saddlepoint {EQL} | R Documentation |
Computes the (normalized) saddlepoint approximation of the mean of n i.i.d. random variables.
saddlepoint(x, n, cumulants, correct = TRUE, normalize = FALSE)
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 . |
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=Y_1+...+Y_n is given by:
f_Z(z) ~ c*(2*pi)^(-1/2)*(n/K''(s))^(1/2) * exp{n*K(s)-n*s*z},
where c is an appropriatly chosen correction term, which is based on higher cumulants. The function K_Y() 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.
saddlepoint
returns an object of class approximation
. See
function approximation
for further details.
Thorn Thaler
Reid, N. (1991). Approximations and Asymptotics. Statistical Theory and Modelling, London: Chapman and Hall.
approximation
, cumulants
, edgeworth
# 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)