Quantifying uncertainty using the bootstrap Reading Efron B

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.

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

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

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

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

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

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

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

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