Nonparametric Bootstrap

Empirical cumulative distribution function (ecdf) Def: For a batch of numbers x_{1}, x_{2}, ..., x_{n} we define the ecdf as

F_{n}(x) = \frac{1}{n}(#x ≤ x).  (1)

Ex. Suppose we have the following batch of numbers: 1, 14, 10, 9, 11, 9. Draw the ecdf for this batch of numbers. Introduction to the Nonparametric Bootstrap The Nonparametric Bootstrap is useful for estimating the variability of a location parameter. Note: nonparametric means distribution-free. Suppose x_{1}, x_{2}, ..., x_{n} are observations from an unknown distribution F, it is appropriate to investigate the variability and sampling distribution of a location estimate calculated from the sample. Let θ̂ = θ̂(x_{1}, x_{2}, ..., x_{n}) be a location estimate. Problems:

1. We do not know F and don’t necessarily want to make a parametric assumption about F.

2. θ̂ may be complicated. (median, trimmed mean, etc.)

The nonparametric bootstrap proceeds as follows:

1. Use the ecdf F_{n}(x) as an approximation to F (x).

2. Take B samples of size n from F_{n}(x), with replacement.

(a) Estimate the sampling distribution of θ̂ by plotting a histogram of

θ̂_{1}^{∗}, θ̂_{2}^{∗}, ..., θ̂_{B}^{∗}  (2)

(b) Estimate the variability by calculating the sd

sθ̂ = \sqrt{\frac{∑_{i=1}^{B}(θ̂_{i}^{∗} − ¯ θ̂^{∗})^{2}}{B − 1}}  (3)

Example: Heat of Sublimation of Platinum. Find the sampling distribution of the mean, median, and 10% trimmed mean. Reference:

Rice, Mathematical Statistics, 3ed, Duxbury, 2007.