TITLE(jackknife @@  jackknife estimation )
USAGE(
jackknife(x,theta,...)
)
ARGUMENTS(
ARG(x@@
a vector containing the data. To jackknife  more complex data structures (e.g
bivariate data) see the last example below.)
ARG(theta@@
function to be jackknifed. Takes x as an argument, and
may take additional arguments (see below and last example).)
ARG(...@@
any additional arguments to be passed to theta)
)
VALUES(
list with the following components
ARG(jack.se@@
The jackknife estimate of standard error of theta.
The leave-one out jackknife is used.)
ARG(jack.bias@@
The jackknife estimate of bias of theta.
The leave-one out jackknife is used.)
ARG(jack.values@@
The n leave-one-out values of theta, 
where n is the number of observations.
That is, theta applied to x with
the 1st observation deleted, theta applied to x with
the 2nd observation deleted, etc.)
)
REFERENCES(
Efron, B. and   Tibshirani, R. (1986).  The Bootstrap
Method for standard errors, confidence intervals,
and other measures of   statistical accuracy.
Statistical Science, Vol 1., No. 1, pp 1-35.
PARA
Efron, B. and Tibshirani, R. (1993) An Introduction to the Bootstrap.
Chapman and Hall, New York, London.
)
EXAMPLES(
# jackknife values for the sample mean 
# (this is for illustration;  # since "mean" is  a 
#  built in function,  jackknife(x,mean) would be simpler!)
x <- rnorm(20)               
theta <- function(x)\{mean(x)\}
                             
results <- jackknife(x,theta)        
                              
# To jackknife functions of more  complex data structures, 
# write theta so that its argument x
#  is the set of observation numbers  
#  and simply  pass as data to jackknife the vector 1,2,..n. 
# For example, to jackknife
# the correlation coefficient from a set of 15 data pairs:      
                        
xdata <- matrix(rnorm(30),ncol=2)
n <- 15
theta <- function(x,xdata)\{ cor(xdata[x,1],xdata[x,2]) \}
results <- jackknife(1:n,theta,xdata)
)

