Fitted Parameters Confidence Interval

Basic Concepts

We can calculate the confidence interval of a fitted distribution parameter using bootstrapping in a manner similar to that used to calculate the standard error (see Fitted Parameters Standard Error). For any parameter, suppose that β-hat is the fitted estimate of the true parameter value β, and suppose further that the distribution of β – β-hat were known. Then for any α, with 0 < α < 1, there would exist δ₁ and δ₂ such that

in which case

and so

yielding a 1–α confidence interval for β of

Since β is not known, we estimate it by β-hat and then generate k bootstrap samples based on this estimate for β from which we generate estimates β₁, …, β_k. Now, let d₁ = the kα/2^th smallest of these values and d₂ = the kα/2^th largest of these values. Thus, we can estimate δ₁ by β-hat – d₁ and δ₂ by β-bar – d₂. The 1–α confidence interval can now be estimated as

Example

Example 1: Based on the data from the bootstrap used in Example 1 of Standard Error of Fitted Parameters, determine the 60% confidence for mu and beta. (Here, we have chosen to ask for a 60% confidence interval rather than the usual 95% confidence interval since the number of bootstraps is so small).

Since k = 10 and 1–α = .6, d₁ = the kα/2 = 2^nd smallest of the parameters and d₂= 2^nd largest of the parameters. Thus, for μ, we can use the SMALL and LARGE functions to find that d₁= 2.049541 and d₂ = 2.197749, and so the 60% confidence interval is

(2 · 2.11322 – 2.197749, 2 · 2.11322 – 2.049541)

i.e. (2.028691, 2.176899). For beta, d₁ = .309857 and d₂ = .371714, and so the 60% confidence interval for β is (.326933, .388790).

Reference

Tibshirani, R. (2014) The bootstrap. Advanced methods for data analysis
https://www.stat.cmu.edu/~ryantibs/advmethods/notes/bootstrap.pdf