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Parametric ABC Confidence Limits

Usage

abcpar(x, tt, S, etahat, mu, n=rep(1,length(x)),lambda=0.001, 
       alpha=c(0.025, 0.05, 0.1, 0.16))

Arguments

x vector of data
tt function of expectation parameter mu defining the parameter of interest
S maximum likelihood estimate of the covariance matrix of x
etahat maximum likelihood estimate of the natural parameter eta
mu function giving expectation of x in terms of eta
n optional argument containing denominators for binomial (vector of length length(x))
lambda optional argument specifying step size for finite difference calculation
alpha optional argument specifying confidence levels desired

Value

list with the following components
call the call to abcpar
limits The nominal confidence level, ABC point, quadratic ABC point, and standard normal point.
stats list consisting of observed value of tt, estimated standard error and estimated bias
constants list consisting of a=acceleration constant, z0=bias adjustment, cq=curvature component

References

Efron, B, and DiCiccio, T. (1992) More accurate confidence intervals in exponential families. Bimometrika 79, pages 231-245.

Efron, B. and Tibshirani, R. (1993) An Introduction to the Bootstrap. Chapman and Hall, New York, London.

Examples

# binomial
# x is a p-vector of successes, n is a p-vector of 
#  number of trials
S <- matrix(0,nrow=p,ncol=p)
S[row(S)==col(S)] <- x*(1-x/n)
mu <- function(eta,n){n/(1+exp(eta))}
etahat <- log(x/(n-x))
#suppose p=2 and we are interested in mu2-mu1
tt <- function(mu){mu[2]-mu[1]}
x <- c(2,4); n <- c(12,12)
a <- abcpar(x, tt, S, etahat,n)