# [R] Possible error in BCa method for confidence intervals in package 'boot'

Robert A. LaBudde ral at lcfltd.com
Tue Oct 2 20:54:35 CEST 2012

```I'm using R 2.15.1 on a 64-bit machine with Windows 7 Home Premium.

Sample problem (screwy subscripted syntax is a relic of edited down a
more complex script):

> N <- 25
> s <- rlnorm(N, 0, 1)
> require("boot")
> v <- NULL # hold sample variance estimates
> i <- 1
> v[i] <- var(s) # get sample variance
> nReal <- 10
> varf <- function (x,i) { var(x[i]) }
> fabc <- function (x, w) { # weighted average (biased) variance
+   sum(x^2 * w) / sum(w) - (sum(x * w) / sum(w))^2
+ }
> p <- c(.25, .75, .2, .8, .15, .85, .1, .9, .05, .95, .025, .975,
.005, .995)
> cl <- c(0.5, 0.6, 0.7, 0.8, 0.9, 0.95, 0.99)
> b <- boot(s, varf, R = nReal) # bootstrap
> bv <- NULL # hold bootstrap mean variance estimates
> bias <- NULL #hold bias estimates
> bv[i] <- mean(b\$t) # bootstrap mean variance
> bias[i] <- bv[i] - v[i] # bias estimate
> bCI90 <- boot.ci(b, conf = 0.90)
Error in bca.ci(boot.out, conf, index[1L], L = L, t = t.o, t0 = t0.o,  :
1: In norm.inter(t, (1 + c(conf, -conf))/2) :
extreme order statistics used as endpoints
2: In boot.ci(b, conf = 0.9) :
bootstrap variances needed for studentized intervals
3: In norm.inter(t, alpha) : extreme order statistics used as endpoints
>
> nReal <- 25
> b <- boot(s, varf, R = nReal) # bootstrap
> bv[i] <- mean(b\$t) # bootstrap mean variance
> bias[i] <- bv[i] - v[i] # bias estimate
> bCI90 <- boot.ci(b, conf = 0.90)
Warning messages:
1: In boot.ci(b, conf = 0.9) :
bootstrap variances needed for studentized intervals
extreme order statistics used as endpoints

The problem is that doing 10 resamples generates an NA in the
estimation of the 'acceleration' in the function abc.ci(), but doing
25 resamples does not. This implies a connection between the number
of resamples and the 'acceleration' which should not exist.
('Acceleration' should be obtained from the original sample via
jackknife as 1/6 the coefficient of skewness.)

Looking at the script for abc.ci(), there is an anomalous reference
to 'n' in the invocation line, yet 'n' is not an argument, so must be
defined more globally before the call. Yet 'n' is defined within the
script as the length of 'data', which is referred to as the
'bootstrap' vector in the comments, yet should be the original sample
data. This confusion, plus the use of an argument 'eps' as a default
0.001/n in the calculations makes me suspect the programming in the script.

The script apparently works correctly if the number of resamples
equals or exceeds the number of original data, but not otherwise.

================================================================
Robert A. LaBudde, PhD, PAS, Dpl. ACAFS  e-mail: ral at lcfltd.com
Least Cost Formulations, Ltd.            URL: http://lcfltd.com/
824 Timberlake Drive                     Tel: 757-467-0954
Virginia Beach, VA 23464-3239            Fax: 757-467-2947

"Vere scire est per causas scire"

```