[R] How to get a point estimate from the studentized bootstrap?

[Andrew Robinson]

Dear Friends and Colleagues,

I'm puzzling over how to interpret or use some bootstrap intervals.  I
think that I know what I should do, but I want to check with
knowledgeable people first!

I'm using a studentized non-parametric bootstrap to estimate 95%
confidence intervals for three parameters.  I estimate the variance of
the bootstrap replicates using another bootstrap.  The script takes
some hours to run, but I am happy to send it if it will help. Also I
am happy to send the boot object if that will help.

In the following object, I am interested in the values 1, 3, and 5.
The estimated variances are in 2, 4, and 6.

The boot object looks like:

> boot.outer

ORDINARY NONPARAMETRIC BOOTSTRAP

Call:
boot(data = err.outer, statistic = boot.fn.outer, R = R.outer, tis =
    tis, tfs = tfs, length.err = length.err, length.tis = length.tis,
    t0 = t0, tp = tp, start.outer = params, max.iter = max.iter,
    pc.ayl = pc.ayl, R.for.sd = R.for.sd)

Bootstrap Statistics :
      original        bias     std. error
t1* 0.62777511  1.898159e+00 8.003005e-01
t2* 0.04067475  1.854990e+00 2.916056e+01
t3* 1.97228885 -3.842986e+00 8.611901e+01
t4* 0.11418095  1.485597e+06 3.321892e+07
t5* 1.44261201  7.520929e-01 5.081878e-01
t6* 0.02751659  1.727005e-01 1.216163e-01



and the 95% CI is 

> boot.ci(boot.outer, type=c("stud", "norm", "basic"), index=1:2)
BOOTSTRAP CONFIDENCE INTERVAL CALCULATIONS
Based on 500 bootstrap replicates

CALL : 
boot.ci(boot.out = boot.outer, type = c("stud", "norm", "basic"), 
    index = 1:2)

Intervals : 
Level      Normal              Basic             Studentized     
95%   (-2.8389,  0.2982 )   (-2.7692,  0.2462 )   (-1.4258,  0.5107 )  
Calculations and Intervals on Original Scale


Note that the original estimate is 0.628 and the bias-corrected
estimate is -1.27.  The diagnostic graphs show no evidence to suggest
any problems with the assumptions.

My question is: if I am willing to believe in the studentized
interval, but I want a point estimate as well, then what should I use
as the point estimate?  

Intuitively I think that I should just use the middle of the
studentized interval.  Is that correct, or at least defensible?

Best wishes to all,

Andrew

-- 
Andrew Robinson  
Department of Mathematics and Statistics            Tel: +61-3-8344-9763
University of Melbourne, VIC 3010 Australia         Fax: +61-3-8344-4599
http://www.ms.unimelb.edu.au/~andrewpr
http://blogs.mbs.edu/fishing-in-the-bay/