# Comparison of different confidence intervals for a distribution mean:
# exponential distribution setting.

set.seed(321)
library(bootstrap)

x <- rexp(25)

# First, Student's t interval.
t.result <- t.test( x, conf.level=0.95 )
t.result$conf.int
# This is not the "gold standard" since the dist'n isn't normal.

# Now the percentile interval.
bss <- matrix( sample( x, 25*3999, replace=T ), nrow=3999 )
mean.rep <- apply( bss, 1, mean )
sorted.mean.rep <- sort( mean.rep )

# UCB for percentile interval:
sorted.mean.rep[3900]

# LCB for percentile interval:
sorted.mean.rep[100]

# Next, the BCA interval.
bca.res <- bcanon( x, 3999, mean, alpha=c( 0.025, 0.975 ) )
bca.res$confpoint
# The confidence bounds above should be close to those of the percentile 
# interval since there is no need for bias correction or acceleration
# adjustments.  Below I'll give the values of the bias correction and 
# acceleration computed by bcanon.
bca.res$z0; bca.res$acc

# Here is the ABC interval.
resamp.mean <- function( p, x ) { sum( p*x )/sum( p ) }
abc.res <- abcnon( x, resamp.mean, alpha=c( 0.025, 0.975 ) )
# LCB & UBC:
abc.res$limits[1,2]; abc.res$limits[2,2]
 
# Now, the bootstrap t interval (which makes use of some results obtained
# in the percentile computations above.
sd.rep <- apply( bss, 1, sd )
mean.x <- mean( x )
z.rep <- ( mean.rep - mean.x )/( sd.rep/5 )
sorted.z.rep <- sort( z.rep )
se.hat <- sd( x )/5

# UCB for bootstap t interval:
mean.x - sorted.z.rep[100]*se.hat 

# LCB for bootstap t interval:
mean.x - sorted.z.rep[3900]*se.hat 
 
# Finally, an exact interval based on the exponential dist'n parametric model
# (obtained from a method covered in STAT 652 / CSI 672). 
numerator <- 2*sum(x)

# UCB for exact interval:
numerator/qchisq( 0.025, 50 )  

# LCB for exact interval:
numerator/qchisq( 0.975, 50 )