

   NNoonnppaarraammeettrriicc AABBCC CCoonnffiiddeennccee LLiimmiittss

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

   AArrgguummeennttss::

          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 propor-
             tions and `x' is the data

    epsilon: optional argument specifying step size for finite
             difference calculations

      alpha: optional argument specifying confidence levels
             desired

   VVaalluuee::

        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 stan-
             dard 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

   RReeffeerreenncceess::

        Efron, B, and DiCiccio, T. (1992) More accurate confi-
        dence 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.

   EExxaammpplleess::

        # 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[, 2] - x1m)^2) *
                      sum(p * (x[, 2] - x1m)^2))
            return(num/den)
        }
        results <- abcnon(x, theta)

