The following dissertation discusses the theoretical study of water on the nanoscale, often involved with essential biological molecules such as DNA and proteins. First I introduce the study of water on the nanoscale and how experimentalists approach confinement with nanopores and nanogaps. Then I discuss the theoretical method we choose for understanding this important biological medium on the molecular level, namely classical molecular dynamics. This leads into transport mechanisms that utilize water on the nanoscale, in our case electronic and ionic transport. On the scale of mere nanometers or less electronic transport in water enters the tunneling regime, requiring the use of a quantum treatment. In addition, I discuss the importance of water in ionic transport and its known effects on biological phenomena such as ion selectivity. Water also has great influence over DNA and proteins, which are both introduced in the context of nanopore sequencing. Several techniques for nanopore sequencing are examined and the importance of protein sequencing is explained. In Chapter 2, we study the effect of volumetric constraints on the structure and electronic transport properties of distilled water in a nanopore with embedded electrodes. Combining classical molecular dynamics simulations with quantum scattering theory, we show that the structural motifs water assumes inside the pore can be probed directly by tunneling. In Chapter 3, we propose an improvement to the original sequencing by tunneling method, in which N pairs of electrodes are built in series along a synthetic nanochannel. Each current time series for each nucleobase is cross-correlated together, reducing noise in the signals. We show using random sampling of data from classical molecular dynamics, that indeed the sequencing error is significantly reduced as the number of pairs of electrodes, N, increases. In Chapter 4, we propose a new technique for de novo protein sequencing that involves translocating a polypeptide through a synthetic nanochannel and measuring the ionic current of each amino acid through an intersecting perpendicular nanochannel. We find that the distribution of ionic currents for each of the 20 proteinogenic amino acids encoded by eukaryotic genes is statistically distinct using our theoretical method.

Fox et al. (1998) carried out a logistic regression analysis with discrete covariates in which one of the covariates was missing for a substantial percentage of respondents. The missing data problem was addressed using the "approximate Bayesian bootstrap." We return to this missing data problem to provide a form of case study. Using the Fox et al. (1998) data for expository purposes we carry out a comparative analysis of eight of the most commonly used techniques for dealing with missing data. We then report on two sets of simulations based on the original data. These suggest, for patterns of missingness we consider realistic, that case deletion and weighted case deletion are inferior techniques, and that common simple alternatives are better. In addition, the simulations do not affirm the theoretical superiority of Bayesian Multiple Imputation. The apparent explanation is that the imputation model, which is the fully saturated interaction model recommended in the literature, was too detailed for the data. This result is cautionary. Even when the analyst of a single body of data is using a missingness technique with desirable theoretical properties, and the missingness mechanism and imputation model are supposedly correctly specified, the technique can still produce biased estimates. This is in addition to the generic problem posed by missing data, which is that usually analysts do not know the missingness mechanism or which among many alternative imputation models is correct.