boot.ci: Nonparametric Bootstrap Confidence Intervals

Description

This function generates 5 different types of equi-tailed two-sided nonparametric confidence intervals. These are the first order normal approximation, the basic bootstrap interval, the studentized bootstrap interval, the bootstrap percentile interval, and the adjusted bootstrap percentile (BCa) interval. All or a subset of these intervals can be generated.

Usage

boot.ci(boot.out, conf=0.95, type="all", 
        index=1:min(2,length(boot.out$t0), var.t0=NULL, 
        var.t=NULL, t0=NULL, t=NULL, L=NULL, h=function(t) t,
        hdot=function(t) rep(1,length(t)), hinv=function(t) t, ...)

Arguments

boot.out

An object of class "boot" containing the output of a bootstrap calculation.

conf

A scalar or vector containing the confidence level(s) of the required interval(s).

type

A vector of character strings representing the type of intervals required. The value should be any subset of the values c("norm","basic", "stud", "perc", "bca") or simply "all" which will compute all five types of intervals.

index

This should be a vector of length 1 or 2. The first element of index indicates the position of the variable of interest in boot.out$t0 and the relevant column in boot.out$t. The second element indicates the positi

var.t0

If supplied, a value to be used as an estimate of the variance of the statistic for the normal approximation and studentized intervals. If it is not supplied and length(index) is 2 then var.t0 defaults to boot.out$t0[inde

var.t

This is a vector (of length boot.out$R) of variances of the bootstrap replicates of the variable of interest. It is used only for studentized intervals. If it is not supplied and length(index) is 2 then var.t defa

The observed value of the statistic of interest. The default value is boot.out$t0[index[1]]. Specification of t0 and t allows the user to get intervals for a transformed statistic which may not be in the bootstra

The bootstrap replicates of the statistic of interest. It must be a vector of length boot.out$R. It is an error to supply one of t0 or t but not the other. Also if studentized intervals are required and t0

L

The empirical influence values of the statistic of interest for the observed
data.  These are used only for BCa intervals.  If a transformation is supplied 
through the parameter h then L should be the influence values for

h

A function defining a transformation.  The intervals are calculated
on the scale of h(t) and the inverse function hinv applied to the resulting
intervals.  It must be a function of one variable only and for a vector 
argument, it

hdot

A function of one argument returning the derivative of h.  It is a required
argument if h is supplied and normal, studentized or BCa intervals are required.  The function is used for approximating the variances of
h(t0)

hinv

A function, like h, which returns the inverse of h.  It is used to transform
the intervals calculated on the scale of h(t) back to the original scale.
The default is the identity function.  If h is suppl

...

Any extra arguments that boot.out$statistic is expecting.  These arguments are
needed only if BCa intervals are required and L is not supplied since in
that case L is calculated through a call to empinf

`Value`

An object of type "bootci" which contains the intervals.  See bootci.object
for further details.

`Details`

The formulae on which the calculations are based can be found in Chapter 5
of Davison and Hinkley (1997).  Function boot must be run prior to running
this function to create the object to be passed as boot.out.  

Variance estimates are required for studentized intervals.  The variance of the 
observed statistic is optional for normal theory intervals.  If it is not 
supplied then the bootstrap estimate of variance is used.  The normal intervals
also use the bootstrap bias correction.  

Interpolation on the normal quantile scale is used when a non-integer order
statistic is required.  If the order statistic used is the smallest or largest
of the R values in boot.out a warning is generated and such intervals should
not be considered reliable.

`References`

Davison, A.C. and Hinkley, D.V. (1997) 
Bootstrap Methods and Their Application, Chapter 5. Cambridge University Press.

DiCiccio, T.J. and Efron  B. (1996) Bootstrap confidence intervals (with 
Discussion). Statistical Science, 11, 189-228.

Efron, B. (1987) Better bootstrap confidence intervals (with Discussion).
Journal of the American Statistical Association, 82, 171-200.

`See Also`

abc.ci, boot, bootci.object, empinf, norm.ci

`Examples`

Run this code# confidence intervals for the city data
data(city)
ratio <- function(d, w)
     sum(d$x * w)/sum(d$u * w)
city.boot <- boot(city, ratio, R=999, stype="w",sim="ordinary")
boot.ci(city.boot, conf=c(0.90,0.95),
     type=c("norm","basic","perc","bca"))


# studentized confidence interval for the two sample 
# difference of means problem using the final two series
# of the gravity data. 
data(gravity)
diff.means <- function(d, f)
{    n <- nrow(d)
     gp1 <- 1:table(as.numeric(d$series))[1]
     m1 <- sum(d[gp1,1] * f[gp1])/sum(f[gp1])
     m2 <- sum(d[-gp1,1] * f[-gp1])/sum(f[-gp1])
     ss1 <- sum(d[gp1,1]^2 * f[gp1]) - 
            (m1 *  m1 * sum(f[gp1]))
     ss2 <- sum(d[-gp1,1]^2 * f[-gp1]) - 
            (m2 *  m2 * sum(f[-gp1]))
     c(m1-m2, (ss1+ss2)/(sum(f)-2))
}
grav1 <- gravity[as.numeric(gravity[,2])>=7,]
grav1.boot <- boot(grav1, diff.means, R=999, stype="f", strata=grav1[,2])
boot.ci(grav1.boot, type=c("stud","norm"))


# Nonparametric confidence intervals for mean failure time 
# of the air-conditioning data as in Example 5.4 of Davison
# and Hinkley (1997)
data(aircondit)
mean.fun <- function(d, i) 
{    m <- mean(d$hours[i])
     n <- length(i)
     v <- (n-1)*var(d$hours[i])/n^2
     c(m, v)
}
air.boot <- boot(aircondit, mean.fun, R=999)
boot.ci(air.boot, type=c("norm", "basic", "perc", "stud"))


# Now using the log transformation
# There are two ways of doing this and they both give the
# same intervals.


# Method 1
boot.ci(air.boot, type=c("norm", "basic", "perc", "stud"), 
     h=log, hdot=function(x) 1/x)


# Method 2
vt0<-air.boot$t0[2]/air.boot$t0[1]^2
vt <- air.boot$t[,2]/air.boot$t[,1]^2
boot.ci(air.boot, type=c("norm", "basic", "perc", "stud"), 
     t0=log(air.boot$t0[1]), t=log(air.boot$t[,1]),
     var.t0=vt0, var.t=vt)
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