

   P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

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

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

          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 parame-
             ter 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

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

        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', esti-
             mated standard error and estimated bias

   constants: list consisting of `a'=acceleration constant,
             `z0'=bias adjustment, `cq'=curvature component

   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. Bimometrika
        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::

        # 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)

