1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
|
\name{norm.ci}
\alias{norm.ci}
\title{
Normal Approximation Confidence Intervals
}
\description{
Using the normal approximation to a statistic, calculate equi-tailed two-sided
confidence intervals.
}
\usage{
norm.ci(boot.out = NULL, conf = 0.95, index = 1, var.t0 = NULL,
t0 = NULL, t = NULL, L = NULL, h = function(t) t,
hdot = function(t) 1, hinv = function(t) t)
}
\arguments{
\item{boot.out}{
A bootstrap output object returned from a call to \code{boot}. If
\code{t0} is missing then \code{boot.out} is a required argument. It is
also required if both \code{var.t0} and \code{t} are missing.
}
\item{conf}{
A scalar or vector containing the confidence level(s) of the required
interval(s).
}
\item{index}{
The index of the statistic of interest within the output of a call to
\code{boot.out$statistic}. It is not used if \code{boot.out} is
missing, in which case \code{t0} must be supplied.
}
\item{var.t0}{
The variance of the statistic of interest. If it is not supplied then
\code{var(t)} is used.
}
\item{t0}{
The observed value of the statistic of interest. If it is missing then it is
taken from \code{boot.out} which is required in that case.
}
\item{t}{
Bootstrap replicates of the variable of interest. These are used to estimate
the variance of the statistic of interest if \code{var.t0} is not
supplied. The default value is \code{boot.out$t[,index]}.
}
\item{L}{
The empirical influence values for the statistic of interest. These are
used to calculate \code{var.t0} if neither \code{var.t0} nor
\code{boot.out} are supplied. If a transformation is supplied through
\code{h} then the influence values must be for the untransformed
statistic \code{t0}.
}
\item{h}{
A function defining a monotonic transformation, the intervals are
calculated on the scale of \code{h(t)} and the inverse function
\code{hinv} is applied to the resulting intervals. \code{h} must be a
function of one variable only and must be vectorized. The default is
the identity function.
}
\item{hdot}{
A function of one argument returning the derivative of \code{h}. It
is a required argument if \code{h} is supplied and is used for
approximating the variance of \code{h(t0)}. The default is the
constant function 1.
}
\item{hinv}{
A function, like \code{h}, which returns the inverse of \code{h}. It is
used to transform the intervals calculated on the scale of \code{h(t)}
back to the original scale. The default is the identity function. If
\code{h} is supplied but \code{hinv} is not, then the intervals returned
will be on the transformed scale.
}
}
\value{
If \code{length(conf)} is 1 then a vector containing the confidence
level and the endpoints of the interval is returned. Otherwise, the
returned value is a matrix where each row corresponds to a different
confidence level.
}
\details{
It is assumed that the statistic of interest has an approximately
normal distribution with variance \code{var.t0} and so a confidence
interval of length \code{2*qnorm((1+conf)/2)*sqrt(var.t0)} is found.
If \code{boot.out} or \code{t} are supplied then the interval is
bias-corrected using the bootstrap bias estimate, and so the interval
would be centred at \code{2*t0-mean(t)}. Otherwise the interval is
centred at \code{t0}.
}
\note{
This function is primarily designed to be called by \code{boot.ci} to
calculate the normal approximation after a bootstrap but it can also be
used without doing any bootstrap calculations as long as \code{t0} and
\code{var.t0} can be supplied. See the examples below.
}
\references{
Davison, A.C. and Hinkley, D.V. (1997)
\emph{Bootstrap Methods and Their Application}. Cambridge University Press.
}
\seealso{
\code{\link{boot.ci}}
}
\examples{
# In Example 5.1 of Davison and Hinkley (1997), normal approximation
# confidence intervals are found for the air-conditioning data.
air.mean <- mean(aircondit$hours)
air.n <- nrow(aircondit)
air.v <- air.mean^2/air.n
norm.ci(t0 = air.mean, var.t0 = air.v)
exp(norm.ci(t0 = log(air.mean), var.t0 = 1/air.n)[2:3])
# Now a more complicated example - the ratio estimate for the city data.
ratio <- function(d, w)
sum(d$x * w)/sum(d$u *w)
city.v <- var.linear(empinf(data = city, statistic = ratio))
norm.ci(t0 = ratio(city,rep(0.1,10)), var.t0 = city.v)
}
\keyword{htest}
% Converted by Sd2Rd version 1.15.
|