Standard error

From a single set of numbers

$$x_1,x_2,\dots ,x_n$$

we can get both a mean:

$$\overline{x}=\frac{1}{n}\sum _{i=1}^n{x}_i$$

This is amazing!

Idea:

• We’d like to get a whole new dataset, and repeat the estimation, to see how different the answer is.

• And, well, our best guess at what the data look like is our dataset itself,

• sooooo, let’s just resample from the dataset, with replacement, to make a “new” dataset!

• If we resample and re-estimate lots of times, this should give us a good idea of the variability of the estimate.

The bootstrap resampling algorithm

To estimate the uncertainty of an estimate:

1. Use the computer to take a random sample of observations from the original data, with replacement.

2. Calculate the estimate from the resampled data set.

3. Repeat 1-2 many times.

4. The standard deviation of these esimates is the bootstrap standard error.

Advantages

• Applies to most any statistic

• Works when there’s no simple formula for the standard error (e.g., median, trimmed mean, eigenvalue, etc)

• Is nonparametric, so doesn’t make specific assumptions about the distribution of the data.

• Applies to even complicated sampling procedures.

Exercise

x <- c(0.6, 1, 3.1, 3.7, 4.8, 6.2, 12.5, 12.5, 13.4, 24.1)

• Use R to make 1000 “pseudo-samples” of size 10 (with replacement),

• and store the mean of each in a vector.

• Plot the histogram of the resampled means, and calculate their standard deviation (with sd()).

• How does this compare to the usual standard error of the mean, sd(x) / sqrt(length(x))?

Confidence intervals?

The 2.5% and 97.5% percentiles of the bootstrap samples estimate a 95% confidence interval. (use the quantile( ) function)