[R] Bootstrap inference for the sample median?

[Emmanuel Charpentier]

Le dimanche 30 août 2009 à 18:43 +0530, Ajay Shah a écrit :
> Folks,
> 
> I have this code fragment:
> 
>   set.seed(1001)
>   x <- c(0.79211363702017, 0.940536712079832, 0.859757602692931, 0.82529998629531, 
>          0.973451006822, 0.92378802164835, 0.996679563355802,
>          0.943347739494445, 0.992873542980045, 0.870624707845108, 0.935917364493788)
>   range(x)
>   # from 0.79 to 0.996
> 
>   e <- function(x,d) {
>     median(x[d])
>   }
> 
>   b <- boot(x, e, R=1000)
>   boot.ci(b)
> 
> The 95% confidence interval, as seen with `Normal' and `Basic'
> calculations, has an upper bound of 1.0028 and 1.0121.
> 
> How is this possible? If I sample with replacement from values which
> are all lower than 1, then any sample median of these bootstrap
> samples should be lower than 1. The upper cutoff of the 95% confidence
> interval should also be below 1.

Nope. "Normal" and "Basic" try to adjust (some form of) normal
distribution to the sample of your statistic. But the median of such a
small sample is quite far from a normal (hint : it is a discrete
distribution with only very few possible values, at most as many value
as the sample. Exercise : derive the distribution of median(x)...).

To convince yourself, look at the histogram of the bootstrap
distribution of median(x). Contrast with the bootstrap distribution of
mean(x). Meditate. Conclude...

HTH,

					Emmanuel Charpentier