We characterize the high-order coverage accuracy of smoothed and unsmoothed Bayesian bootstrap confidence intervals for population quantiles. Although the original (Rubin 1981) unsmoothed intervals have the same O(n−1/2) coverage error as the standard empirical bootstrap, the smoothed Bayesian bootstrap of Banks (1988) has much smaller O(n−3/2[log(n)]3) coverage error and is exact in special cases, without requiring any smoothing parameter. It automatically removes an error term of order 1/n that other approaches need to explicitly correct for. This motivates further study of the smoothed Bayesian bootstrap in more complex settings and models.
This dissertation consists of three chapters. In the first chapter, I analyze credible intervals for quantiles constructed using Bayesian bootstrap techniques and show that credible intervals constructed using the "continuity-corrected" Bayesian bootstrap (Banks, 1988) have frequentist coverage probability error of only O(n [superscript -1]). In addition, I show that these "continuity-corrected" Bayesian bootstrap credible intervals achieve the same frequentist coverage probability as the frequentist confidence intervals of Goldman and Kaplan (2017), up to some error term of magnitude O(n [superscript -1]). Furthermore, I demonstrate that credible intervals constructed using the "continuity-corrected" Bayesian bootstrap have less frequentist coverage probability error than those constructed using the Bayesian bootstrap (Rubin, 1981). In the second chapter, I investigate three strikes laws, which mandate sharply increased sentences for criminals who commit a specific number of felonies. Specifically, I analyze the effect of these laws on violent crime rates using municipal-level data from the FBI. I compare violent crime rates of border municipalities in states with differing treatment statuses using a difference-in-differences specification with a sample matched on pre-treatment outcomes. I find no statistical evidence that three strikes laws reduce violent crime rates. I rule out reductions in violent crime rates greater than 1.3 [percent] and reject the hypothesis that three strikes laws reduce violent crime rates at the 5 [percent] significance level. Additional analyses and robustness checks support my main findings. In the third chapter, I examine medical marijuana laws (MMLs), which legalize the use, possession, and cultivation of marijuana by individuals with qualifying medical conditions. Namely, I employ municipal-level data from the FBI to analyze the effect of MMLs on violent crime rates. I compare municipalities in border regions with different treatments statuses using a difference-in-differences specification with a sample matched on pre-treatment outcomes. I find a lack of evidence for MMLs increasing violent crime rates, but I cannot eliminate the possibility of small-to-medium positive effects. However, I rule out increases in violent crime rates greater than 9.9 [percent] and reject the hypothesis that MMLs increase violent crime at the 10 [percent] significance level.
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