David Lartey's Ph.D. Thesis Proposal (Dept. of Mathematical Sciences, MSU)

05/18/2022

Abstract:

The importance of capturing uncertainty cannot be overemphasized in today’s scientific world (Gardner and Alman, 1986; Young and Lewis, 1997). The International Committee of Medical Journal Editors in their guidelines for statistical reporting stated that: “When possible, quantify findings and present them with appropriate indicators of measurement error or uncertainty (such as confidence intervals)”, and “Avoid sole reliance on statistical hypothesis testing such as the use of P-values, which fails to convey important quantitative information.” (Bailar and Mosteller, 1988). Confidence intervals are used to quantify the uncertainty of a point estimate associated with a single parameter. For multivariate situations, confidence regions can be used to quantify uncertainty in the simultaneous estimation of a p dimensional parameter vector (Krishnamoorthy and Mathew, 2009). Most methods published in the statistical literature that have been developed for constructing confidence regions require distributional assumptions (such as multivariate normality) (Fuchs and Sampson, 1987). However, in practice, these assumptions are not always be satisfied which led to development of non-parametric methods which relax assumptions required by parametric methods while still providing valid results. In particular, bootstrap methods exist for standard error and confidence interval estimation of univariate parameters (e.g., for parameters in linear and non-linear models). Currently, statistical software (including bootstrap software) only produces confidence intervals (CIs) for individual model parameters. However, it is inappropriate to generate a multidimensional confidence region (CR) for two or more parameters by combining the individual CIs for simultaneous inference. This research aims to develop a methodology that extends the bootstrap technique by using Particle Swarm Optimization (PSO) algorithms to construct confidence regions that relax asymptotic and parametric assumptions of existing methods. The proposed PSO algorithms will find a minimum volume ellipsoid that captures 95% (or any confidence level) of these multi-parameter bootstrap estimates. The research outcomes are to:

 

  1. Develop simulation studies that compare the accuracy of the newly proposed hybrid bootstrap/PSO method with existing methods for various applications in linear and nonlinear models, as well as, develop confidence regions for parameters defining multivariate distributions.
  2. Develop software implementation of the proposed method with divulgation through via publication of an R package.