Bayesian analysis often requires the researcher to employ Markov Chain Monte Carlo (MCMC) techniques to draw samples from a posterior distribution which in turn is used to make inferences. Currently, several approaches to determine convergence of the chain as well as sensitivities of the resulting inferences have been developed. This work develops a Hellinger distance approach to MCMC diagnostics. An approximation to the Hellinger distance between two distributions f and g based on sampling is introduced. This approximation is studied via simulation to determine the accuracy. A criterion for using this Hellinger distance for determining chain convergence is proposed as well as a criterion for sensitivity studies. These criteria are illustrated using a dataset concerning the Anguilla australis, an eel native to New Zealand.

Firefighters and police have different physical job requirements and selection processes. Firefighters have on-duty exercise time and police do not. This could affect body composition and strength. The purpose of this investigation was to compare body composition and bench press strength between firefighters and police officers in the Charlotte-Mecklenburg departments. Current 2006 fitness records were analyzed to compare firefighters and police (N = 2330). Variables included: gender, age, body mass, percent body-fat, fat mass, lean mass, bench press strength, bench press/lean mass, bench press/body mass, and percent that were obese. Significant (p ≤ 0.05) differences were found between male firefighters and police in age (37.7 vs. 36.9 yrs), body mass (91.5 vs. 93.2 kg), percent body-fat (17.8% vs.18.5%), fat mass (16.9 vs. 18.0 kg), bench press strength (93.4 vs. 96.3 kg), bench press/body mass (1.03 vs.1.05) and percent that were obese (10% vs.17%), respectively. Significant differences between female firefighters and police were found in age (42.4 vs. 37.4 yrs), body mass (77.5 vs. 71.8 kg), lean mass (54.4 vs. 51.2 kg), bench press strength (52.3 vs. 43.9 kg), bench press/lean mass (0.99 vs. 0.86) and bench press/body mass (0.69 vs. 0.62), respectively. Male police were younger, weighed more, had higher percent body fat, higher percent of obesity, and had greater upper body strength than firefighters. Female police were younger, weighed less, had less lean mass and less upper body strength than their firefighter counterparts. Contributing factors might include pre-employment selection, recruit training, in-service physical fitness testing, and counseling provided by departments.

Fractional Stochastic Differential Equations are becoming more popular in the literature as they can model phenomena in financial data that typical Stochastic Differential Equations models cannot. In the formulation considered here, the Hurst parameter, H, controls the Fraction of Differentiation, which needs to be estimated from the data. Fortunately, the covariance structure among observations in time is easily expressed in terms of the Hurst parameter which means that a likelihood is easily defined. This work derives the Maximum Likelihood Estimator for H, which shows that it is biased and is not a consistent estimator. Simulation data used to understand the bias of the estimator is used to create an empirical bias correction function and a bias-corrected estimator is proposed and studied. Via simulation, the bias-corrected estimator is shown to be minimally biased and its simulation-based standard error is created, which is then used to create a 95% confidence interval for H. A simulation study shows that the 95% confidence intervals have decent coverage probabilities for large n. This method is then applied to the S&P500 and VIX data before and after the 2008 financial crisis.