Coverage probabilities of the basic bootstrap confidence Interval

Question

I have the following question for a course I'm working on:

Conduct a Monte Carlo study to estimate the coverage probabilities of the standard normal bootstrap confidence interval and the basic bootstrap confidence interval. Sample from a normal population and check the empirical coverage rates for the sample mean.

Coverage probabilities for the standard normal bootstrap CI are easy:

n = 1000;
alpha = c(0.025, 0.975);
x = rnorm(n, 0, 1);
mu = mean(x);
sqrt.n = sqrt(n);

LNorm = numeric(B);
UNorm = numeric(B);

for(j in 1:B)
{
    smpl = x[sample(1:n, size = n, replace = TRUE)];
    xbar = mean(smpl);
    s = sd(smpl);

    LNorm[j] = xbar + qnorm(alpha[1]) * (s / sqrt.n);
    UNorm[j] = xbar + qnorm(alpha[2]) * (s / sqrt.n);
}

mean(LNorm < 0 & UNorm > 0); # Approximates to 0.95
# NOTE: it is not good enough to look at overall coverage
# Must compute separately for each tail

From what I've been taught for this course, the basic bootstrap confidence interval can be calculated like this:

# Using x from previous...
R = boot(data = x, R=1000, statistic = function(x, i){ mean(x[i]); });
result = 2 * mu - quantile(R$t, alpha, type=1);

That makes sense. What I don't understand is how to calculate coverage probabilities for the basic bootstrap CI. I understand that the coverage probability would represent the number of times that the CI contains the true value (in this case mu). Do I simply run the boot function many times?

How can I approach this question differently?

Answer 1

The terminology is probably not used consistently, so the following is only how I understand the original question. From my understanding, the normal CIs you computed are not what was asked for. Each set of bootstrap replicates gives you one confidence interval, not many. The way to compute different CI-types from the results of a set of bootstrap replicates is as follows:

B    <- 999                  # number of replicates
muH0 <- 100                  # for generating data: true mean
sdH0 <- 40                   # for generating data: true sd
N    <- 200                  # sample size
DV   <- rnorm(N, muH0, sdH0) # simulated data: original sample

Since I want to compare the calculations against the results from package boot, I first define a function that will be called for each replicate. Its arguments are the original sample, and an index vector specifying the cases for a single replicate. It returns $M^{\star}$, the plug-in estimate for $\mu$, as well as $S_{M}^{2\star}$, the plug-in estimate for the variance of the mean $\sigma_{M}^{2}$. The latter will be required only for the bootstrap $t$-CI.

> getM <- function(orgDV, idx) {
+     bsM   <- mean(orgDV[idx])                       # M*
+     bsS2M <- (((N-1) / N) * var(orgDV[idx])) / N    # S^2*(M)
+     c(bsM, bsS2M)
+ }

> library(boot)                                       # for boot(), boot.ci()
> bOut <- boot(DV, statistic=getM, R=B)
> boot.ci(bOut, conf=0.95, type=c("basic", "perc", "norm", "stud"))
BOOTSTRAP CONFIDENCE INTERVAL CALCULATIONS
Based on 999 bootstrap replicates
CALL : 
boot.ci(boot.out = bOut, conf = 0.95, type = c("basic", "perc", "norm", "stud"))

Intervals : 
Level      Normal            Basic         Studentized        Percentile    
95%   ( 95.6, 106.0 )   ( 95.7, 106.2 )  ( 95.4, 106.2 )   ( 95.4, 106.0 )  
Calculations and Intervals on Original Scale

Without using package boot you can simply use replicate() to get a set of bootstrap replicates.

boots <- t(replicate(B, getM(DV, sample(seq(along=DV), replace=TRUE))))

But let's stick with the results from boot.ci() to have a reference.

boots   <- bOut$t                     # estimates from all replicates
M       <- mean(DV)                   # M from original sample
S2M     <- (((N-1)/N) * var(DV)) / N  # S^2(M) from original sample
Mstar   <- boots[ , 1]                # M* for each replicate
S2Mstar <- boots[ , 2]                # S^2*(M) for each replicate
biasM   <- mean(Mstar) - M            # bias of estimator M

The basic, percentile, and $t$-CI rely on the empirical distribution of bootstrap estimates. To get the $\alpha/2$ and $1 - \alpha/2$ quantiles, we find the corresponding indices to the sorted vector of bootstrap estimates (note that boot.ci() will do a more complicated interpolation to find the empirical quantiles when the indices are not natural numbers).

(idx <- trunc((B + 1) * c(0.05/2, 1 - 0.05/2)) # indices for sorted vector of estimates
[1] 25 975

> (ciBasic <- 2*M - sort(Mstar)[idx])          # basic CI
[1] 106.21826  95.65911

> (ciPerc <- sort(Mstar)[idx])                 # percentile CI
[1] 95.42188 105.98103

For the $t$-CI, we need the bootstrap $t^{\star}$ estimates to calculate the critical $t$-values. For the standard normal CI, the critical value will just be the $z$-value from the standard normal distribution.

# standard normal CI with bias correction
> zCrit   <- qnorm(c(0.025, 0.975))   # z-quantiles from std-normal distribution
> (ciNorm <- M - biasM + zCrit * sqrt(var(Mstar)))
[1] 95.5566 106.0043

> tStar <- (Mstar-M) / sqrt(S2Mstar)  # t*
> tCrit <- sort(tStar)[idx]           # t-quantiles from empirical t* distribution
> (ciT  <- M - tCrit * sqrt(S2M))     # studentized t-CI
[1] 106.20690  95.44878

In order to estimate the coverage probabilities of these CI-types, you will have to run this simulation many times. Just wrap the code into a function, return a list with the CI-results and run it with replicate() like demonstrated in this gist.