) saddlepoint approximation of the mean of \(n\) i.i.d. random variables.

Usage

saddlepoint(x, n, cumulants, correct = TRUE, normalize = FALSE)

Arguments

x

a numeric vector or array with 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.

cumulants

a cumulants object giving the cumulant functions and the saddlepoint function. See cumulants for further information.

correct

logical. If TRUE (the default) the correction term involving the 3rd and the 4th standardized cumulant functions is included.

normalize

logical. If TRUE the renormalized version of the saddlepoint approximation is calculated. The renormalized version does neither make use of the 3rd nor of the 4th cumulant function so setting correct=TRUE will result in a warning. The default is FALSE.

Value

saddlepoint returns an object of class approximation. See function approximation for further details.

Details

The saddlepoint approximation (SA) for the density of the mean \(Z=S_n/n\) of i.i.d. random variables \(Y_i\) with \(S_n=\sum_{i=1}^n Y_i\) is given by: $$f_Z(z) \approx c\sqrt{\frac{n}{2\pi K_{Y}''(s)}} \exp\{n[K_{Y}(s)- sz]\},$$ where \(c\) is an appropriatly chosen correction term, which is based on higher cumulants. The function \(K_Y(\cdot)\) denotes the cumulant generating function and \(s\) denotes the saddlepoint which is the solution of the saddlepoint function: $$K'(s)=z.$$

For the renormalized version of the SA one chooses \(c\) such that \(f_Z(z)\) integrates to one, otherwise it includes the 3rd and the 4th standardized cumulant.

The saddlepoint approximation is an improved version of the Edgeworth approximation and makes use of ‘exponential tilted’ densities. The weakness of the Edgeworth method lies in the approximation in the tails of the density. Thus, the saddlepoint approximation embed the original density in the “conjugate exponential family” with parameter \(\theta\). The mean of the embeded density depends on \(\theta\) which allows for evaluating the Edgeworth approximation at the mean, where it is known to give reasonable results.

References

Reid, N. (1991). Approximations and Asymptotics. Statistical Theory and Modelling, London: Chapman and Hall.

See Also

approximation, cumulants, edgeworth

Examples

# NOT RUN {
# Saddlepoint approximation for the density of the mean of n Gamma
# variables with shape=1 and scale=1
n <- 10
shape <- scale <- 1
x <- seq(0, 3, length=1000) 
sp <- saddlepoint(x, n, gammaCumulants(shape, scale))
plot(sp, lwd=2)

# Mean of n Gamma(1,1) variables is n*Gamma(n,1) distributed
lines(x, n*dgamma(n*x, shape=n*shape, scale=scale), col=2)

# }