Computational Graphics and Statistical Analysis: Mixed Type Random Variables, Confidence Regions, and Golden Quantile Rank Sets

This dissertation has three principle areas of research: mixed type random variables, confidence regions, and golden quantile rank sets. While each offers a specific focus, some common themes persist; broadly stated, there are three. First, computational graphics play a critical role. Second, software development facilitates implementation and accessibility. Third, statistical analysis---often attributable to the aforementioned automation---provides valuable insights and applications. Each of the principle research areas are briefly summarized next. Mixed type random variables are a hybrid of continuous and discrete random variables, having components of both continuous probability density and discrete probability mass. This dissertation illustrates the challenges inherent in plotting mixed type distributions, and introduces an algorithm that addresses those issues. It considers sums and products of mixed type random variables, and supports its conclusions using Monte Carlo simulation experiments. Lastly, it introduces MixedAPPL, a computer algebra system software package designed for manipulating mixed type random variables. Confidence regions are a multi-dimensional version of a confidence interval. They are helpful to visualize and quantify uncertainty surrounding a point estimate. We begin by developing efficient plot algorithms for two-dimensional confidence regions. This research focuses specifically on likelihood-ratio based confidence regions for two-parameter univariate probability models, although the plot techniques are transferable to any two-dimensional setting. The R package \u27conf\u27 is introduced, which automates these confidence region plot algorithms for complete and right-censored data sets. Among its benefits, \u27conf\u27 provides access to Monte Carlo simulation experiments for confidence region coverage to an extent not possible previously. The corresponding coverage analysis results include reference tables for the Weibull, normal, and log-logistic distributions. These reference tables yield confidence region plots with exact coverage. The final topic is the introduction and analysis of a golden quantile rank set (GQRS). The term quantile rank set is used here to denote the population cumulative distribution function values corresponding to a sample. A GQRS can be thought of as perfectly representative of their population distribution because samples corresponding to a GQRS result in an estimator(s) matching the associated true population parameter(s). This unique characteristic is not applicable for all estimators and/or distributions, but when present, provides valuable insights and applications. Specifically, applications include an alternative (and at times computationally superior) method for parameter estimation and an exact actual coverage methodology for confidence regions (at times in which currently only estimates exist). Distributions with a GQRS associated with maximum likelihood estimation include the normal, exponential, Weibull, log logistic, and one-parameter exponential power distributions

Accurate estimation with one order statistic

Estimating parameters from certain survival distributions is shown to suffer little loss of accuracy in the presence of left censoring. The variance of maximum likelihood estimates (MLE) in the presence of type II right-censoring is almost un-degraded if there also is heavy left-censoring when estimating certain parameters. In fact, if only a single data point, the rth recorded failure time, is available, the MLE estimates using the one data point are similar in variance to the estimates using all r failure points for all but the most extreme values of r. Analytic results are presented for the case of the exponential and Rayleigh distributions, to include the exact distributions of the estimators for the parameters. Simulated results are also presented for the gamma distribution. Implications in life test design and cost savings are explained as a result. Also computational considerations for finding analytic results as well as simulated results in a computer algebra system are discussed.Order statistics Maximum likelihood Type II right censoring Computational probability