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This thesis presents simple efficient algorithms to estimate distribution parameters and to construct prediction intervals for location-scale families. Specifically, we study two scenarios: one is a frequentist method for a general location--scale family and then extend to a 3-parameter distribution, another is a Bayesian method for the Gumbel distribution. At the end of the thesis, a generalized bootstrap resampling scheme is proposed to construct prediction intervals for data with an unknown distribution. Our estimator construction begins with the equivariance principle, and then makes use of unbiasedness principle. These two estimates have closed form and are functions of the sample mean, sample standard deviation, sample size, as well as the mean and variance of a corresponding standard distribution. Next, we extend the previous result to estimate a 3-parameter distribution which we call a mixed method. A central idea of the mixed method is to estimate the location and scale parameters as functions of the shape parameter. The sample mean is a popular estimator for the population mean. The mean squared error (MSE) of the sample mean is often large, however, when the sample size is small or the scale parameter is greater than the location parameter. To reduce the MSE of our location estimator, we introduce an adaptive estimator. We will illustrate this by the example of the power Gumbel distribution. The frequentist approach is often criticized as failing to take into account the uncertainty of an unknown parameter, whereas a Bayesian approach incorporates such uncertainty. The present Bayesian analysis for the Gumbel data is achieved numerically as it is hard to obtain an explicit form. We tackle the problem by providing an approximation to the exponential sum of Gumbel random variables. Next, we provide two efficient methods to construct prediction intervals. The first one is a Monte Carlo method for a general location-scale family, based on our previous parameter estimation. Another is the Gibbs sampler, a special case of Markov Chain Monte Carlo. We derive the predictive distribution by making use of an approximation to the exponential sum of Gumbel random variables . Finally, we present a new generalized bootstrap and show that Efron's bootstrap re-sampling is a special case of the new re-sampling scheme. Our result overcomes the issue of the bootstrap of its ``inability to draw samples outside the range of the original dataset.'' We give an applications for constructing prediction intervals, and simulation shows that generalized bootstrap is better than that of the bootstrap when the sample size is small. The last contribution in this thesis is an improved GRS method used in nuclear engineering for construction of non-parametric tolerance intervals for percentiles of an unknown distribution. Our result shows that the required sample size can be reduced by a factor of almost two when the distribution is symmetric. The confidence level is computed for a number of distributions and then compared with the results of applying the generalized bootstrap. We find that the generalized bootstrap approximates the confidence level very well. / Dissertation / Doctor of Philosophy (PhD)

Hearkening to calls from life scientists for aid in interpreting rapidly-growing repositories of data, the fields of bioinformatics and computational systems biology continue to bear increasingly sophisticated methods capable of summarizing and distilling pertinent phenomena captured by high-throughput experiments. Techniques in analysis of genome-wide gene expression (e.g., microarray) data, for example, have moved beyond simply detecting individual genes perturbed in treatment-control experiments to reporting the collective perturbation of biologically-related collections of genes, or "processes". Recent expression analysis methods have focused on improving comprehensibility of results by reporting concise, non-redundant sets of processes by leveraging statistical modeling techniques such as Bayesian networks. Simultaneously, integrating gene expression measurements with gene interaction networks has led to computation of response networks--subgraphs of interaction networks in which genes exhibit strong collective perturbation or co-expression. Methods that integrate process annotations of genes with interaction networks identify high-level connections between biological processes, themselves. To identify context-specific changes in these inter-process connections, however, techniques beyond process-based expression analysis, which reports only perturbed processes and not their relationships, response networks, composed of interactions between genes rather than processes, and existing techniques in process connection detection, which do not incorporate specific biological context, proved necessary. We present two novel methods which take inspiration from the latest techniques in process-based gene expression analysis, computation of response networks, and computation of inter-process connections. We motivate the need for detecting inter-process connections by identifying a collection of processes exhibiting significant differences in collective expression in two liver tissue culture systems widely used in toxicological and pharmaceutical assays. Next, we identify perturbed connections between these processes via a novel method that integrates gene expression, interaction, and annotation data. Finally, we present another novel method that computes non-redundant sets of perturbed inter-process connections, and apply it to several additional liver-related data sets. These applications demonstrate the ability of our methods to capture and report biologically relevant high-level trends. / Ph. D.