Normal Approximation Confidence Intervals

Using the normal approximation to a statistic, calculate equi-tailed two-sided confidence intervals.

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

boot.out: A bootstrap output object returned from a call to boot. If t0 is missing then boot.out is a required argument. It is also required if both var.t0 and t are missing.
conf: A scalar or vector containing the confidence level(s) of the required interval(s).
index: The index of the statistic of interest within the output of a call to boot.out$statistic. It is not used if boot.out is missing, in which case t0 must be supplied.
var.t0: The variance of the statistic of interest. If it is not supplied then var(t) is used.
t0: The observed value of the statistic of interest. If it is missing then it is taken from boot.out which is required in that case.
t: Bootstrap replicates of the variable of interest. These are used to estimate the variance of the statistic of interest if var.t0 is not supplied. The default value is boot.out$t[,index].
L: The empirical influence values for the statistic of interest. These are used to calculate var.t0 if neither var.t0 nor boot.out are supplied. If a transformation is supplied through h then the influence values must be for the untransformed statistic t0.
h: A function defining a monotonic transformation, the intervals are calculated on the scale of h(t) and the inverse function hinv is applied to the resulting intervals. h must be a function of one variable only and must be vectorized. The default is the identity function.
hdot: A function of one argument returning the derivative of h. It is a required argument if h is supplied and is used for approximating the variance of h(t0). The default is the constant function 1.
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 supplied but hinv is not, then the intervals returned will be on the transformed scale.

Value

If 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 var.t0 and so a confidence interval of length 2*qnorm((1+conf)/2)*sqrt(var.t0) is found. If boot.out or t are supplied then the interval is bias-corrected using the bootstrap bias estimate, and so the interval would be centred at 2*t0-mean(t). Otherwise the interval is centred at t0.

Note

This function is primarily designed to be called by boot.ci to calculate the normal approximation after a bootstrap but it can also be used without doing any bootstrap calculations as long as t0 and var.t0 can be supplied. See the examples below.

References

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

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)
#>      conf                  
#> [1,] 0.95 46.93055 169.2361
exp(norm.ci(t0 = log(air.mean), var.t0 = 1/air.n)[2:3])
#> [1]  61.38157 190.31782

# 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)
#>      conf                  
#> [1,] 0.95 1.167046 1.873579