So far, we have discussed the idea behind the bootstrap and how it can be used to estimate standard errors Standard errors are often used to construct confidence intervals based on the estimate having a normal sampling distribution: θˆ ± z1−α/2 SE; alternatively, the interval could be based on the t distribution The bootstrap SE can be used in this way as well
Patrick Breheny
STA 621: Nonparametric Statistics
Bootstrap confidence intervals
Introduction
However, recall that the bootstrap can also be used to estimate the CDF G of θˆ Thus, with the bootstrap, we do not need to make assumptions/approximations concerning the sampling distribution of θˆ – we can estimate it as part of the confidence interval procedure This has been an active area of theoretical research into the bootstrap
Patrick Breheny
STA 621: Nonparametric Statistics
Bootstrap confidence intervals
The bootstrap-t interval
For example, suppose that the standard error of an estimate varies with the size of the estimate If this is so, then our confidence interval should be wider on the right side than it is on the left One way to implement this idea is to estimate the SE separately for each bootstrap replication
Patrick Breheny
STA 621: Nonparametric Statistics
Bootstrap confidence intervals
The bootstrap-t interval: Procedure The procedure of the bootstrap-t interval is as follows: For each bootstrap sample, compute zb∗ =
θˆb∗ − θˆ d∗b SE
d∗b is an estimate of the standard error of θˆ∗ based on where SE the data in the bth bootstrap sample Estimate the αth percentile of z ∗ by the value tˆα such that X I(zb∗ ≤ tˆα ) = α B −1 b
A 1 − α confidence interval for θ is then d θˆ − tˆα/2 SE) d (θˆ − tˆ1−α/2 SE, Patrick Breheny
STA 621: Nonparametric Statistics
Bootstrap confidence intervals
The bootstrap-t interval: Example
As a small example, the survival times of 9 rats were 10, 27, 30, 40, 46, 51, 52, 104, and 146 days (mean=56.2, SE=14.1) The 1 − α percentile points: Normal -1.96 1.96 t -2.31 2.31 Bootstrap-t -4.86 1.61 This translates into the following confidence intervals: Normal 28.5 84.0 t 23.6 88.9 Bootstrap-t 33.4 125.1
Patrick Breheny
STA 621: Nonparametric Statistics
Bootstrap confidence intervals
The bootstrap-t interval: R If you want to implement this confidence interval in R, your function that you pass to boot will need to return two things: ˆ θˆ∗ ) θˆ∗ and V( b
