Quantifying uncertainty using the bootstrap

1. Quantifying uncertainty using the bootstrap Reading Efron, B. and R. Tibishirani, (1993), An Introduction to the Bootstrap, Chapman Hall, New York, 436 p. Chapters 1, 2, 6.

2. Approaches to uncertainty estimation • Use statistical theory • Bootstrapping e.g. Standard Error Confidence Intervals:

3. Bootstrapping • Motivated by the absence of equations for other accuracy measures (bias, prediction error, confidence intervals) for statistics of interest (correlation, regressions, ACF) • Definition: “The bootstrap is a data-based simulation method for statistical inference.” • Principle: resample with replacement from data. After Efron and Tibshirani, An Introduction to the Bootstrap, 1993

4. from Efron and Tibshirani, An Introduction to the Bootstrap, 1993

5. Schematic of Bootstrap Process from Efron and Tibshirani, An Introduction to the Bootstrap, 1993

6. BOOTSTRAP WORLD F * x* = {x*1, x *2, …, x *n} REAL WORLD F x = {x1, x2, …, xn} Empirical Distribution Bootstrap Sample Bootstrap Replication Bootstrapping Unknown Probability Distribution Observed Random Sample Sampling with replacement Statistic of Interest After Efron and Tibshirani, An Introduction to the Bootstrap, 1993

7. from Efron and Tibshirani, An Introduction to the Bootstrap, 1993

8. Bootstrap Algorithm for Standard Error from Efron and Tibshirani, An Introduction to the Bootstrap, 1993

9. Hillsborough River at Zephyr Hills, September flows Mean = 8621 mgal S = 8194 mgal N = 31 Uncertainty on estimates of the mean One and two standard errors 95% CI and interquartile range from 500 bootstrap samples Millions of gallons