Nonparametric ABC Confidence Limits

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Description

See Efron and Tibshirani (1993) for details on this function.

Usage

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abcnon(x, tt, epsilon=0.001, 
       alpha=c(0.025, 0.05, 0.1, 0.16, 0.84, 0.9, 0.95, 0.975))

Arguments

x

the data. Must be either a vector, or a matrix whose rows are the observations

tt

function defining the parameter in the resampling form tt(p,x), where p is the vector of proportions and x is the data

epsilon

optional argument specifying step size for finite difference calculations

alpha

optional argument specifying confidence levels desired

Value

list with following components

limits

The estimated confidence points, from the ABC and standard normal methods

stats

list consisting of t0=observed value of tt, sighat=infinitesimal jackknife estimate of standard error of tt, bhat=estimated bias

constants

list consisting of a=acceleration constant, z0=bias adjustment, cq=curvature component

tt.inf

approximate influence components of tt

pp

matrix whose rows are the resampling points in the least favourable family. The abc confidence points are the function tt evaluated at these points

call

The deparsed call

References

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

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

Examples

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# compute abc intervals for the mean
x <- rnorm(10)
theta <- function(p,x) {sum(p*x)/sum(p)}
results <- abcnon(x, theta)  
# compute abc intervals for the correlation
x <- matrix(rnorm(20),ncol=2)
theta <- function(p, x)
{
    x1m <- sum(p * x[, 1])/sum(p)
    x2m <- sum(p * x[, 2])/sum(p)
    num <- sum(p * (x[, 1] - x1m) * (x[, 2] - x2m))
    den <- sqrt(sum(p * (x[, 1] - x1m)^2) *
              sum(p * (x[, 2] - x2m)^2))
    return(num/den)
}
results <- abcnon(x, theta)