Step density estimation and bootstrap resampling

Some failures of the nonparametric bootstrap resampling originate from the discreteness of the empirical distribution function used in the resampling process. Density estimation with smoothing kernel functions is the most suitable method to solve the problem; yet in reality density estimation had not been widely applied due to its tedious fine-tuning of smoothing width in addition to the ad hoc selection of smoothing kernel from many candidate functions. With the above restrictions in mind a novel Step Density Estimation has been devised from simple step-functions in this thesis. The step density function has been constructed and shown to be MLE and UMVUE as an estimator of the underlying distribution with a clear goal to make the density estimation objective as possible while keeping the smoothed bootstrap still as simple as it is. A well known bootstrap bias problem in small sample cases was chosen to test the success of the approach of bootstrap resampling drawn from the step density function.